Concepts in Programming Languages

Concepts in Programming Languages
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Concepts in Programming Languages
This textbook for undergraduate and beginning graduate students explains and examines the central concepts used in modern programming
languages, such as functions, types, memory management, and control.
The book is unique in its comprehensive presentation and comparison
of major object-oriented programming languages. Separate chapters examine the history of objects, Simula and Smalltalk, and the prominent
languages C++ and Java.
The author presents foundational topics, such as lambda calculus and
denotational semantics, in an easy-to-read, informal style, focusing on the
main insights provided by these theories. Advanced topics include concurrency and concurrent object-oriented programming. A chapter on logic
programming illustrates the importance of specialized programming methods for certain kinds of problems.
This book will give the reader a better understanding of the issues
and trade-offs that arise in programming language design and a better
appreciation of the advantages and pitfalls of the programming languages
they use.
John C. Mitchell is Professor of Computer Science at Stanford University,
where he has been a popular teacher for more than a decade. Many of his
former students are successful in research and private industry. He received his Ph.D. from MIT in 1984 and was a Member of Technical Staff at
AT&T Bell Laboratories before joining the faculty at Stanford. Over the
past twenty years, Mitchell has been a featured speaker at international
conferences; has led research projects on a variety of topics, including
programming language design and analysis, computer security, and applications of mathematical logic to computer science; and has written more
than 100 research articles. His previous textbook, Foundations for Programming Languages (MIT Press, 1996), covers lambda calculus, type
systems, logic for program verification, and mathematical semantics of
programming languages. Professor Mitchell was a member of the programming language subcommittee of the ACM/IEEE Curriculum 2001
standardization effort and the 2002 Program Chair of the ACM Principles
of Programming Languages conference.
CONCEPTS IN
PROGRAMMING
LANGUAGES
John C. Mitchell
Stanford University
         
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
  
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
Ruiz de Alarcón 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
© Cambridge University Press 2004
First published in printed format 2002
ISBN 0-511-03492-X eBook (Adobe Reader)
ISBN 0-521-78098-5 hardback
Contents
Preface
Part 1
1
2
3
4
page ix
Functions and Foundations
Introduction
1.1 Programming Languages
1.2 Goals
1.3 Programming Language History
1.4 Organization: Concepts and Languages
Computability
2.1 Partial Functions and Computability
2.2 Chapter Summary
Exercises
3
3
5
6
8
10
10
16
16
Lisp: Functions, Recursion, and Lists
3.1 Lisp History
3.2 Good Language Design
3.3 Brief Language Overview
3.4 Innovations in the Design of Lisp
3.5 Chapter Summary: Contributions of Lisp
Exercises
18
Fundamentals
4.1 Compilers and Syntax
4.2 Lambda Calculus
4.3 Denotational Semantics
4.4 Functional and Imperative Languages
4.5 Chapter Summary
Exercises
48
18
20
22
25
39
40
48
57
67
76
82
83
v
vi
Contents
Part 2
5
6
7
8
The Algol Family and ML
5.1 The Algol Family of Programming Languages
5.2 The Development of C
5.3 The LCF System and ML
5.4 The ML Programming Language
5.5 Chapter Summary
Exercises
93
99
101
103
121
122
Type Systems and Type Inference
6.1 Types in Programming
6.2 Type Safety and Type Checking
6.3 Type Inference
6.4 Polymorphism and Overloading
6.5 Type Declarations and Type Equality
6.6 Chapter Summary
Exercises
129
132
135
145
151
155
156
Scope, Functions, and Storage Management
7.1 Block-Structured Languages
7.2 In-Line Blocks
7.3 Functions and Procedures
7.4 Higher-Order Functions
7.5 Chapter Summary
Exercises
162
165
170
182
190
191
Control in Sequential Languages
8.1 Structured Control
8.2 Exceptions
8.3 Continuations
8.4 Functions and Evaluation Order
8.5 Chapter Summary
Exercises
Part 3
9
10
Procedures, Types, Memory Management, and Control
93
129
162
204
204
207
218
223
227
228
Modularity, Abstraction, and Object-Oriented Programming
Data Abstraction and Modularity
9.1 Structured Programming
9.2 Language Support for Abstraction
9.3 Modules
9.4 Generic Abstractions
9.5 Chapter Summary
Exercises
Concepts in Object-Oriented Languages
10.1 Object-Oriented Design
10.2 Four Basic Concepts in Object-Oriented Languages
235
235
242
252
259
269
271
277
277
278
Contents
10.3
10.4
10.5
10.6
Program Structure
Design Patterns
Chapter Summary
Looking Forward: Simula, Smalltalk,
C++, Java
Exercises
11
12
13
History of Objects: Simula and Smalltalk
11.1 Origin of Objects in Simula
11.2 Objects in Simula
11.3 Subclasses and Subtypes in Simula
11.4 Development of Smalltalk
11.5 Smalltalk Language Features
11.6 Smalltalk Flexibility
11.7 Relationship between Subtyping and
Inheritance
11.8 Chapter Summary
Exercises
Objects and Run-Time Efficiency: C++
12.1 Design Goals and Constraints
12.2 Overview of C++
12.3 Classes, Inheritance, and Virtual Functions
12.4 Subtyping
12.5 Multiple Inheritance
12.6 Chapter Summary
Exercises
Portability and Safety: Java
13.1 Java Language Overview
13.2 Java Classes and Inheritance
13.3 Java Types and Subtyping
13.4 Java System Architecture
13.5 Security Features
13.6 Java Summary
Exercises
Part 4
14
288
290
292
293
294
300
300
303
308
310
312
318
322
326
327
337
337
340
346
355
359
366
367
384
386
389
396
404
412
417
420
Concurrency and Logic Programming
Concurrent and Distributed Programming
14.1 Basic Concepts in Concurrency
14.2 The Actor Model
14.3 Concurrent ML
14.4 Java Concurrency
14.5 Chapter Summary
Exercises
431
433
441
445
454
466
469
vii
viii
Contents
15
The Logic Programming Paradigm and Prolog
15.1 History of Logic Programming
15.2 Brief Overview of the Logic Programming Paradigm
15.3 Equations Solved by Unification as Atomic Actions
15.4 Clauses as Parts of Procedure Declarations
15.5 Prolog’s Approach to Programming
15.6 Arithmetic in Prolog
15.7 Control, Ambivalent Syntax, and Meta-Variables
15.8 Assessment of Prolog
15.9 Bibliographic Remarks
15.10 Chapter Summary
475
475
476
478
482
486
492
496
505
507
507
Appendix A Additional Program Examples
A.1 Procedural and Object-Oriented Organization
509
Glossary
521
Index
525
509
Preface
A good programming language is a conceptual universe for
thinking about programming.
Alan Perlis, NATO Conference on Software
Engineering Techniques, Rome, 1969
Programming languages provide the abstractions, organizing principles, and control
structures that programmers use to write good programs. This book is about the
concepts that appear in programming languages, issues that arise in their implementation, and the way that language design affects program development. The text is
divided into four parts:
Part 1:
Part 2:
Part 3:
Part 4:
Functions and Foundations
Procedures, Types, Memory Management, and Control
Modularity, Abstraction, and Object-Oriented Programming
Concurrency and Logic Programming
Part 1 contains a short study of Lisp as a worked example of programming language
analysis and covers compiler structure, parsing, lambda calculus, and denotational
semantics. A short Computability chapter provides information about the limits of
compile-time program analysis and optimization.
Part 2 uses procedural Algol family languages and ML to study types, memory
management, and control structures.
In Part 3 we look at program organization using abstract data types, modules, and
objects. Because object-oriented programming is the most prominent paradigm in
current practice, several different object-oriented languages are compared. Separate
chapters explore and compare Simula, Smalltalk, C++, and Java.
Part 4 contains chapters on language mechanisms for concurrency and on logic
programming.
The book is intended for upper-level undergraduate students and beginning
graduate students with some knowledge of basic programming. Students are expected to have some knowledge of C or some other procedural language and some
ix
x
Preface
acquaintance with C++ or some form of object-oriented language. Some experience
with Lisp, Scheme, or ML is helpful in Parts 1 and 2, although many students have
successfully completed the course based on this book without this background. It
is also helpful if students have some experience with simple analysis of algorithms
and data structures. For example, in comparing implementations of certain constructs, it will be useful to distinguish between algorithms of constant-, polynomial-,
and exponential-time complexity.
After reading this book, students will have a better understanding of the range of
programming languages that have been used over the past 40 years, a better understanding of the issues and trade-offs that arise in programming language design, and a
better appreciation of the advantages and pitfalls of the programming languages they
use. Because different languages present different programming concepts, students
will be able to improve their programming by importing ideas from other languages
into the programs they write.
Acknowledgments
This book developed as a set of notes for Stanford CS 242, a course in programming
languages that I have taught since 1993. Each year, energetic teaching assistants
have helped debug example programs for lectures, formulate homework problems,
and prepare model solutions. The organization and content of the course have been
improved greatly by their suggestions. Special thanks go to Kathleen Fisher, who
was a teaching assistant in 1993 and 1994 and taught the course in my absence in
1995. Kathleen helped me organize the material in the early years and, in 1995,
transcribed my handwritten notes into online form. Thanks to Amit Patel for his
initiative in organizing homework assignments and solutions and to Vitaly Shmatikov
for persevering with the glossary of programming language terms. Anne Bracy, Dan
Bentley, and Stephen Freund thoughtfully proofread many chapters.
Lauren Cowles, Alan Harvey, and David Tranah of Cambridge University Press
were encouraging and helpful. I particularly appreciate Lauren’s careful reading and
detailed comments of twelve full chapters in draft form. Thanks also are due to the
reviewers they enlisted, who made a number of helpful suggestions on early versions
of the book. Zena Ariola taught from book drafts at the University of Oregon several
years in a row and sent many helpful suggestions; other test instructors also provided
helpful feedback.
Finally, special thanks to Krzystof Apt for contributing a chapter on logic
programming.
John Mitchell
PART 1
Functions and Foundations
1
Introduction
“The Medium Is the Message”
Marshall McLuhan
1.1 PROGRAMMING LANGUAGES
Programming languages are the medium of expression in the art of computer programming. An ideal programming language will make it easy for programmers to
write programs succinctly and clearly. Because programs are meant to be understood, modified, and maintained over their lifetime, a good programming language
will help others read programs and understand how they work. Software design and
construction are complex tasks. Many software systems consist of interacting parts.
These parts, or software components, may interact in complicated ways. To manage complexity, the interfaces and communication between components must be
designed carefully. A good language for large-scale programming will help programmers manage the interaction among software components effectively. In evaluating
programming languages, we must consider the tasks of designing, implementing, testing, and maintaining software, asking how well each language supports each part of
the software life cycle.
There are many difficult trade-offs in programming language design. Some language features make it easy for us to write programs quickly, but may make it harder
for us to design testing tools or methods. Some language constructs make it easier for
a compiler to optimize programs, but may make programming cumbersome. Because
different computing environments and applications require different program characteristics, different programming language designers have chosen different tradeoffs. In fact, virtually all successful programming languages were originally designed
for one specific use. This is not to say that each language is good for only one purpose.
However, focusing on a single application helps language designers make consistent,
purposeful decisions. A single application also helps with one of the most difficult
parts of language design: leaving good ideas out.
3
4
Introduction
THE AUTHOR
I hope you enjoy using this book. At the beginning of each chapter, I have
included pictures of people involved in the development or analysis of
programming languages. Some of these people are famous, with major
awards and published biographies. Others are less widely recognized.
When possible, I have tried to include some personal information based
on my encounters with these people. This is to emphasize that programming languages are developed by real human beings. Like most human
artifacts, a programming language inevitably reflects some of the personality of its designers.
As a disclaimer, let me point out that I have not made an attempt
to be comprehensive in my brief biographical comments. I have tried
to liven up the text with a bit of humor when possible, leaving serious
biography to more serious biographers. There simply is not space to
mention all of the people who have played important roles in the history
of programming languages.
Historical and biographical texts on computer science and computer
scientists have become increasingly available in recent years. If you like
reading about computer pioneers, you might enjoy paging through Out of
Their Minds: The Lives and Discoveries of 15 Great Computer Scientists
by Dennis Shasha and Cathy Lazere or other books on the history of
computer science.
John Mitchell
Even if you do not use many of the programming languages in this book, you
may still be able to put the conceptual framework presented in these languages to
good use. When I was a student in the mid-1970s, all “serious” programmers (at my
university, anyway) used Fortran. Fortran did not allow recursion, and recursion was
generally regarded as too inefficient to be practical for “real programming.” However,
the instructor of one course I took argued that recursion was still an important idea
and explained how recursive techniques could be used in Fortran by managing data
in an array. I am glad I took that course and not one that dismissed recursion as an
impractical idea. In the 1980s, many people considered object-oriented programming
too inefficient and clumsy for real programming. However, students who learned
about object-oriented programming in the 1980s were certainly happy to know about
1.2 Goals
these “futuristic” languages in the 1990s, as object-oriented programming became
more widely accepted and used.
Although this is not a book about the history of programming languages, there is
some attention to history throughout the book. One reason for discussing historical
languages is that this gives us a realistic way to understand programming language
trade-offs. For example, programs were different when machines were slow and
memory was scarce. The concerns of programming language designers were therefore
different in the 1960s from the current concerns. By imaging the state of the art in
some bygone era, we can give more serious thought to why language designers made
certain decisions. This way of thinking about languages and computing may help
us in the future, when computing conditions may change to resemble some past
situation. For example, the recent rise in popularity of handheld computing devices
and embedded processors has led to renewed interest in programming for devices
with limited memory and limited computing power.
When we discuss specific languages in this book, we generally refer to the original
or historically important form of a language. For example, “Fortran” means the
Fortran of the 1960s and early 1970s. These early languages were called Fortran I,
Fortran II, Fortran III, and so on. In recent years, Fortran has evolved to include
more modern features, and the distinction between Fortran and other languages has
blurred to some extent. Similarly, Lisp generally refers to the Lisps of the 1960s,
Smalltalk to the language of the late 1970s and 1980s, and so on.
1.2 GOALS
In this book we are concerned with the basic concepts that appear in modern programming languages, their interaction, and the relationship between programming
languages and methods for program development. A recurring theme is the trade-off
between language expressiveness and simplicity of implementation. For each programming language feature we consider, we examine the ways that it can be used
in programming and the kinds of implementation techniques that may be used to
compile and execute it efficiently.
1.2.1 General Goals
In this book we have the following general goals:
To understand the design space of programming languages. This includes concepts and constructs from past programming languages as well as those that may
be used more widely in the future. We also try to understand some of the major conflicts and trade-offs between language features, including implementation
costs.
To develop a better understanding of the languages we currently use by comparing
them with other languages.
To understand the programming techniques associated with various language
features. The study of programming languages is, in part, the study of conceptual
frameworks for problem solving, software construction, and development.
5
6
Introduction
Many of the ideas in this book are common knowledge among professional programmers. The material and ways of thinking presented in this book should be
useful to you in future programming and in talking to experienced programmers
if you work for a software company or have an interview for a job. By the end of the
course, you will be able to evaluate language features, their costs, and how they fit
together.
1.2.2 Specific Themes
Here are some specific themes that are addressed repeatedly in the text:
Computability: Some problems cannot be solved by computer. The undecidability of the halting problem implies that programming language compilers and
interpreters cannot do everything that we might wish they could do.
Static analysis: There is a difference between compile time and run time. At
compile time, the program is known but the input is not. At run time, the program
and the input are both available to the run-time system. Although a program
designer or implementer would like to find errors at compile time, many will not
surface until run time. Methods that detect program errors at compile time are
usually conservative, which means that when they say a program does not have
a certain kind of error this statement is correct. However, compile-time errordetection methods will usually say that some programs contain errors even if
errors may not actually occur when the program is run.
Expressiveness versus efficiency: There are many situations in which it would be
convenient to have a programming language implementation do something automatically. An example discussed in Chapter 3 is memory management: The Lisp
run-time system uses garbage collection to detect memory locations no longer
needed by the program. When something is done automatically, there is a cost.
Although an automatic method may save the programmer from thinking about
something, the implementation of the language may run more slowly. In some
cases, the automatic method may make it easier to write programs and make programming less prone to error. In other cases, the resulting slowdown in program
execution may make the automatic method infeasible.
1.3 PROGRAMMING LANGUAGE HISTORY
Hundreds of programming languages have been designed and implemented over
the last 50 years. As many as 50 of these programming languages contained new
concepts, useful refinements, or innovations worthy of mention. Because there are
far too many programming languages to survey, however, we concentrate on six
programming languages: Lisp, ML, C, C++, Smalltalk, and Java. Together, these
languages contain most of the important language features that have been invented
since higher-level programming languages emerged from the primordial swamp of
assembly language programming around 1960.
The history of modern programming languages begins around 1958–1960 with
the development of Algol, Cobol, Fortran, and Lisp. The main body of this book
1.3 Programming Language History
covers Lisp, with a shorter discussion of Algol and subsequent related languages. A
brief account of some earlier languages is given here for those who may be curious
about programming language prehistory.
In the 1950s, a number of languages were developed to simplify the process
of writing sequences of computer instructions. In this decade, computers were very
primitive by modern standards. Most programming was done with the native machine
language of the underlying hardware. This was acceptable because programs were
small and efficiency was extremely important. The two most important programming
language developments of the 1950s were Fortan and Cobol.
Fortran was developed at IBM around 1954–1956 by a team led by John
Backus. The main innovation of Fortran (a contraction of formula translator) was
that it became possible to use ordinary mathematical notation in expressions. For
example, the Fortran expression for adding the value of i to twice the value of
j is i + 2∗ j. Before the development of Fortran, it might have been necessary to
place i in a register, place j in a register, multiply j times 2 and then add the
result to i. Fortran allowed programmers to think more naturally about numerical calculation by using symbolic names for variables and leaving some details of
evaluation order to the compiler. Fortran also had subroutines (a form of procedure or function), arrays, formatted input and output, and declarations that gave
programmers explicit control over the placement of variables and arrays in memory. However, that was about it. To give you some idea of the limitations of
Fortran, many early Fortran compilers stored numbers 1, 2, 3 . . . in memory locations, and programmers could change the values of numbers if they were not
careful! In addition, it was not possible for a Fortran subroutine to call itself, as
this required memory management techniques that had not been invented yet (see
Chapter 7).
Cobol is a programming language designed for business applications. Like
Fortran programs, many Cobol programs are still in use today, although current
versions of Fortran and Cobol differ substantially from forms of these languages
of the 1950s. The primary designer of Cobol was Grace Murray Hopper, an important computer pioneer. The syntax of Cobol was intended to resemble that
of common English. It has been suggested in jest that if object-oriented Cobol
were a standard today, we would use “add 1 to Cobol giving Cobol” instead of
“C++”.
The earliest languages covered in any detail in this book are Lisp and Algol,
which both came out around 1960. These languages have stack memory management and recursive functions or procedures. Lisp provides higher-order functions
(still not available in many current languages) and garbage collection, whereas the
Algol family of languages provides better type systems and data structuring. The
main innovations of the 1970s were methods for organizing data, such as records (or
structs), abstract data types, and early forms of objects. Objects became mainstream
in the 1980s, and the 1990s brought increasing interest in network-centric computing,
interoperability, and security and correctness issues associated with active content
on the Internet. The 21st century promises greater diversity of computing devices,
cheaper and more powerful hardware, and increasing interest in correctness, security,
and interoperability.
7
8
Introduction
1.4 ORGANIZATION: CONCEPTS AND LANGUAGES
There are many important language concepts and many programming languages.
The most natural way to summarize the field is to use a two-dimensional matrix, with
languages along one axis and concepts along the other. Here is a partial sketch of
such a matrix:
Heap
Language Expressions Functions storage Exceptions Modules Objects Threads
Lisp
C
Algol 60
Algol 68
Pascal
Modula-2
Modula-3
ML
Simula
Smalltalk
C++
Objective
C
Java
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Although this matrix lists only a fraction of the languages and concepts that might
be covered in a basic text or course on the programming languages, one general
characteristic should be clear. There are some basic language concepts, such as expressions, functions, local variables, and stack storage allocation that are present in
many languages. For these concepts, it makes more sense to discuss the concept in
general than to go through a long list of similar languages. On the other hand, for
concepts such as objects and threads, there are relatively few languages that exhibit
these concepts in interesting ways. Therefore, we can study most of the interesting
aspects of objects by comparing a few languages. Another factor that is not clear from
the matrix is that, for some concepts, there is considerable variation from language to
language. For example, it is more interesting to compare the way objects have been
integrated into languages than it is to compare integer expressions. This is another
reason why competing object-oriented languages are compared, but basic concepts
related to expressions, statements, functions, and so on, are covered only once, in a
concept-oriented way.
Most courses and texts on programming languages use some combination of
language-based and concept-based presentation. In this book a concept-oriented
organization is followed for most concepts, with a language-based organization used
to compare object-oriented features.
The text is divided into four parts:
Part 1: Functions and Foundations
(Chapters 1–4)
Part 2: Procedures, Types, Memory Management, and Control
(5–8)
Part 3: Modularity, Abstraction and Object-Oriented Programming
(9–13)
1.4 Organization: Concepts and Languages
Part 4: Concurrency and Logic Programming
(14 and 15)
In Part 1 a short study of Lisp is presented, followed by a discussion of compiler
structure, parsing, lambda calculus, and denotational semantics. A short chapter provides a brief discussion of computability and the limits of compile-time program
analysis and optimization. For C programmers, the discussion of Lisp should provide
a good chance to think differently about programming and programming languages.
In Part 2, we progress through the main concepts associated with the conventional
languages that are descended in some way from the Algol family. These concepts
include type systems and type checking, functions and stack storage allocation, and
control mechanisms such as exceptions and continuations. After some of the history
of the Algol family of languages is summarized, the ML programming language is
used as the main example, with some discussion and comparisons using C syntax.
Part 3 is an investigation of program-structuring mechanisms. The important language advances of the 1970s were abstract data types and program modules. In the
late 1980s, object-oriented concepts attained widespread acceptance. Because objectoriented programming is currently the most prominent programming paradigm,
in most of Part 3 we focus on object-oriented concepts and languages, comparing
Smalltalk, C++, and Java.
Part 4 contains chapters on language mechanisms for concurrent and distributed
programs and on logic programming.
Because of space limitations, a number of interesting topics are not covered.
Although scripting languages and other “special-purpose” languages are not covered
explicitly in detail, an attempt has been made to integrate some relevant language
concepts into the exercises.
9
2
Computability
Some mathematical functions are computable and some are not. In all generalpurpose programming languages, it is possible to write a program for each function
that is computable in principle. However, the limits of computability also limit the
kinds of things that programming language implementations can do. This chapter
contains a brief overview of computability so that we can discuss limitations that
involve computability in other chapters of the book.
2.1 PARTIAL FUNCTIONS AND COMPUTABILITY
From a mathematical point of view, a program defines a function. The output of
a program is computed as a function of the program inputs and the state of the
machine before the program starts. In practice, there is a lot more to a program than
the function it computes. However, as a starting point in the study of programming
languages, it is useful to understand some basic facts about computable functions.
The fact that not all functions are computable has important ramifications for
programming language tools and implementations. Some kinds of programming
constructs, however useful they might be, cannot be added to real programming
languages because they cannot be implemented on real computers.
2.1.1 Expressions, Errors, and Nontermination
In mathematics, an expression may have a defined value or it may not. For example,
the expression 3 + 2 has a defined value, but the expression 3/0 does not. The reason
that 3/0 does not have a value is that division by zero is not defined: division is defined
to be the inverse of multiplication, but multiplication by zero cannot be inverted.
There is nothing to try to do when we see the expression 3/0; a mathematician
would just say that this operation is undefined, and that would be the end of the
discussion.
In computation, there are two different reasons why an expression might not
have a value:
10
2.1 Partial Functions and Computability
ALAN TURING
Alan Turing was a British mathematician. He is known for his early work
on computability and his work for British Intelligence on code breaking
during the Second World War. Among computer scientists, he is best
known for the invention of the Turing machine. This is not a piece of
hardware, but an idealized computing device. A Turing machine consists
of an infinite tape, a tape read–write head, and a finite-state controller.
In each computation step, the machine reads a tape symbol and the
finite-state controller decides whether to write a different symbol on the
current tape square and then whether to move the read–write head one
square left or right. The importance of this idealized computer is that it
is both very simple and very powerful.
Turing was a broad-minded individual with interests ranging from
relativity theory and mathematical logic to number theory and the engineering design of mechanical computers. There are numerous published
biographies of Alan Turing, some emphasizing his wartime work and others calling attention to his sexuality and its impact on his professional
career.
The ACM Turing Award is the highest scientific honor in computer
science, equivalent to a Nobel Prize in other fields.
Error termination: Evaluation of the expression cannot proceed because of a
conflict between operator and operand.
Nontermination: Evaluation of the expression proceeds indefinitely.
An example of the first kind is division by zero. There is nothing to compute in this
case, except possibly to stop the computation in a way that indicates that it could
not proceed any further. This may halt execution of the entire program, abort one
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12
Computability
thread of a concurrent program, or raise an exception if the programming language
provides exceptions.
The second case is different: There is a specific computation to perform, but the
computation may not terminate and therefore may not yield a value. For example,
consider the recursive function defined by
f(x:int) = if x = 0 then 0 else x + f(x-2)
This is a perfectly meaningful definition of a partial function, a function that has a
value on some arguments but not on all arguments. The expression f(4) calling the
function f above has value 4 + 2 + 0 = 6, but the expression f(5) does not have a
value because the computation specified by this expression does not terminate.
2.1.2 Partial Functions
A partial function is a function that is defined on some arguments and undefined on
others. This is ordinarily what is meant by function in programming, as a function
declared in a program may return a result or may not if some loop or sequence
of recursive calls does not terminate. However, this is not what a mathematician
ordinarily means by the word function.
The distinction can be made clearer by a look at the mathematical definitions. A
reasonable definition of the word function is this: A function f : A → B from set A
to set B is a rule associating a unique value y = f (x) in B with every x in A. This is
almost a mathematical definition, except that the word rule does not have a precise
mathematical meaning. The notation f : A → B means that, given arguments in the
set A, the function f produces values from set B. The set A is called the domain of
f , and the set B is called the range or the codomain of f .
The usual mathematical definition of function replaces the idea of rule with a
set of argument–result pairs called the graph of a function. This is the mathematical
definition:
A function f : A → B is a set of ordered pairs f ⊆ A × B that satisfies the following conditions:
1. If x, y ∈ f and x, z ∈ f , then y = z.
2. For every x ∈ A, there exists y ∈ B with x, y ∈ f.
When we associate a set of ordered pairs with a function, the ordered pair x, y
is used to indicate that y is the value of the function on argument x. In words, the
preceding two conditions can be stated as (1) a function has at most one value for
every argument in its domain, and (2) a function has at least one value for every
argument in its domain.
A partial function is similar, except that a partial function may not have a value
for every argument in its domain. This is the mathematical definition:
A partial function f : A → B is a set of ordered pairs f ⊆ A × B satisfying the
preceding condition
1. If x, y ∈ f and x, z ∈ f , then y = z.
2.1 Partial Functions and Computability
In words, a partial function is single valued, but need not be defined on all elements
of its domain.
Programs Define Partial Functions
In most programming languages, it is possible to define functions recursively. For
example, here is a function f defined in terms of itself:
f(x:int) = if x = 0 then 0 else x + f(x-2);
If this were written as a program in some programming language, the declaration
would associate the function name f with an algorithm that terminates on every even
x –> 0, but diverges (does not halt and return a value) if x is odd or negative. The
algorithm for f defines the following mathematical function f , expressed here as a
set of ordered pairs:
f = {x, y | x is positive and even, y = 0 + 2 + 4 + · · · + x}.
This is a partial function on the integers. For every integer x, there is at most one
y with f (x) = y. However, if x is an odd number, then there is no y with f (x) = y.
Where the algorithm does not terminate, the value of the function is undefined.
Because a function call may not terminate, this program defines a partial function.
2.1.3 Computability
Computability theory gives us a precise characterization of the functions that are
computable in principle. The class of functions on the natural numbers that are
computable in principle is often called the class of partial recursive functions, as
recursion is an essential part of computation and computable functions are, in general,
partial rather than total. The reason why we say “computable in principle” instead of
“computable in practice” is that some computable functions might take an extremely
long time to compute. If a function call will not return for an amount of time equal
to the length of the entire history of the universe, then in practice we will not be able
to wait for the computation to finish. Nonetheless, computability in principle is an
important benchmark for programming languages.
Computable Functions
Intuitively, a function is computable if there is some program that computes it. More
specifically, a function f : A → B is computable if there is an algorithm that, given
any x ∈ A as input, halts with y = f (x) as output.
One problem with this intuitive definition of computable is that a program has
to be written out in some programming language, and we need to have some implementation to execute the program. It might very well be that, in one programming
language, there is a program to compute some mathematical function and in another
language there is not.
In the 1930s, Alonzo Church of Princeton University proposed an important principle, called Church’s thesis. Church’s thesis, which is a widely held belief about the
relation between mathematical definitions and the real world of computing, states
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Computability
that the same class of functions on the integers can be computed by any general
computing device. This is the class of partial recursive functions, sometimes called
the class of computable functions. There is a mathematical definition of this class of
functions that does not refer to programming languages, a second definition that uses
a kind of idealized computing device called a Turing machine, and a third (equivalent) definition that uses lambda calculus (see Section 4.2). As mentioned in the
biographical sketch on Alan Turing, a Turing machine consists of an infinite tape, a
tape read–write head, and a finite-state controller. The tape is divided into contiguous cells, each containing a single symbol. In each computation step, the machine
reads a tape symbol and the finite-state controller decides whether to write a different symbol on the current tape square and then whether to move the read–write
head one square left or right. Part of the evidence that Church cited in formulating this thesis was the proof that Turing machines and lambda calculus are equivalent. The fact that all standard programming languages express precisely the class
of partial recursive functions is often summarized by the statement that all programming languages are Turing complete. Although it is comforting to know that
all programming languages are universal in a mathematical sense, the fact that all
programming languages are Turing complete also means that computability theory
does not help us distinguish among the expressive powers of different programming
languages.
Noncomputable Functions
It is useful to know that some specific functions are not computable. An important
example is commonly referred to as the halting problem. To simplify the discussion
and focus on the central ideas, the halting problem is stated for programs that require
one string input. If P is such a program and x is a string input, then we write P(x)
for the output of program P on input x.
Halting Problem: Given a program P that requires exactly one string input
and a string x, determine whether P halts on input x.
We can associate the halting problem with a function fhalt by letting fhalt (P, x) =
“halts” if P halts on input and fhalt (P, x) = “does not halt” otherwise. This function
fhalt can be considered a function on strings if we write each program out as a sequence
of symbols.
The undecidability of the halting problem is the fact that the function fhalt is not
computable. The undecidability of the halting problem is an important fact to keep
in mind in designing programming language implementations and optimizations. It
implies that many useful operations on programs cannot be implemented, even in
principle.
Proof of the Undecidability of the Halting Problem. Although you will not need to
know this proof to understand any other topic in the book, some of you may be interested in proof that the halting function is not computable. The proof is surprisingly
short, but can be difficult to understand. If you are going to be a serious computer
scientist, then you will want to look at this proof several times, over the course of
several days, until you understand the idea behind it.
Step 1: Assume that there is a program Q that solves the halting problem. Specifically, assume that program Q reads two inputs, both strings, and has the
2.1 Partial Functions and Computability
following output:
halts
Q(P, x) =
does not halt
if P(x) halts
.
if P(x) does not
An important part of this specification for Q is that Q(P, x) always halts for
every P and x.
Step 2: Using program Q, we can build a program D that reads one string input
and sometimes does not halt. Specifically, let D be a program that works as
follows:
D(P) = if Q(P, P) = halts then run forever else halt.
Note that D has only one input, which it gives twice to Q. The program D
can be written in any reasonable language, as any reasonable language should
have some way of programming if-then-else and some way of writing a loop or
recursive function call that runs forever. If you think about it a little bit, you
can see that D has the following behavior:
halt
if P(P) runs forever
D(P) =
.
run forever if P(P) halts
In this description, the word halt means that D(P) comes to a halt, and runs
forever means that D(P) continues to execute steps indefinitely. The program
D(P) halts or does not halt, but does not produce a string output in any case.
Step 3: Derive a contradiction by considering the behavior D(D) of program D
on input D. (If you are starting to get confused about what it means to run
a program with the program itself as input, assume that we have written the
program D and stored it in a file. Then we can compile D and run D with the
file containing a copy of D as input.) Without thinking about how D works or
what D is supposed to do, it is clear that either D(D) halts or D(D) does not
halt. If D(D) halts, though, then by the property of D given in step 2, this must
be because D(D) runs forever. This does not make any sense, so it must be that
D(D) runs forever. However, by similar reasoning, if D(D) runs forever, then
this must be because D(D) halts. This is also contradictory. Therefore, we have
reached a contradiction.
Step 4: Because the assumption in step 1 that there is a program Q solving the
halting problem leads to a contradiction in step 3, it must be that the assumption
is false. Therefore, there is no program that solves the halting problem.
Applications
Programming language compilers can often detect errors in programs. However,
the undecidability of the halting problem implies that some properties of programs
cannot be determined in advance. The simplest example is halting itself. Suppose
someone writes a program like this:
i = 0;
while (i != f(i)) i = g(i);
printf(. . . i . . .);
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Computability
It seems very likely that the programmer wants the while loop to halt. Otherwise,
why would the programmer have written a statement to print the value of i after the
loop halts? Therefore, it would be helpful for the compiler to print a warning message
if the loop will not halt. However useful this might be, though, it is not possible for
a compiler to determine whether the loop will halt, as this would involve solving the
halting problem.
2.2 CHAPTER SUMMARY
Computability theory establishes some important ground rules for programming
language design and implementation. The following main concepts from this short
overview should be remembered:
Partiality: Recursively defined functions may be partial functions. They are not
always total functions. A function may be partial because a basic operation is not
defined on some argument or because a computation does not terminate.
Computability: Some functions are computable and others are not. Programming
languages can be used to define computable functions; we cannot write programs
for functions that are not computable in principle.
Turing completeness: All standard general-purpose programming languages give
us the same class of computable functions.
Undecidability: Many important properties of programs cannot be determined
by any computable function. In particular, the halting problem is undecidable.
When the value of a function or the value of an expression is undefined because a basic
operation such as division by zero does not make sense, a compiler or interpreter can
cause the program to halt and report the error. However, the undecidability of the
halting problem implies that there is no way to detect and report an error whenever
a program is not going to halt.
There is a lot more to computability and complexity theory than is summarized in
the few pages here. For more information, see one of the many books on computability and complexity theory such as Introduction to Automata Theory, Languages, and
Computation by Hopcroft, Motwani, and Ullman (Addison Wesley, 2001) or Introduction to the Theory of Computation by Sipser (PWS, 1997).
EXERCISES
2.1 Partial and Total Functions
For each of the following function definitions, give the graph of the function. Say
whether this is a partial function or a total function on the integers. If the function is
partial, say where the function is defined and undefined.
For example, the graph of f(x) = if x > 0 then x + 2 else x/0 is the set of ordered pairs
{x, x + 2 | x > 0}. This is a partial function. It is defined on all integers greater than
0 and undefined on integers less than or equal to 0.
Functions:
(a) f(x) = if x + 2 > 3 then x ∗ 5 else x/0
(b) f(x) = if x < 0 then 1 else f(x - 2)
(c) f(x) = if x = 0 then 1 else f(x - 2)
Exercises
2.2 Halting Problem on No Input
Suppose you are given a function Halt∅ that can be used to determine whether a
program that requires no input halts. To make this concrete, assume that you are
writing a C or Pascal program that reads in another program as a string. Your program
is allowed to call Halt∅ with a string input. Assume that the call to Halt∅ returns true if
the argument is a program that halts and does not read any input and returns false if
the argument is a program that runs forever and does not read any input. You should
not make any assumptions about the behavior of Halt∅ on an argument that is not a
syntactically correct program.
Can you solve the halting problem by using Halt∅ ? More specifically, can you
write a program that reads a program text P as input, reads an integer n as input,
and then decides whether P halts when it reads n as input? You may assume that
any program P you are given begins with a read statement that reads a single integer
from standard input. This problem does not ask you to write the program to solve
the halting problem. It just asks whether it is possible to do so.
If you believe that the halting problem can be solved if you are given Halt∅ , then
explain your answer by describing how a program solving the halting problem would
work. If you believe that the halting problem cannot be solved by using Halt∅ , then
explain briefly why you think not.
2.3 Halting Problem on All Input
Suppose you are given a function Halt∀ that can be used to determine whether a
program halts on all input. Under the same conditions as those of problem 2.2, can
you solve the halting problem by using Halt∀ ?
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3
Lisp: Functions, Recursion, and Lists
Lisp is the medium of choice for people who enjoy free
style and flexibility.
Gerald J. Sussman
A Lisp programmer knows the value of everything, but the
cost of nothing.
Alan Perlis
Lisp is a historically important language that is good for illustrating a number of
general points about programming languages. Because Lisp is very different from
procedure-oriented and object-oriented languages you may use more frequently, this
chapter may help you think about programming in a different way. Lisp shows that
many goals of programming language design can be met in a simple, elegant way.
3.1 LISP HISTORY
The Lisp programming language was developed at MIT in the late 1950s for research
in artificial intelligence (AI) and symbolic computation. The name Lisp is an acronym
for the LISt Processor. Lists comprise the main data structure of Lisp.
The strength of Lisp is its simplicity and flexibility. It has been widely used for
exploratory programming, a style of software development in which systems are built
incrementally and may be changed radically as the result of experimental evaluation.
Exploratory programming is often used in the development of AI programs, as a
researcher may not know how the program should accomplish a task until several
unsuccessful programs have been tested. The popular text editor emacs is written in
Lisp, as is the linux graphical toolkit gtk and many other programs in current use in
a variety of computing environments.
Many different Lisp implementations have been built over the years, leading
to many different dialects of the language. One influential dialect was Maclisp,
18
3.1 Lisp History
JOHN MCCARTHY
A programming language designer and a central figure in the field of
artificial intelligence, John McCarthy led the original Lisp effort at MIT in
the late 1950s and early 1960s. Among other seminal contributions to
the field, McCarthy participated in the design of Algol 60 and formulated
the concept of time sharing in a 1959 memo to the director of the MIT
Computation Center. McCarthy moved to Stanford in 1962, where he
has been on the faculty ever since.
Throughout his career, John McCarthy has advocated using formal
logic and mathematics to understand programming languages and systems, as well as common-sense reasoning and other topics in artificial
intelligence. In the early 1960s, he wrote a series of papers on what he
called a Mathematical Theory of Computation. These identified a number
of important problems in understanding and reasoning about computer
programs and systems. He supported political freedom for scientists
abroad during the Cold War and has been an advocate of free speech in
electronic media.
Now a lively person with graying hair and beard, McCarthy is an
independent thinker who suggests creative solutions to bureaucratic as
well as technical problems. He has won a number of important prizes
and honors, including the ACM Turing Award in 1971.
developed in the 1960s at MIT’s Project MAC. Another was Scheme, developed at
MIT in the 1970s by Guy Steele and Gerald Sussman. Common Lisp is a modern
Lisp with complex forms of object-oriented primitives.
McCarthy’s 1960 paper on Lisp, called “Recursive functions of symbolic expressions and their computation by machine” [Communications of the Association for
Computing Machinery, 3(4), 184–195 (1960)] is an important historical document
with many good ideas. In addition to the value of the programming language ideas
it contains, the paper gives us some idea of the state of the art in 1960 and provides
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Lisp: Functions, Recursion, and Lists
some useful insight into the language design process. You might enjoy reading the
first few sections of the paper and skim the other parts briefly to see what they contain. The journal containing the article will be easy to find in many computer science
libraries or you can find a retypeset version of the paper in electronic form on the
Web.
3.2 GOOD LANGUAGE DESIGN
Most successful language design efforts share three important characteristics with
the Lisp project:
Motivating Application: The language was designed so that a specific kind of
program could be written more easily.
Abstract Machine: There is a simple and unambiguous program execution model.
Theoretical Foundations: Theoretical understanding was the basis for including
certain capabilities and omitting others.
These points are elaborated in the subsequent subsections.
Motivating Application
An important programming problem for McCarthy’s group was a system called
Advice Taker. This was a common-sense reasoning system based on logic. As the
name implies, the program was supposed to read statements written in a specific
input language, perform logical reasoning, and answer simple questions. Another
important problem used in the design of Lisp was symbolic calculation. For example,
McCarthy’s group wanted to be able to write a program that could find a symbolic
expression for the indefinite integral (as in calculus) for a function, given a symbolic
description of the function as input.
Most good language designs start from some specific need. For comparison, here
are some motivating problems that were influential in the design of other programming languages:
Lisp
C
Simula
PL/1
Symbolic computation, logic, experimental programming
Unix operating system
Simulation
Tried to solve all programming problems; not successful or influential
A specific purpose provides focus for language designers. It helps us to set criteria for
making design decisions. A specific, motivating application also helps us to solve one
of the hardest problems in programming language design: deciding which features
to leave out.
Program Execution Model
A language design must be specific about how all basic operations are done. The
language design may either be very concrete, prescribing exactly how the parts of
the language must be implemented, or more abstract, specifying only certain properties that must be satisfied in any implementation. It is possible to err in either
direction. A language that is too closely tied to one machine will lead to programs
3.2 Good Language Design
that are not portable. When new technology leads to faster machine architectures,
programs written in the language may become obsolete. At the other extreme, it
is possible to be too abstract. If a language design specifies only what the eventual value of an expression must be, without any information about how it is to be
evaluated, it may be difficult for programmers to write efficient code. Most programmers find it important to have a good understanding of how programs will be
executed, with enough detail to predict program running time. Lisp was designed
for a specific machine, the IBM 704. However, if the designers had built the language around a lot of special features of a particular computer, the language would
not have survived as well as it has. Instead, by luck or by design, they identified a
useful set of simple concepts that map easily onto the IBM 704 architecture, and
also onto other computers. The Lisp execution model is discussed in more detail in
Subsection 3.4.3.
A systematic, predictable machine model is critical to the success of a programming language. For comparison, here are some execution models associated with the
design of other programming languages:
Fortran
Algol family
Smalltalk
Flat register machine
No stacks, no recursion
Memory arranged as linear array
Stack of activation records
Heap storage
Objects, communicating by messages
Theoretical Foundations
McCarthy described Lisp as a “scheme for representing the partial recursive functions
of a certain class of symbolic expressions.” We discussed computability and partial
recursive functions in Chapter 2. Here are the main points about computability
theory that are relevant to the design of Lisp:
Lisp was designed to be Turing complete, meaning that all partial recursive
functions may be written in Lisp. The phrase “Turing complete” refers to a characterization of computability proposed by the mathematician A.M. Turing; see
Chapter 2.
The use of function expressions and recursion in Lisp take direct advantage
of a mathematical characterization of computable functions based on lambda
calculus.
Today it is unlikely that a team of programming language designers would advertise
that their language is sufficient to define all partial recursive functions. Most computer scientists nowadays know about computability theory and assume that most
languages intended for general programming are Turing complete. However, computability theory and other theoretical frameworks such as type theory continue to
have important consequences for programming language design.
The connection between Lisp and lambda calculus is important, and lambda calculus remains an important tool in the study of programming languages. A summary
of lambda calculus appears in Section 4.2.
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Lisp: Functions, Recursion, and Lists
3.3 BRIEF LANGUAGE OVERVIEW
The topic of this chapter is a language that might be called Historical Lisp. This is
essentially Lisp 1.5, from the early 1960s, with one or two minor changes. Because
there are several different versions of Lisp in common use, it is likely that some
function names used in this book will differ from those you may have used in previous
Lisp programming.
An engaging book that captures some of the spirit of contemporary Lisp is
the Scheme-based paperback by D.P. Friedman and M. Felleisen, titled The Little
Schemer (MIT Press, Cambridge, MA, 1995). This is similar to an earlier book by the
same authors entitled The Little LISPer. Lisp syntax is extremely simple. To make
parsing (see Section 4.1) easy, all operations are written in prefix form, with the operator in front of all the operands. Here are some examples of Lisp expressions, with
corresponding infix form for comparison.
Lisp prefix notation
Infix notation
(+ 1 2 3 4 5)
(∗ (+ 2 3) (+ 4 5))
(f x y)
(1 + 2 + 3 + 4 + 5)
((2 + 3) ∗ (4 + 5))
f(x, y)
Atoms
Lisp programs compute with atoms and cells. Atoms include integers, floating-point
numbers, and symbolic atoms. Symbolic atoms may have more than one letter. For
example, the atom duck is printed with four letters, but it is atomic in the sense that
there is no Lisp operation for taking the atom apart into four separate atoms.
In our discussion of Historical Lisp, we use only integers and symbolic atoms.
Symbolic atoms are written with a sequence of characters and digits, beginning with
a character. The atoms, symbols, and numbers are given by the following Backus
normal form (BNF) grammar (see Section 4.1 if you are not familiar with grammars):
<atom> ::= <smbl> | <num>
<smbl> ::= <char> | <smbl><char> | <smbl><digit>
<num> ::= <digit> | <num><digit>
An atom that is used for some special purposes is the atom nil.
S-Expressions and Lists
The basic data structures of Lisp are dotted pairs, which are pairs written with a
dot between the two parts of the pair. Putting atoms or pairs together, we can write
symbolic expressions in a form traditionally called S-expressions. The syntax of Lisp
S-expressions is given by the following grammar:
<sexp> ::= <atom> | (<sexp> . <sexp>)
Although S-expressions are the basic data of Historical Lisp, most Lisp programs
3.3 Brief Language Overview
actually use lists. Lisp lists are built out of pairs in a particular way, as described in
Subsection 3.4.3.
Functions and Special Forms
The basic functions of Historical Lisp are the operations
cons
car cdr
eq
atom
on pairs and atoms, together with the general programming functions
cond
lambda
define quote
eval
We also use numeric functions such as +, –, and ∗, writing these in the usual Lisp prefix
notation. The function cons is used to combine two atoms or lists, and car and cdr take
lists apart. The function eq is an equality test and atom tests whether its argument is
an atom. These are discussed in more detail in Subsection 3.4.3 in connection with
the machine representation of lists and pairs.
The general programming functions include cond for a conditional test (if. . .
then. . .else. . .), lambda for defining functions, define for declarations, quote to delay
or prevent evaluation, and eval to force evaluation of an expression.
The functions cond, lambda, define, and quote are technically called special forms
since an expression beginning with one of these special functions is evaluated without
evaluating all of the parts of the expression. More about this below.
The language summarized up to this point is called pure Lisp. A feature of pure
Lisp is that expressions do not have side effects. This means that evaluating an expression only produces the value of that expression; it does not change the observable
state of the machine. Some basic functions that do have side effects are
rplaca
rplacd
set setq
We discuss these in Subsection 3.4.9. Lisp with one or more of these functions is
sometimes called impure Lisp.
Evaluation of Expressions
The basic structure of the Lisp interpreter or compiler is the read-eval-print loop.
This means that the basic action of the interpreter is to read an expression, evaluate
it, and print the value. If the expression defines the meaning of some symbol, then
the association between the symbol and its value is saved so that the symbol can be
used in expressions that are typed in later.
In general, we evaluate a Lisp expression
(function arg1 . . . argn )
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Lisp: Functions, Recursion, and Lists
by evaluating each of the arguments in turn, then passing the list of argument values
to the function. The exceptions to this rule are called special forms. For example, we
evaluate a conditional expression
(cond (p1 e1 ) . . . (pn en ))
by proceeding from left to right, finding the first pi with a value different from nil.
This involves evaluating p1 . . . pn and one ei if pi is nonnil. We return to this below.
Lisp uses the atoms T and nil for true and false, respectively. In this book, true
and false are often written in Lisp code, as these are more intuitive and more understandable if you are have not done a lot of Lisp programming. You may read Lisp
examples that contain true and false as if they appear inside a program for which we
have already defined true and false as synonyms for T and nil, respectively.
A slightly tricky point is that the Lisp evaluator needs to distinguish between a
string that is used to name an atom and a string that is used for something else, such
as the name of a function. The form quote is used to write atoms and lists directly:
(quote cons)
(cons a b)
(cons (quote A) (quote B))
expression whose value is the atom “cons”
expression whose value is the pair containing
the values of a and b
expression whose value is the pair containing
the atoms “A” and “B”
In most dialects of Lisp, it is common to write ’bozo instead of (quote bozo). You can
see from the preceding brief description that quote must be a special form. Here are
some additional examples of Lisp expressions and their values:
(+ 4 5)
(+ (+ 1 2) (+ 4 5))
(quote (+ 1 2))
’(+ 1 2)
expression with value 9
first evaluate 1+2, then 4+5, then 3+9 to get value 12
evaluates to a list (+ 1 2)
same as (quote (+ 1 2))
Example. Here is a slightly longer Lisp program example, the definition of a function
that searches a list. The find function takes two arguments, x and y, and searches the
list y for an occurrence of x. The declaration begins with define, which indicates
that this is a declaration. Then follows the name find that is being defined, and the
expression for the find function:
(define find (lambda (x y)
(cond ((equal y nil) nil)
((equal x (car y)) x)
(true (find x (cdr y)))
)))
3.4 Innovations in the Design of Lisp
Lisp function expressions begin with lambda. The function has two arguments, x
and y, which appear in a list immediately following lambda. The return value of the
function is given by the expression that follows the parameters. The function body
is a conditional expression, which returns nil, the empty list, if y is the empty list.
Otherwise, if x is the first element (car) of the list y, then the function returns the
element x. Otherwise the function makes a recursive call to see if x is in the cdr of the
list y. The cdr of a list is the list of all elements that occur after the first element. We
can use this function to find ’apple in the list ’(pear peach apple fig banana) by writing
the Lisp expression
(find ’apple ’(pear peach apple fig banana))
Static and Dynamic Scope
Historically, Lisp was a dynamically scoped language. This means that a variable
inside an expression could refer to a different value if it is passed to a function that
declared this variable differently. When Scheme was introduced in 1978, it was a
statically scoped variant of Lisp. As discussed in Chapter 7, static scoping is common
in most modern programming languages. Following the widespread acceptance of
Scheme, most modern Lisps have become statically scoped. The difference between
static and dynamic scope is not covered in this chapter.
Lisp and Scheme
If you want to try writing Lisp programs by using a Scheme compiler, you will want
to know that the names of some functions and special forms differ in Scheme and
Lisp. Here is a summary of some of the notational differences:
Lisp
Scheme
Lisp
Scheme
defun
defvar
car, cdr
cons
null
atom
eq, equal
Setq
cond . . . t
define
define
car, cdr
cons
null?
atom?
eq?, equal?
set!
cond . . . else
rplacaset
rplacdset
mapcar
t
nil
nil
nil
progn
car!
cdr!
map
#t
#f
nil
’()
begin
3.4 INNOVATIONS IN THE DESIGN OF LISP
3.4.1 Statements and Expressions
Just as virtually all natural languages have certain basic parts of speech, such as
nouns, verbs, and adjectives, there are programming language parts of speech that
occur in most languages. The most basic programming language parts of speech are
expressions, statements, and declarations. These may be summarized as follows:
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Lisp: Functions, Recursion, and Lists
Expression: a syntactic entity that may be evaluated to determine its value. In
some cases, evaluation may not terminate, in which case the expression has no
value. Evaluation of some expressions may change the state of the machine,
causing a side effect in addition to producing a value for the expression.
Statement : a command that alters the state of the machine in some explicit
way. For example, the machine language statement load 4094 r1 alters the
state of the machine by placing the contents of location 4094 into register r1.
The programming language statement x := y + 3 alters the state of the machine
by adding 3 to the value of variable y and storing the result in the location
associated with variable x.
Declaration: a syntactic entity that introduces a new identifier, often specifying one or more attributes. For example, a declaration may introduce a
variable i and specify that it is intended to have only integer values.
Errors and termination may depend on the order in which parts of expressions
are evaluated. For example, consider the expression
if f(2)=2 or f(3)=3 then 4 else 4
where f is a function that halts on even arguments but runs forever on odd arguments.
In many programming languages, a Boolean expression A or B would be evaluated
from left to right, with B evaluated only if A is false. In this case, the value of the
preceding expression would be 4. However, if we evaluate the test A or B from right
to left or evaluate both A and B regardless of the value of A, then the value of the
expression is undefined.
Traditional machine languages and assembly languages are based on statements.
Lisp is an expression-based language, meaning that the basic constructs of the language are expressions, not statements. In fact, pure Lisp has no statements and no
expressions with side effects. Although it was known from computability theory that
it was possible to define all computable functions without using statements or side effects, Lisp was the first programming language to try to put this theoretical possibility
into practice.
3.4.2 Conditional Expressions
Fortran and assembly languages used before Lisp had conditional statements. A
typical statement might have the form
if (condition) go to 112
If the condition is true when this command is executed, then the program jumps to
the statement with the label 112. However, conditional expressions that produce a
value instead of causing a jump were new in Lisp. They also appeared in Algol 60, but
this seems to have been the result of a proposal by McCarthy, modified by a syntactic
suggestion of Backus.
3.4 Innovations in the Design of Lisp
The Lisp conditional expression
(cond (p1 e1 ) . . . (pn en ))
could be written as
if p1 then e1
else if p2 then e2
...
else if pn then en
else no value
in an Algol-like notation, except that most programming languages do not have
a direct way of specifying the absence of a value. In brief, the value of (cond (p1
e1 ) . . . (pn en )) is the first ei , proceeding from left to right, with pi nonnil and pj nil
(representing false) for all j < i. If there is no such ei then the conditional expression
has no value. If any of the expressions p1 . . . pn have side effects, then these will occur
from left to right as the conditional expression is evaluated.
The Lisp conditional expression would now be called a sequential conditional
expression. The reason it is called sequential is that the parts of this expression are
evaluated in sequence from left to right, with evaluation terminating as soon as a
value for the expression can be determined.
It is worth noting that (cond (p1 e1 ) . . . (pn en )) is undefined if
p1 , . . . ,pn are all nil
p1 , . . . ,pi false and pi+1 undefined
p1 , . . . ,pi false, pi+1 true, and ei+1 undefined
Here are some example conditional expressions and their values:
(cond ((< 2 1) 2) ((< 1 2) 1))
(cond ((< 2 1) 2) ((< 3 2) 3))
(cond (diverge 1) (true 0))
(cond (true 0) (diverge 1))
has value 1
is undefined
is undefined, if diverge does not terminate
has value 0
Strictness. An important part of the Lisp cond expression is that a conditional
expression may have a value even if one or more subexpressions do not. For example, (cond (true e1 ) (false e2 )) may be defined even if e2 is undefined. In contrast,
e1 + e2 is undefined if either e1 or e2 is undefined. In standard programming language
terminology, an operator or expression form is strict if all operands or subexpressions
are evaluated. Lisp cond is not strict, but addition is. (Some operators from C that
are not strict are && and .)
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Lisp: Functions, Recursion, and Lists
3.4.3 The Lisp Abstract Machine
What is an Abstract Machine?
The phrase abstract machine is generally used to refer to an idealized computing
device that can execute a specific programming language directly. Typically an abstract machine may not be fully implementable: An abstract machine may provide
infinitely many memory locations or infinite-precision arithmetic. However, as we
use the phrase in this book, an abstract machine should be sufficiently realistic to
provide useful information about the real execution of real programs on real hardware. Our goal in discussing abstract machines is to identify the mental model of the
computer that a programmer uses to write and debug programs. For this reason, there
is a tendency to refer to the abstract machine associated with a specific programming
language.
The Abstract Machine for Lisp
The abstract machine for Pure Lisp has four parts:
A Lisp expression to be evaluated.
A continuation, which is a function representing the remaining program to evaluate when done with the current expression.
An association list, commonly called the A-list in much of the literature on Lisp
and called the run-time stack in the literature on Algol-based languages. The
purpose of the A-list is to store the values of variables that may occur either in
the current expression to be evaluated or in the remaining expressions in the
program.
A heap, which is a set of cons cells (pairs stored in memory) that might be pointed
to by pointers in the A-list.
The structure of this machine is not investigated in detail. The main idea is that when
a Lisp expression is evaluated some bindings between identifiers and values may be
created. These are stored on the A-list. Some of these values may involve cons cells
that are placed in the heap. When the evaluation of an expression is completed, the
value of that expression is passed to the continuation, which represents the work to
be done by the program after that expression is evaluated.
This abstract machine is similar to a standard register machine with a stack, if
we think of the current expression as representing the program counter and the
continuation as representing the remainder of the program.
There are four main equality functions in Lisp: eq, eql, equal, and =. The function
eq tests whether its arguments are represented by the same sequence of memory
locations, and = is numeric equality. The function eql tests whether its arguments are
the same symbol or number, and equal is a recursive equality test on lists or atoms that
is implemented by use of eq and =. For simplicity, we generally use equal in sample
code.
Cons Cells
Cons cells (or dotted pairs) are the basic data structure of the Lisp abstract machine.
Cons cells have two parts, historically called the address part and the decrement part.
The words address and decrement come from the IBM 704 computer and are hardly
3.4 Innovations in the Design of Lisp
ever used today. Only the letters a and d remain in the acronyms car (for “contents
of the address register”) and cdr (for “contents of the decrement register”).
We draw cons cells as follows:
car
cdr
Cons cells
provide a simple model of memory in the machine,
are efficiently implementable, and
are not tightly linked to particular computer architecture.
We may represent an atom may be represented with a cons cell by putting a “tag”
that tells what kind of atom it is in the address part and the actual atom value in
the decrement part. For example, the letter “a” could be represented as a Lisp atom
as
atm
a
where atm indicates that the cell represents an atom and a indicates that the atom is
the letter a.
Because putting a pointer in one or both parts of a cons cell represents lists, the bit
pattern used to indicate an atom must be different from every pointer value. There
are five basic functions on cons cells, which are evaluated as follows:
atom, a function with one argument: If a value is an atom, then the word
storing the value has a special bit pattern in its address part that flags the value
as being an atom. The atom function returns true if this pattern indicates that
the function argument is an atom. (In Scheme, the function atom is written
atom?, which reminds us that the function value will be true or false.)
eq, a function with two arguments: compares two arguments for equality by
checking to see if they are stored in the same location. This is meaningful for
atoms as well as for cons cells because conceptually the Lisp compiler behaves
as if each atom (including every number) is stored once in a unique location.
cons, a function with two arguments: The expression (cons x y) is evaluated
as follows:
1.
2.
3.
4.
Allocate new cell c.
Set the address part of c to point to the value of x.
Set the decrement part of c to point to the value of y.
Return a pointer to c.
car, a function with one argument: If the argument is a cons cell c, then return
the contents of the address r egister of c. Otherwise the application of car
results in an error.
cdr, a function with one argument: If the argument is a cons cell c, then return
the contents of the decrement r egister of c. Otherwise the application of cdr
results in an error.
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Lisp: Functions, Recursion, and Lists
In drawing cons cells, we draw the contents of a cell as either a pointer to another
cell or as an atom. For our purposes, it is not important how an atom is represented
inside a cons cell. (It could be represented as either a specific bit pattern inside a cell
or as a pointer to a location that contains the representation of an atom.)
Example 3.1
We evaluate the expression (cons ’A ’B) by creating a new cons cell and then setting
the car of this cell to the atom ’A and the cdr of the cell to ’B. The expression ’(A . B)
would have the same effect, as this is the syntax for a dotted pair of atom A and atom
B. Although this dotted-pair notation was a common part of early Lisp, Scheme and
later Lisps emphasize lists over pairs.
Example 3.2
When the expression (cons (cons ’A ’B) (cons ’A ’B)) is evaluated, a new structure of
the following form is created:
A
B
A
B
The reason that there are two cons cells with the same parts is that each evaluation
of (cons ’A ’B) creates a new cell. This structure is printed as ((A . B) . (A . B)).
Example 3.3
It is also possible to write a Lisp expression that creates the following structure, which
is also printed ((A . B) . (A . B)):
A
B
One expression whose evaluation produces this structure is ((lambda (x) (cons x x))
(cons ’A ’B)). We proceed with the evaluation of this expression by first evaluating
the function argument (cons ’A ’B) to produce the cons cell drawn here, then passing
the cell to the function (lambda (x) (cons x x)) that creates the upper cell with two
pointers to the (A.B) cell. Lisp lambda expressions are described in this chapter in
Subsection 3.4.5.
Representation of Lists by Cons Cells
Because a cons cell may contain two pointers, cons cells may be used to construct
trees. Because Lisp programs often use lists, there are conventions for representing
3.4 Innovations in the Design of Lisp
lists as a certain form of trees. Specifically, the list a1 , a2 , . . . an is represented by a
cons cell whose car is a1 and whose cdr points to the cells representing list a2 , . . . an .
For the empty list, we use a pointer set to NIL.
For example, here is representation for the list (A B C), also written as (A . (B . (C .
NIL))):
atm
a
nil
atm
b
atm
c
In this illustration, atoms are shown as cons cells, with a bit pattern indicating atom
in the first part of the cell and the actual atom value in the second. For simplicity, we
often draw lists by placing an atom inside half a cons cells. For example, we could
write A in the address part of the top-left cell instead of drawing a pointer to the
cell representing atom A, and similarly for list cells pointing to atoms B and C. In the
rest of the book, we use the simpler form of drawing; the illustration here is just to
remind us that atoms as well as lists must be represented inside the machine in some
way.
3.4.4 Programs as Data
Lisp data and Lisp programs have the same syntax and internal representation. This
makes it easy to manipulate Lisp programs as data. One feature that sets Lisp apart
from many other languages is that it is possible for a program to build a data structure
that represents an expression and then evaluates the expression as if it were written
as part of the program. This is done with the function eval.
Example. We can write a Lisp function to substitute expression x for all occurrences
of y in expression z and then evaluate the resulting expression. To make the logical
structure of the substitute-and-eval function clear, we define substitute first and then
use it in substitute-and-eval. The substitute function has three arguments, exp1, var,
and exp2. The result of (substitute exp1 var exp2) is the expression obtained from
exp2 when all occurrences of var are replaced with exp1:
(define substitute (lambda (exp1 var exp2)
(cond ((atom exp2) (cond ((eq exp2 var) exp1) (true exp2)))
(true (cons (substitute exp1 var (car exp2))
(substitute exp1 var (cdr exp2)))))))
(define substitute-and-eval (lambda (x y z) (eval (substitute x y z))))
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Lisp: Functions, Recursion, and Lists
The ability to use list operations to build programs at run time and then execute
them is a distinctive feature of Lisp. You can appreciate the value of this feature if
you think about how you would write a C program to read in a few C expressions
and execute them. To do this, you would have to write some kind of C interpreter or
compiler. In Lisp, you can just read in an expression and apply the built-in function
eval to it.
3.4.5 Function Expressions
Lisp computation is based on functions and recursive calls instead of on assignment
and iterative loops. This was radically different from other languages in 1960, and is
fundamentally different from many languages in common use now.
A Lisp function expression has the form
(lambda ( parameters ) function body)
where parameters is a list of identifiers and function body is an expression. For
example, a function that places its argument in a list followed by atoms A and B may
be written as
(lambda (x) (cons x ’(A B)))
Another example is this function that, with a primitive function used for addition,
adds its two arguments:
(lambda (x y) (+ x y))
The idea of writing expressions for functions may be traced to lambda calculus, which
was developed by Alonzo Church and others, beginning in the 1930s. One fact about
lambda calculus that was known in the 1930s was that every function computable by
a Turing machine could also be written in the lambda calculus and conversely.
In lambda calculus, function expressions are written with the Greek lowercase
letter lambda (λ), rather than with the word lambda, as in Lisp. Lambda calculus
also uses different conventions about parenthesization. For example, the function
that squares its argument and adds the result to y is written as
λx.(x 2 + y)
in lambda calculus, but as
(lambda (x) (+ (square x) y))
in Lisp. In this function, x is called the formal parameter; this means that x is a
3.4 Innovations in the Design of Lisp
placeholder in the function definition that will refer to the actual parameter when
the function is applied. More specifically, consider the expression
((lambda (x) (+ (square x) y)) 4)
that applies a function to the integer 4. This expression can be evaluated only in a
context in which y already has a value. The value of this expression will be 16 plus the
value of y. This will be computed by the evaluation of (plus (square x) y) with x set to
4. The identifier y is a said to be a global variable in this function expression because
its value must be given a value by some definition outside the function expression.
More information about variables, binding, and lambda calculus may be found in
Section 4.2.
3.4.6 Recursion
Recursive functions were new at the time Lisp appeared. McCarthy, in addition to
including them in Lisp, was the main advocate for allowing recursive functions in
Algol 60. Fortran, by comparison, did not allow a function to call itself.
Members of the Algol 60 committee later wrote that they had no idea what they
were in for when they agreed to include recursive functions in Algol 60. (They might
have been unhappy for a few years, but they would probably agree now that it was a
visionary decision.)
Lisp lambda makes it possible to write anonymous functions, which are functions
that do not have a declared name. However, it is difficult to write recursive functions
with this notation. For example, suppose we want to write an expression for the
function f such that
(f x) = (cond
((eq x 0) 0)
(true (+ x (f ( - x 1))))
)
A first attempt might be
(lambda (x) (cond ((eq x 0) 0) (true (+ x (f ( - x 1))))))
However, this does not make any sense because the f inside the function expression
is not defined anywhere. McCarthy’s solution in 1960 was to add an operator called
label so that
(label f (lambda (x) (cond ((eq x 0) 0) (true (+ x (f (- x 1)))))))
defines the recursive f suggested previously. In later Lisps, it became accepted style
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Lisp: Functions, Recursion, and Lists
just to declare a function by use of the define declaration form. In particular, a
recursive function f can be declared by
(define f (lambda (x) (cond ((eq x 0) 0) (true (+ x (f (- x 1)))))))
Another notation in some versions of Lisps is defun for “define function,” which
eliminates the need for lambda:
(defun f (x) (cond ((eq x 0) 0) (true (+ x (f (- x 1))))))
McCarthy’s 1960 paper comments that the lambda notation is inadequate for expression recursive functions. This statement is false. Lambda calculus, and therefore
Lisp, is capable of expressing recursive functions without any additional operator
such as label. This was known to experts in lambda calculus in the 1930s, but apparently not known to McCarthy and his group in the 1950s. (See Subsection 4.2.3 for
an explanation of how to do it.)
3.4.7 Higher-Order Functions
The phrase higher-order function means a function that either takes a function as
an argument or returns a function as a result (or both). This use of higher-order
comes from a convention that calls a function whose arguments and results are not
functions a first-order function. A function that takes a first-order function as an
argument is called a second-order function, functions on second-order functions are
called third-order functions, and so on.
Example 3.4
If f and g are mathematical functions, say functions on the integers, then their
composition f ◦ g is the function such that for every integer x, we have
( f ◦ g)(x) = f (g(x)).
We can write composition as a Lisp function compose that takes two functions as
arguments and returns their composition:
(define compose (lambda (f g) (lambda (x) (f (g x)))))
The first lambda is used to identify the arguments to compose. The second lambda is
used to define the return value of the function, which is a function. You might enjoy
calculating the value of the expression
(compose (lambda (x) (+ x x)) (lambda (x) (∗ x x)))
3.4 Innovations in the Design of Lisp
Example 3.5
A maplist is a function that takes a function and list and applies the function to
every element in the list. The result is a list that contains all the results of function
application. Using define to define a recursive function, we can write the maplist as
follows:
(define maplist (lambda (f x)
(cond ((eq x nil) nil) (true (cons (f (car x)) (maplist f (cdr x)))))))
We cannot say whether the maplist is a second-order or third-order function, as
the elements of the list might be atoms, functions, or higher-order functions. As an
example of the use of this function, we have
(maplist square ’(1 2 3 4 5)) ⇒ (1 4 9 16 25),
where the symbol ⇒ means “evaluates to.”
Higher-order functions require more run-time support than first-order functions,
as discussed in some detail in Chapter 7.
3.4.8 Garbage Collection
In computing, garbage refers to memory locations that are not accessible to a program. More specifically, we define garbage as follows:
At a given point in the execution of a program P, a memory location m is
garbage if no completed execution of P from this point can access location m.
In other words, replacing the contents of m or making this location inaccessible
to P cannot affect any further execution of the program.
Note that this definition does not give an algorithm for finding garbage. However, if
we could find all locations that are garbage (by this definition), at some point in the
suspended execution of a program, it would be safe to deallocate these locations or
use them for some other purpose.
In Lisp, the memory locations that are accessible to a program are cons cells.
Therefore the garbage associated with a running Lisp program will be a set of cons
cells that are not needed to complete the execution of the program. Garbage collection is the process of detecting garbage during the execution of a program and
making it available for other uses. In garbage-collected languages, the run-time system receives requests for memory (as when Lisp cons cells are created) and allocates
memory from some list of available space. The list of available memory locations is
called the free list. When the run-time system detects that the available space is below
some threshold, the program may be suspended and the garbage collector invoked.
In Lisp and other garbage-collected languages, it is generally not necessary for the
program to invoke the garbage collector explicitly. (In some modern implementations, the garbage collector may run in parallel with the program. However, because
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Lisp: Functions, Recursion, and Lists
concurrent garbage collection raises some additional considerations, we will assume
that the program is suspended when the garbage collector is running.)
The idea and implementation of automatic garbage collection appear to have
originated with Lisp.
Here is an example of garbage. After the expression
(car (cons e1 e2 ))
is evaluated, any cons cells created by evaluation of e2 will typically be garbage.
However, it is not always correct to deallocate the locations used in a list after
applying car to the list. For example, consider the expression
((lambda (x) (car (cons x x))) ’(A B))
When this expression is evaluated, the function car will be applied to a cons cell
whose “a” and “d” parts both point to the same list.
Many algorithms for garbage collection have been developed over the years. Here
is a simple example called mark-and-sweep. The name comes from the fact that the
algorithm first marks all of the locations reachable from the program, then “sweeps”
up all the unmarked locations as garbage. This algorithm assumes that we can tell
which bit sequences in memory are pointers and which are atoms, and it also assumes
that there is a tag bit in each location that can be switched to 0 or 1 without destroying
the data in that location.
Mark-and-Sweep Garbage Collection
1. Set all tag bits to 0.
2. Start from each location used directly in the program. Follow all links, changing
the tag bit of each cell visited to 1.
3. Place all cells with tags still equal to 0 on the free list.
Garbage collection is a very useful feature, at least as far as programmer convenience
goes. There is some debate about the efficiency of garbage-collected languages, however. Some researchers have experimental evidence showing that garbage collection
adds of the order of 5% overhead to program execution time. However, this sort
of measurement depends heavily on program design. Some simple programs could
be written in C without the use of any user-allocated memory, but when translated
into Lisp could create many cons cells during expression evaluation and therefore
involve a lot of garbage-collection overhead. On the other hand, explicit memory
management in C and C++ (in place of garbage collection) can be cumbersome and
error prone, so that for certain programs it is highly advantageous to have automatic
garbage collection. One phenomenon that indicates the importance and difficulty of
memory management in C programs is the success of program analysis tools that are
aimed specifically at detecting memory management errors.
Example. In Lisp, we can write a function that takes a list lst and an entry x, returning
the part of the list that follows x, if any. This function, which we call select, can be
3.4 Innovations in the Design of Lisp
written as follows:
(define select (lambda (x lst)
(cond ((equal lst nil) nil)
((equal x (car lst)) (cdr lst))
(true (select x (cdr lst)))
)))
Here are two analogous C programs that have different effects on the list they are
passed. The first one leaves the list alone, returning a pointer to the cdr of the first
cell that has its car equal to x:
typedef struct cell cell;
struct cell {
cell ∗ car, ∗ cdr;
};
cell ∗ select (cell ∗ x, cell ∗ lst) {
cell ∗ ptr;
for (ptr=lst; ptr != 0; ) {
if (ptr->car = = x) return(ptr->cdr);
else ptr = ptr->cdr;
};
};
A second C program might be more appropriate if only the part of the list that follows
x will be used in the rest of the program. In this case, it makes sense to free the cells
that will no longer be used. Here is a C function that does just this:
cell ∗ select1 (cell ∗ x; cell ∗ lst) {
cell ∗ ptr, ∗ previous;
for (ptr=lst; ptr != 0; ) {
if (ptr->car = = x) return(ptr->cdr);
else previous = ptr;
ptr = ptr->cdr;
free(previous);
}
}
An advantage of Lisp garbage collection is that the programmer does not have to
decide which of these two functions to call. In Lisp, it is possible to just return a
pointer to the part of the list that you want and let the garbage collector figure out
whether you may need the rest of the list ever again. In C, on the other hand, the
programmer must decide, while traversing the list, whether this is the last time that
these cells will be referenced by the program.
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Lisp: Functions, Recursion, and Lists
Question to Ponder. It is interesting to observe that programming languages such
as Lisp, in which most computation is expressed by recursive functions and linked
data structures, provide automatic garbage collection. In contrast, simple imperative
languages such as C require the programmer to free locations that are no longer
needed. Is it just a coincidence that function-oriented languages have garbage collection and assignment-oriented languages do not? Or is there something intrinsic
to function-oriented programming that makes garbage collection more appropriate
for these languages? Part of the answer lies in the preceding example. Another part
of the answer seems to lie in the problems associated with storage management for
higher-order functions, studied in Section 7.4.
3.4.9 Pure Lisp and Side Effects
Pure Lisp expressions do not have side effects, which are visible changes in the state
of the machine as the result of evaluating an expression. However, for efficiency,
even early Lisp had expressions with side effects. Two historical functions with side
effects are rplaca and rplacd:
(rplaca x y) − replace the address field of cons cell x with y,
(rplacd x y) − replace the decrement field of cons cell x with y.
In both cases, the value of the expression is the cell that has been modified. For
example, the value of
(rplaca (cons ’A ’B) ’C)
is the cons cell with car ’C and cdr ’B produced when a new cons cell is allocated in the
evaluation of (cons ’A ’B) and then the car ’A is replaced with ’C.
With these constructs, two occurrences of the same expression may have different
values. (This is really what side effect means.) For example, consider the following
code:
(lambda (x) (cons (car x) (cons (rplaca x c) (car x)))) (cons a b)
The expression (car x) occurs twice within the function expression, but there will be
two different values in the two places this expression is evaluated. When rplaca and
rplacd are used, it is possible to create circular list structures, something that is not
possible in pure Lisp.
One situation in which rplaca and rplacd may increase efficiency is when a program
modifies a cell in the middle of a list. For example, consider the following list of four
elements:
A
B
C
D
Suppose we call this list x and we want to change the third element of list x to ’. In
pure Lisp, we cannot change any of the cells of this list, but we can define a new list
3.5 Chapter Summary: Contributions of Lisp
with elements A, B, y, D. The new list can be defined with the following expression,
where cadr x means “car of the cdr of x” and cdddr x means “cdr of cdr of cdr of x”:
(cons (car x) (cons (cadr x) (cons y (cdddr x))))
Note that evaluating this expression will involve creating new cons cells for the first
three elements of the list and, if there is no further use for them, eventually garbage
collecting the old cells used for the first three elements of x. In contrast, in impure
Lisp we can change the third cell directly by using the expression
(rplaca (cddr x) y)
If all we need is the list we obtained by replacing the third element of x with y, then
this expression gets the result we want far more efficiently. In particular, there is no
need to allocate new memory or free memory used for the original list.
Although this example may suggest that side effects lead to efficiency, the larger
picture is more complicated. In general, it is difficult to compare the efficiency of
different languages if the same problem would be best solved in very different ways.
For example, if we write a program by using pure Lisp, we might be inclined to
use different algorithms than those for impure Lisp. Once we begin to compare
the efficiency of different solutions for the same problem, we should also take into
account the amount of effort a programmer must spend writing the program, the ease
of debugging, and so on. These are complex properties of programming languages
that are difficult to quantify.
3.5 CHAPTER SUMMARY: CONTRIBUTIONS OF LISP
Lisp is an elegant programming language designed around a few simple ideas. The
language was intended for symbolic computation, as opposed to the kind of numeric
computation that was dominant in most programming outside of artificial intelligence
research in 1960. This innovative orientation can be seen in the basic data structure,
lists, and in the basic control structures, recursion and conditionals. Lists can be used
to store sequences of symbols or represent trees or other structures. Recursion is a
natural way to proceed through lists that may contain atomic data or other lists.
Three important aspects of programming language design contributed to the
success of Lisp: a specific motivation application, an unambiguous program execution model, and attention to theoretical considerations. Among the main theoretical
considerations, Lisp was designed with concern for the mathematical class of partial recursive functions. Lisp syntax for function expressions is based on lambda
calculus.
The following contributions are some that are important to the field of programming languages:
Recursive functions. Lisp programming is based on functions and recursion instead of assignment and while loops. Lisp introduces recursive functions and
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Lisp: Functions, Recursion, and Lists
supports functions with function arguments and functions that return functions
as results.
Lists. The basic data structure in early Lisp was the cons cell. The main use of cons
cells in modern forms of Lisp is for building lists, and lists are used for everything.
The list data structure is extremely useful. In addition, the Lisp presentation
of memory as an unlimited supply of cons cells provides a more useful abstract
machine for nonnumerical programming than do arrays, which were primary data
structures in other languages of the early days of computing.
Programs as data. This is still a revolutionary concept 40 years after its introduction in Lisp. In Lisp, a program can build the list representation of a function or
other forms of expression and then use the eval function to evaluate the expression.
Garbage collection. Lisp was the first language to manage memory for the programmer automatically. Garbage collection is a useful feature that eliminates the
program error of using a memory location after freeing it.
In the years since 1960, Lisp has continued to be successful for symbolic mathematics
and exploratory programming, as in AI research projects and other applications of
symbolic computation or logical reasoning. It has also been widely used for teaching
because of the simplicity of the language.
EXERCISES
3.1 Cons Cell Representations
(a) Draw the list structure created by evaluating (cons ’A (cons
’B ’C)).
(b) Write a pure Lisp expression that will result in this representation, with no sharing of the (B . C) cell. Explain why
your expression produces this structure.
(c) Write a pure Lisp expression that will result in this representation, with sharing of the (B . C) cell. Explain why your
expression produces this structure.
B C
A
B C
A
B C
While writing your expressions, use only these Lisp constructs: lambda abstraction,
function application, the atoms A B C, and the basic list functions (cons, car, cdr,
atom, eq). Assume a simple-minded Lisp implementation that does not try to do
any clever detection of common subexpressions or advanced memory allocation
optimizations.
3.2 Conditional Expressions in Lisp
The semantics of the Lisp conditional expression
(cond ( p1 e1 ) . . . ( pn en ))
is explained in the text. This expression does not have a value if p1 , . . . , pk are false
and pk+1 does not have a value, regardless of the values of pk+2 , . . . , pn .
Imagine you are an MIT student in 1958 and you and McCarthy are considering
alternative interpretations for conditionals in Lisp:
(a) Suppose McCarthy suggests that the value of (cond ( p1 e1 ) . . . ( pn en )) should be
Exercises
the value of ek if pk is true and if, for every i < k, the value of expression pi is
either false or undefined. Is it possible to implement this interpretation? Why
or why not? (Hint: Remember the halting problem.)
(b) Another design for conditional might allow any of several values if more than
one of the guards ( p1 , . . . , pn ) is true. More specifically (and be sure to read
carefully), suppose someone suggests the following meaning for conditional:
i. The conditional’s value is undefined if none of the pk is true.
ii. If some pk are true, then the implementation must return the value of e j for
some j with p j true. However, it need not be the first such e j .
Note that in (cond (a b) (c d) (e f )), for example, if a runs forever, c evaluates to
true, and e halts in error, the value of this expression should be the value of d,
if it has one. Briefly describe a way to implement conditional so that properties
i and ii are true. You need to write only two or three sentences to explain the
main idea.
(c) Under the original interpretation, the function
(defun odd (x) (cond ((eq x 0) nil)
((eq x 1) t)
((> x 0) (odd (- x 2)))
(t (odd (+ x 2)))))
would give us t for odd numbers and nil for even numbers. Modify this expression
so that it would always give us t for odd numbers and nil for even numbers under
the alternative interpretation described in part (b).
(d) The normal implementation of Boolean or is designed not to evaluate a subexpression unless it is necesary. This is called the short-circuiting or, and it may be
defined as follows:

true
if e1 = true



true
if e1 = false and e2 = true
.
Scor(e1 , e2 ) =
false
if e1 = e2 = false



undefined otherwise
It allows e2 to be undefined if e1 is true.
The parallel or is a related construct that gives an answer whenever possible (possibly doing some unnecessary subexpression evaluation). It is defined
similarly:

true
if e1 = true



true
if e2 = true
.
Por(e1 , e2 ) =
false
if e1 = e2 = false



undefined otherwise
It allows e2 to be undefined if e1 is true and also allows e1 to be undefined if e2
is true. You may assume that e1 and e2 do not have side effects.
Of the original interpretation, the interpretation in part (a), and the interpretation in part (b), which ones would allow us to implement Scor most
easily? What about Por? Which interpretation would make implementations of
short-circuiting or difficult? Which interpretation would make implementation
of parallel or difficult? Why?
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3.3 Detecting Errors
Evaluation of a Lisp expression can either terminate normally (and return a value),
terminate abnormally with an error, or run forever. Some examples of expressions
that terminate with an error are (/ 3 0), division by 0; (car ’a), taking the car of an
atom; and (+ 3 “a”), adding a string to a number. The Lisp system detects these errors,
terminates evaluation, and prints a message to the screen. Your boss wants to handle
errors in Lisp programs without terminating the computation, but doesn’t know how,
so your boss asks you to. . .
(a) . . . implement a Lisp construct (error? E) that detects whether an expression E
will cause an error. More specifically, your boss wants the evaluation of (error?
E) to halt with the value true if the evaluation of E terminates in error and to halt
with the value false otherwise. Explain why it is not possible to implement the
error? construct as part of the Lisp environment.
(b) . . . implement a Lisp construct (guarded E) that either executes E and returns its
value, or, if E halts with an error, returns 0 without performing any side effects.
This could be used to try to evaluate E and, if an error occurs, just use 0 instead.
For example,
(+ (guarded E) E’)
; just E’ if E halts with an error; E+E’ otherwise
will have the value of E’ if the evaluation of E halts in error and the value of
E + E’ otherwise. How might you implement the guarded construct? What difficulties might you encounter? Note that, unlike that of (error? E), evaluation of
(guarded E) does not need to halt if evaluation of E does not halt.
3.4 Lisp and Higher-Order Functions
Lisp functions compose, mapcar, and maplist are defined as follows, with #t written for
true and () for the empty list. Text beginning with ;; and continuing to the end of a
line is a comment.
(define compose
(lambda (f g)
(lambda (x) (f (g x)))))
(define mapcar
(lambda (f xs)
(cond
((eq? xs ()) ()) ;; If the list is empty, return the empty list
(#t
;; Otherwise, apply f to the first element . . .
(cons (f (car xs))
;; and map f on the rest of the list
(mapcar f (cdr xs))
)))))
(define maplist
(lambda (f xs)
(cond
((eq? xs ()) ()) ;; If the list is empty, return the empty list
(#t
;; Otherwise, apply f to the list . . .
(cons (f xs)
;; and map f on the rest of the list
Exercises
(maplist f (cdr xs))
)))))
The difference between maplist and mapcar is that maplist applies f to every sublist,
whereas mapcar applies f to every element. (The two function expressions differ in
only the sixth line.) For example, if inc is a function that adds one to any number,
then
(mapcar inc ’(1 2 3 4)) = (2 3 4 5)
whereas
(maplist (lambda (xs) (mapcar inc xs)) ’(1 2 3 4))
= ((2 3 4 5) (3 4 5) (4 5) (5))
However, you can almost get mapcar from maplist by composing with the car function.
In particular, note that
(mapcar f ’(1 2 3 4))
= ((f (car (1 2 3 4))) (f (car (2 3 4))) (f (car (3 4))) (f (car (4))))
Write a version of compose that lets us define mapcar from maplist. More specifically,
write a definition of compose2 so that
((compose2 maplist car) f xs) = (mapcar f xs)
for any function f and list xs.
(a) Fill in the missing code in the following definition. The correct answer is short
and fits here easily. You may also want to answer parts (b) and (c) first.
(define compose2
(lambda (g h)
(lambda (f xs)
(g (lambda (xs) (
)) xs )
)))
(b) When (compose2 maplist car) is evaluated, the result is a function defined by
(lambda (f xs) (g . . . )) above, with
i. which function replacing g?
ii. and which function replacing h?
(c) We could also write the subexpression (lambda (xs) ( . . . )) as (compose (. . . ) (. . . ))
for two functions. Which two functions are these? (Write them in the correct
order.)
3.5 Definition of Garbage
This question asks you to think about garbage collection in Lisp and compare our
definition of garbage in the text to the one given in McCarthy’s 1960 paper on Lisp.
McCarthy’s definition is written for Lisp specifically, whereas our definition is stated
generally for any programming language. Answer the question by comparing the
definitions as they apply to Lisp only. Here are the two definitions.
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Lisp: Functions, Recursion, and Lists
Garbage, our definition: At a given point in the execution of a program P, a
memory location m is garbage if no continued execution of P from this point can
access location m.
Garbage, McCarthy’s definition: “Each register that is accessible to the program
is accessible because it can be reached from one or more of the base registers
by a chain of car and cdr operations. When the contents of a base register are
changed, it may happen that the register to which the base register formerly
pointed cannot be reached by a car–cdr chain from any base register. Such a
register may be considered abandoned by the program because its contents can
no longer be found by any possible program.”
(a) If a memory location is garbage according to our definition, is it necessarily
garbage according to McCarthy’s definition? Explain why or why not.
(b) If a location is garbage according to McCarthy’s definition, is it garbage by our
definition? Explain why or why not.
(c) There are garbage collectors that collect everything that is garbage according
to McCarthy’s definition. Would it be possible to write a garbage collector to
collect everything that is garbage according to our definition? Explain why or
why not.
3.6 Reference Counting
This question is about a possible implementation of garbage collection for Lisp. Both
impure and pure Lisp have lambda abstraction, function application, and elementary
functions atom, eq, car, cdr, and cons. Impure Lisp also has rplaca, rplacd, and other
functions that have side effects on memory cells.
Reference counting is a simple garbage-collection scheme that associates a reference count with each datum in memory. When memory is allocated, the associated
reference count is set to 0. When a pointer is set to point to a location, the count
for that location is incremented. If a pointer to a location is reset or destroyed, the
count for the location is decremented. Consequently, the reference count always
tells how many pointers there are to a given datum. When a count reaches 0, the
datum is considered garbage and is returned to the free-storage list. For example,
after evaluation of (cdr (cons (cons ’A ’B) (cons ’C ’D))), the cell created for (cons ’A ’B)
is garbage, but the cell for (cons ’C ’D) is not.
(a) Describe how reference counting could be used for garbage collection in evaluating the following expression:
(car (cdr (cons (cons a b) (cons c d))))
where a, b, c, and d are previously defined names for cells. Assume that the
reference counts for a, b, c, and d are initially set to some numbers greater than
0, so that these do not become garbage. Assume that the result of the entire
expression is not garbage. How many of the three cons cells generated by the
evaluation of this expression can be returned to the free-storage list?
(b) The “impure” Lisp function rplaca takes as arguments a cons cell c and a value
v and modifies c ’s address field to point to v . Note that this operation does not
produce a new cons cell; it modifies the one it receives as an argument. The
function rplacd performs the same function with respect the decrement portion
of its argument cons cell.
Exercises
Lisp programs that use rplaca or rplacd may create memory structures that
cannot be garbage collected properly by reference counting. Describe a configuration of cons cells that can be created by use of operations of pure Lisp and
rplaca and rplacd. Explain why the reference-counting algorithm deos not work
properly on this structure.
3.7 Regions and Memory Management
There are a wide variety of algorithms to chose from when implementing garbage
collection for a specific language. In this problem, we examine one algorithm for
finding garbage in pure Lisp (Lisp without side effects) based on the concept of
regions. Generally speaking, a region is a section of the program text. To make things
simple, we consider each function as a separate region. Region-based collection
reclaims garbage each time program execution leaves a region. Because we are
treating functions as regions in this problems, our version of region-based collection
will try to find garbage each time a program returns from a function call.
(a) Here is a simple idea for region-based garbage collection:
When a function exits, free all the memory that was allocated during execution of the function.
However, this is not correct as some memory locations that are freed may still be
accessible to the program. Explain the flaw by describing a program that could
possibly access a previously freed piece of memory. You do not need to write
more than four or five sentences; just explain the aspects of an example program
that are relevant to the question.
(b) Fix the method in part (a) to work correctly. It is not necessary for your method
to find all garbage, but the locations that are freed should really be garbage.
Your answer should be in the following form:
When a function exits, free all memory allocated by the function except. . . .
Justify your answer. (Hint: Your statement should not be more than a sentence
or two. Your justification should be a short paragraph.)
(c) Now assume that you have an correctly functioning region-based garbage collector. Does your region-based collector have any advantages or disadvantages
over a simple mark-and-sweep collector?
(d) Could a region-based collector like the one described in this problem work
for impure Lisp? If you think the problem is more complicated for impure
Lisp, briefly explain why. You may consider the problem for C instead of for
impure Lisp if you like, but do not give an answer that depends on specific
properties of C such as pointer arithmetic. The point of this question is to explore the relationship between side effects and a simple form of region-based
collection.
3.8 Concurrency in Lisp
The concept of future was popularized by R. Halstead’s work on the language Multilisp for concurrent Lisp programming. Operationally, a future consists of a location
in memory (part of a cons cell) and a process that is intended to place a value in this
location at some time “in the future.” More specifically, the evaluation of (future e)
proceeds as follows:
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Lisp: Functions, Recursion, and Lists
(i) The location that will contain the value of (future e) is identified (if the value
is going to go into an existing cons cell) or created if needed.
(ii) A process is created to evaluate e.
(iii) When the process evaluating e completes, the value of e is placed in the
location .
(iv) The process that invoked (future e) continues in parallel with the new process.
If the originating process tries to read the contents of location while it is still
empty, then the process blocks until the location has been filled with the value
of e.
Other than this construct, all other operations in this problem are defined as in pure
Lisp. For example, if expression e evaluates to the list (1 2 3), then the expression
(cons ’a (future e))
produces a list whose first element is the atom ’a and whose tail becomes (1 2 3)
when the process evaluating e terminates. The value of the future construct is that
the program can operate on the car of this list while the value of the cdr is being
computed in parallel. However, if the program tries to examine the cdr of the list
before the value has been placed in the empty location, then the computation will
block (wait) until the data is available.
(a) Assuming an unbounded number of processors, how much time would you expect the evaluation of the following fib function to take on positive-integer
argument n?
(defun fib (n)
(cond ( (eq n 0) 1)
((eq n 1) 1)
(T (plus (future (fib (minus n 1)))
(future (fib (minus n 2)))))))
We are interested only in time up to a multiplicative constant; you may use “big
Oh” notation if you wish. If two instructions are done at the same time by two
processors, count that as one unit of time.
(b) At first glance, we might expect that two expressions
( ... e...)
( . . . (future e) . . . )
which differ only because an occurrence of a subexpression e is replaced with
(future e), would be equivalent. However, there are some circumstances in which
the result of evaluating one might differ from the other. More specifically, side
effects may cause problems. To demonstrate this, write an expression of the
form (. . . e . . . ) so that when the e is changed to (future e), the expression’s value
or behavior might be different because of side effects, and explain why. Do
not be concerned with the efficiency of either computation or the degree of
parallelism.
(c) Side effects are not the only cause for different evaluation results. Write a
pure Lisp expression of the form (. . . e’ . . . ) so that when the e’ is changed to
(future e’), the expression’s value or behavior might be different, and explain
why.
(d) Suppose you are part of a language design team that has adopted futures as
an approach to concurrency. The head of your team suggests an error-handling
Exercises
feature called a try block. The syntactic form of a try block is
(try e
(error-1 handler-1)
(error-2 handler-2)
...
(error-n handler-n))
This construct would have the following characteristics:
i. Errors are programmer defined and occur when an expression of the form
(raise error-i) is evaluated inside e, the main expression of the try block.
ii. If no errors occur, then (try e (error-1 handler-1) . . . ) is equivalent to e.
iii. If the error named error-i occurs during the evaluation of e, the rest of the
computation of e is aborted, the expression handler-i is evaluated, and this
becomes the value of the try block.
The other members of the team think this is a great idea and, claiming that it is
a completely straightforward construct, ask you to go ahead and implement it.
You think the construct might raise some tricky issues. Name two problems or
important interactions between error handling and concurrency that you think
need to be considered. Give short code examples or sketches to illustrate your
point(s). (Note: You are not being asked to solve any problems associated with
futures and try blocks; just identify the issues.) Assume for this part that you
are using pure Lisp (no side effects).
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4
Fundamentals
In this chapter some background is provided on programming language implementation through brief discussions of syntax, parsing, and the steps used in conventional
compilers. We also look at two foundational frameworks that are useful in programming language analysis and design: lambda calculus and denotational semantics.
Lambda calculus is a good framework for defining syntactic concepts common to
many programming languages and for studying symbolic evaluation. Denotational
semantics shows that, in principle, programs can be reduced to functions.
A number of other theoretical frameworks are useful in the design and analysis
of programming languages. These range from computability theory, which provides
some insight into the power and limitations of programs, to type theory, which includes aspects of both syntax and semantics of programming languages. In spite of
many years of theoretical research, the current programming language theory still
does not provide answers to some important foundational questions. For example, we
do not have a good mathematical theory that includes higher-order functions, state
transformations, and concurrency. Nonetheless, theoretical frameworks have had an
impact on the design of programming languages and can be used to identify problem
areas in programming languages. To compare one aspect of theory and practice, we
compare functional and imperative languages in Section 4.4.
4.1 COMPILERS AND SYNTAX
A program is a description of a dynamic process. The text of a program itself is
called its syntax; the things a program does comprise its semantics. The function of a
programming language implementation is to transform program syntax into machine
instructions that can be executed to cause the correct sequence of actions to occur.
4.1.1 Structure of a Simple Compiler
Programming languages that are convenient for people to use are built around concepts and abstractions that may not correspond directly to features of the underlying
machine. For this reason, a program must be translated into the basic instruction
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4.1 Compilers and Syntax
JOHN BACKUS
An early pioneer, John Backus became a computer programmer at IBM
in 1950. In the 1950s, Backus developed Fortran, the first high-level
computer language, which became commercially available in 1957. The
language is still widely used for numerical and scientific programming.
In 1959, John Backus invented Backus naur form (BNF), the standard
notation for defining the syntax of a programming language. In later
years, he became an advocate of pure functional programming, devoting
his 1977 ACM Turing Award lecture to this topic.
I met John Backus through IFIP WG 2.8, a working group of the
International Federation of Information Processing on functional programming. Backus continued to work on functional programming at
IBM Almaden through the 1980s, although his group was disbanded after his retirement. A mild-mannered and unpretentious individual, here
is a quote that gives some sense of his independent, pioneering spirit:
“I really didn’t know what the hell I wanted to do with my life. I
decided that what I wanted was a good hi fi set because I liked music.
In those days, they didn’t really exist so I went to a radio technicians’
school. I had a very nice teacher – the first good teacher I ever had – and
he asked me to cooperate with him and compute the characteristics of
some circuits for a magazine.”
“I remember doing relatively simple calculations to get a few points
on a curve for an amplifier. It was laborious and tedious and horrible,
but it got me interested in math. The fact that it had an application –
that interested me.”
set of the machine before it can be executed. This can be done by a compiler, which
translates the entire program into machine code before the program is run, or an interpreter, which combines translation and program execution. We discuss programming
language implementation by using compilers, as this makes it easier to separate the
main issues and to discuss them in order.
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50
Fundamentals
The main function of a compiler is illustrated in this simple diagram:
source
program
target
program
compiler
Given a program in some source language, the compiler produces a program in a
target language, which is usually the instruction set, or machine language, of some
machine.
Most compilers are structured as a series of phases, with each phase performing
one step in the translation of source program to target program. A typical compiler
might consist of the phases shown in the following diagram:
Source Program
Lexical Analyzer
Syntax Analyzer
Semantic Analyzer
Intermediate Code
Generator
Code Optimizer
Code Generator
Target Program
Each of these phases is discussed briefly. Our goal with this book is only to
understand the parts of a compiler so that we can discuss how different programming
language features might be implemented. We do not discuss how to build a compiler.
That is the subject of many books on compiler construction, such as Compilers:
Principles, Techniques and Tools by Aho, Sethi, and Ullman (Addison-Wesley, 1986),
and Modern Compiler Implementation in Java/ML/C by Appel (Cambridge Univ.
Press, 1998).
Lexical Analysis
The input symbols are scanned and grouped into meaningful units called tokens.
For example, lexical analysis of the expression temp := x+1, which uses Algol-style
notation := for assignment, would divide this sequence of symbols into five tokens:
the identifier temp, the assignment “symbol” :=, the variable x, the addition symbol
+, and the number 1. Lexical analysis can distinguish numbers from identifiers. However, because lexical analysis is based on a single left-to-right (and top-to-bottom)
scan, lexical analysis does not distinguish between identifiers that are names of variables and identifiers that are names of constants. Because variables and constants
are declared differently, variables and constants are distinguished in the semantic
analysis phase.
4.1 Compilers and Syntax
Syntax Analysis
In this phase, tokens are grouped into syntactic units such as expressions, statements,
and declarations that must conform to the grammatical rules of the programming
language. The action performed during this phase, called parsing, is described in
Subsection 4.1.2. The purpose of parsing is to produce a data structure called a
parse tree, which represents the syntactic structure of the program in a way that is
convenient for subsequent phases of the compiler. If a program does not meet the
syntactic requirements to be a well-formed program, then the parsing phase will
produce an error message and terminate the compiler.
Semantic Analysis
In this phase of a compiler, rules and procedures that depend on the context surrounding an expression are applied. For example, returning to our sample expression temp := x+1, we find that it is necessary to make sure that the types match. If
this assignment occurs in a language in which integers are automatically converted
to floats as needed, then there are several ways that types could be associated with
parts of this expression. In standard semantic analysis, the types of temp and x would
be determined from the declarations of these identifiers. If these are both integers,
then the number 1 could be marked as an integer and + marked as integer addition,
and the expression would be considered correct. If one of the identifiers, say x, is a
float, then the number 1 would be marked as a float and + marked as a floating-point
addition. Depending on whether temp is a float or an integer, it might also be necessary to insert a conversion around the subexpression x+1. The output of this phase is
an augmented parse tree that represents the syntactic structure of the program and
includes additional information such as the types of identifiers and the place in the
program where each identifier is declared.
Although the phase following parsing is commonly called semantic analysis, this
use of the word semantic is different from the standard use of the term for meaning.
Some compiler writers use the word semantic because this phase relies on context
information, and the kind of grammar used for syntactic analysis does not capture
context information. However, in the rest of this book, the word semantics is used
to refer to how a program executes, not the essentially syntactic properties that arise
in the third phase of a compiler.
Intermediate Code Generation
Although it might be possible to generate a target program from the results of syntactic and semantic analysis, it is difficult to generate efficient code in one phase.
Therefore, many compilers first produce an intermediate form of code and then optimize this code to produce a more efficient target program.
Because the last phase of the compiler can translate one set of instructions to
another, the intermediate code does not need to be written with the actual instruction set of the target machine. It is important to use an intermediate representation
that is easy to produce and easy to translate into the target language. The intermediate representation can be some form of generic low-level code that has properties
common to several computers. When a single generic intermediate representation is
used, it is possible to use essentially the same compiler to generate target programs
for several different machines.
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Fundamentals
Code Optimization
There are a variety of techniques that may be used to improve the efficiency of a
program. These techniques are usually applied to the intermediate representation.
If several optimization techniques are written as transformations of the intermediate representation, then these techniques can be applied over and over until some
termination condition is reached.
The following list describes some standard optimizations:
Common Subexpression Elimination: If a program calculates the same value more
than once and the compiler can detect this, then it may be possible to transform
the program so that the value is calculated only once and stored for subsequent
use.
Copy Propagation: If a program contains an assignment such as x=y, then it may
be possible to change subsequent statements to refer to y instead of to x and to
eliminate the assignment.
Dead-Code Elimination: If some sequence of instructions can never be reached,
then it can be eliminated from the program.
Loop Optimizations: There are several techniques that can be applied to remove
instructions from loops. For example, if some expression appears inside a loop
but has the same value on each pass through the loop, then the expression can be
moved outside the loop.
In-Lining Function Calls: If a program calls function f, it is possible to substitute
the code for f into the place where f is called. This makes the target program more
efficient, as the instructions associated with calling a function can be eliminated,
but it also increases the size of the program. The most important consequence
of in-lining function calls is usually that they allow other optimizations to be
performed by removing jumps from the code.
Code Generation
The final phase of a standard compiler is to convert the intermediate code into
a target machine code. This involves choosing a memory location, a register, or
both, for each variable that appears in the program. There are a variety of register
allocation algorithms that try to reuse registers efficiently. This is important because
many machines have a fixed number of registers, and operations on registers are
more efficient than transferring data into and out of memory.
4.1.2 Grammars and Parse Trees
We use grammars to describe various languages in this book. Although we usually
are not too concerned about the pragmatics of parsing, we take a brief look in this
subsection at the problem of producing a parse tree from a sequence of tokens.
Grammars
Grammars provide a convenient method for defining infinite sets of expressions. In
addition, the structure imposed by a grammar gives us a systematic way of processing
expressions.
A grammar consists of a start symbol, a set of nonterminals, a set of terminals,
and a set of productions. The nonterminals are symbols that are used to write out
4.1 Compilers and Syntax
the grammar, and the terminals are symbols that appear in the language generated
by the grammar. In books on automata theory and related subjects, the productions
of a grammar are written in the form s → tu, with an arrow, meaning that in a string
containing the symbol s, we can replace s with the symbols tu. However, here we use
a more compact notation, commonly referred to as BNF.
The main ideas are illustrated by example. A simple language of numeric expressions is defined by the following grammar:
e ::= n | e+e | e-e
n ::= d | nd
d ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
where e is the start symbol, symbols e, n, and d are nonterminals, and 0, 1, 2, 3, 4, 5,
6, 7, 8, 9, +, and - are the terminals. The language defined by this grammar consists of
all the sequences of terminals that we can produce by starting with the start symbol e
and by replacing nonterminals according to the preceding productions. For example,
the first preceding production means that we can replace an occurrence of e with
the symbol n, the three symbols e+e, or the three symbols e-e. The process can be
repeated with any of the preceding three lines.
Some expressions in the language given by this grammar are
0, 1 + 3 + 5, 2 + 4 - 6 - 8
Sequences of symbols that contain nonterminals, such as
e, e + e, e + 6 - e
are not expressions in the language given by the grammar. The purpose of nonterminals is to keep track of the form of an expression as it is being formed. All
nonterminals must be replaced with terminals to produce a well-formed expression
of the language.
Derivations
A sequence of replacement steps resulting in a string of terminals is called a
derivation.
Here are two derivations in this grammar, the first given in full and the second
with a few missing steps that can be filled in by the reader (be sure you understand
how!):
e → n → nd → dd → 2d → 25
e → e - e → e - e+e → · · · → n-n+n → · · · → 10-15+12
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Fundamentals
Parse Trees and Ambiguity
It is often convenient to represent a derivation by a tree. This tree, called the parse
tree of a derivation, or derivation tree, is constructed with the start symbol as the root
of the tree. If a step in the derivation is to replace s with x1 , . . . , xn , then the children
of s in the tree will be nodes labeled x1 , . . . , xn .
The parse tree for the derivation of 10-15+12 in the preceding subsection has some
useful structure. Specifically, because the first step yields e-e, the parse tree has the
form
e
e
−
e
10
e
+
15
e
12
where we have contracted the subtrees for each two-digit number to a single node.
This tree is different from
e
e
10
e
+
e
−
e
12
15
which is another parse tree for the same expression. An important fact about parse
trees is that each corresponds to a unique parenthesization of the expression. Specifically, the first tree corresponds to 10-(15+12) whereas the second corresponds to
(10-15)+12. As this example illustrates, the value of an expression may depend on
how it is parsed or parenthesized.
A grammar is ambiguous if some expression has more than one parse tree. If
every expression has at most one parse tree, the grammar is unambiguous.
Example 4.1
There is an interesting ambiguity involving if-then-else. This can be illustrated by the
following simple grammar:
s ::= v := e | s;s | if b then s | if b then s else s
v ::= x | y | z
e ::= v | 0 | 1 | 2 | 3 | 4
b ::= e=e
where s is the start symbol, s, v, e, and b are nonterminals, and the other symbols
are terminals. The letters s, v, e, and b stand for statement, variable, expression, and
Boolean test, respectively. We call the expressions of the language generated by this
4.1 Compilers and Syntax
grammar statements. Here is an example of a well-formed statement and one of its
parse trees:
s
s
v
x
:=
s
s
e
1
v
y
:=
s
e
if
s
b
then
=
e
v
y
y
2
e
x
:=
e
3
x := 1; y := 2; if x=y then y := 3
This statement also has another parse tree, which we obtain by putting two assignments to the left of the root and the if-then statement to the right. However, the
difference between these two parse trees will not affect the behavior of code generated by an ordinary compiler. The reason is that s1 ;s2 is normally compiled to the
code for s1 followed by the code for s2 . As a result, the same code would be generated
whether we consider s1 ;s2 ;s3 as (s1 ;s2 );s3 or s1 ;(s2 ;s3 ).
A more complicated situation arises when if-then is combined with if-then-else in
the following way:
if b1 then if b2 then s1 else s2
What should happen if b1 is true and b2 is false? Should s2 be executed or not? As
you can see, this depends on how the statement is parsed. A grammar that allows this
combination of conditionals is ambiguous, with two possible meanings for statements
of this form.
4.1.3 Parsing and Precedence
Parsing is the process of constructing parse trees for sequences of symbols. Suppose
we define a language L by writing out a grammar G. Then, given a sequence of
symbols s, we would like to determine if s is in the language L. If so, then we would
like to compile or interpret s, and for this purpose we would like to find a parse tree
for s. An algorithm that decides whether s is in L, and constructs a parse tree if it is,
is called a parsing algorithm for G.
There are many methods for building parsing algorithms from grammars. Many
of these work for only particular forms of grammars. Because parsing is an important
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Fundamentals
part of compiling programming languages, parsing is usually covered in courses and
textbooks on compilers. For most programming languages you might consider, it is
either straightforward to parse the language or there are some changes in syntax that
do not change the structure of the language very much but make it possible to parse
the language efficiently.
Two issues we consider briefly are the syntactic conventions of precedence and
right or left associativity. These are illustrated briefly in the following example.
Example 4.2
A programming language designer might decide that expressions should include
addition, subtraction, and multiplication and write the following grammar:
e ::= 0 | 1 | e+e | e-e | e∗e
This grammar is ambiguous, as many expressions have more than one parse tree.
For expressions such as 1-1+1, the value of the expression will depend on the way it
is parsed. One solution to this problem is to require complete parenthesization. In
other words, we could change the grammar to
e ::= 0 | 1 | (e+e) | (e-e) | (e∗e)
so that it is no longer ambiguous. However, as you know, it can be awkward to write
a lot of parentheses. In addition, for many expressions, such as 1+2+3+4, the value of
the expression does not depend on the way it is parsed. Therefore, it is unnecessarily
cumbersome to require parentheses for every operation.
The standard solution to this problem is to adopt parsing conventions that specify
a single parse tree for every expression. These are called precedence and associativity. For this specific grammar, a natural precedence convention is that multiplication
(∗) has a higher precedence than addition (+) and subtraction (−). We incorporate
precedence into parsing by treating an unparenthesized expression e op1 e op2 e as
if parentheses are inserted around the operator of higher precedence. With this rule
in effect, the expression 5∗4-3 will be parsed as if it were written as (5∗4)-3. This
coincides with the way that most of us would ordinarily think about the expression
5∗4-3. Because there is no standard way that most readers would parse 1+1-1, we
might give addition and subtraction equal precedence. In this case, a compiler could
issue an error message requiring the programmer to parenthesize 1+1-1. Alternatively, an expression like this could be disambiguated by use of an additional
convention.
Associativity comes into play when two operators of equal precedence appear
next to each other. Under left associativity, an expression e op1 e op2 e would be
parsed as (e op1 e) op2 e, if the two operators have equal precedence. If we adopted a right-associativity convention instead, e op1 e op2 e would be parsed as e op1
(e op2 e).
4.2 Lambda Calculus
Expression
Precedence
Left Associativity
Right Associativity
5∗4-3
1+1-1
2+3-4∗5+2
(5∗4)-3
(no change)
(no change)
(no change)
(1+1)-1
((2+3)-(4∗ 5))+2
1+(1-1)
2+(3-((4∗ 5)+2))
2+3-(4∗5)+2
4.2 LAMBDA CALCULUS
Lambda calculus is a mathematical system that illustrates some important programming language concepts in a simple, pure form. Traditional lambda calculus has
three main parts: a notation for defining functions, a proof system for proving equations between expressions, and a set of calculation rules called reduction. The first
word in the name lambda calculus comes from the use of the Greek letter lambda
(λ) in function expressions. (There is no significance to the letter λ.) The second
word comes from the way that reduction may be used to calculate the result of
applying a function to one or more arguments. This calculation is a form of symbolic evaluation of expressions. Lambda calculus provides a convenient notation
for describing programming languages and may be regarded as the basis for many
language constructs. In particular, lambda calculus provides fundamental forms of
parameterization (by means of function expressions) and binding (by means of declarations). These are basic concepts that are common to almost all modern programming languages. It is therefore useful to become familiar enough with lambda
calculus to regard expressions in your favorite programming language as essentially
a form of lambda expression. For simplicity, the untyped lambda calculus is discussed; there are also typed versions of lambda calculus. In typed lambda calculus,
there are additional type-checking constraints that rule out certain forms of expressions. However, the basic concepts and calculation rules remain essentially the
same.
4.2.1 Functions and Function Expressions
Intuitively, a function is a rule for determining a value from an argument. This view
of functions is used informally in most mathematics. (See Subsection 2.1.2 for a discussion of functions in mathematics.) Some examples of functions studied in mathematics are
f (x) = x 2 + 3,
g(x + y) = x 2 + y2 .
In the simplest, pure form of lambda calculus, there are no domain-specific operators
such as addition and exponentiation, only function definition and application. This
allows us to write functions such as
h(x) = f (g(x))
because h is defined solely in terms of function application and other functions that
we assume are already defined. It is possible to add operations such as addition
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Fundamentals
and exponentiation to pure lambda calculus. Although purists stick to pure lambda
calculus without addition and multiplication, we will use these operations in examples
as this makes the functions we define more familiar.
The main constructs of lambda calculus are lambda abstraction and application.
We use lambda abstraction to write functions: If M is some expression, then λx.M is
the function we get by treating M as a function of the variable x. For example,
λx.x
is a lambda abstraction defining the identity function, the function whose value at x
is x. A more familiar way of defining the identity function is by writing
I(x) = x.
However, this way of defining a function forces us to make up a name for every
function we want. Lambda calculus lets us write anonymous functions and use them
inside larger expressions.
In lambda notation, it is traditional to write function application just by putting
a function expression in front of one or more arguments; parentheses are optional. For example, we can apply the identity function to the expression M by
writing
(λx.x)M.
The value of this application is the identity function, applied to M, that just ends up
being M. Thus, we have
(λx.x)M = M.
Part of lambda calculus is a set of rules for deducing equations such as this. Another
example of a lambda expression is
λ f.λg.λx. f (g x).
Given functions f and g, this function produces the composition λx. f (g x) of f
and g.
We can extend pure lambda calculus by adding a variety of other constructs. We
call an extension of pure lambda calculus with extra operations an applied lambda
calculus. A basic idea underlying the relation between lambda calculus and computer
science is the slogan
Programming language = applied λ-calculus
= pure λ-calculus + additional data types.
This works even for programming languages with side effects, as the way a program
depends on the state of the machine can be represented by explicit data structures for
the machine state. This is one of the basic ideas behind Scott–Strachey denotational
semantics, as discussed in Section 4.3.
4.2 Lambda Calculus
4.2.2 Lambda Expressions
Syntax of Expressions
The syntax of untyped lambda expressions may be defined with a BNF grammar.
We assume we have some infinite set V of variables and use x, y, z, . . . , to stand for
arbitrary variables. The grammar for lambda expressions is
M ::= x|MM | λx.M
where x may be any variable (element of V). An expression of the form M1 M2 is
called an application, and an expression of the form λx.M is called a lambda abstraction. Intuitively, a variable x refers to some function, the particular function being
determined by context; M1 M2 is the application of function M1 to argument M2 ; and
λx.M is the function that, given argument x, returns the value M. In the lambda
calculus literature, it is common to refer to the expressions of lambda calculus as
lambda terms.
Here are some example lambda terms:
λx.x
λx.( f (gx))
(λx.x)5
a lambda abstraction called the identity function
another lambda abstraction
an application
There are a number of syntactic conventions that are generally convenient for lambda
calculus experts, but they are confusing to learn. We can generally avoid these by
writing enough parentheses. One convention that we will use, however, is that in an
expression containing a λ, the scope of λ extends as far to the right as possible. For
example, λx.xy should be read as λx.(xy), not (λx.x)y.
Variable Binding
An occurrence of a variable in an expression may be either free or bound. If a variable is free in some expression, this means that the variable is not declared in the
expression. For example, the variable x is free in the expression x + 3. We cannot
evaluate the expression x + 3 as it stands, without putting it inside some larger expression that will associate some value with x. If a variable is not free, then that must
be because it is bound.
The symbol λ is called a binding operator, as it binds a variable within a specific
scope (part of an expression). The variable x is bound in λx.M. This means that x is
just a place holder, like x in the integral ∫ f (x) dx or the logical formula ∀x.P(x), and
the meaning of λx.M does not depend on x. Therefore, just as ∫f (x) dx and ∫f (y) dy
describe the same integral, we can rename a λ-bound x to y without changing the
meaning of the expression. In particular,
λx.x
defines the same function as λy.y.
Expressions that differ in only the names of bound variables are called α equivalent. When we want to emphasize that two expressions are α equivalent, we write
λx.x = α λy.y, for example.
In λx.M, the expression M is called the scope of the binding λx. A variable x
appearing in an expression M is bound if it appears in the scope of some λx and is
free otherwise. To be more precise about this, we may define the set FV(M) of free
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Fundamentals
variables of M by induction on the structure of expressions, as follows:
FV(x) = {x},
FV(MN) = FV(M) ∪ FV(N),
FV(λx.M) = FV(M) − x,
where − means set difference, that is, S − x = {y ∈ S | y = x}. For example,
FV(λx.x) = because there are no free variables in λx.x, whereas FV (λ f.λx.
( f (g(x))) = {g}.
It is sometimes necessary to talk about different occurrences of a variable in an
expression and to distinguish free and bound occurrences. In the expression
λx.(λy.xy)y,
the first occurrence of x is called a binding occurrence, as this is where x becomes
bound. The other occurrence of x is a bound occurrence. Reading from left to
right, we see that the first occurrence of y is a binding occurrence, the second is
a bound occurrence, and the third is free as it is outside the scope of the λy in
parentheses.
It can be confusing to work with expressions that use the same variable in more
than one way. A useful convention is to rename bound variables so that all bound
variables are different from each other and different from all of the free variables.
Following this convention, we will write (λy.(λz.z)y)x instead of (λx.(λx.x)x)x. The
variable convention will be particularly useful when we come to equational reasoning
and reduction.
Lambda Abstraction in Lisp and Algol
Lambda calculus has an obvious syntactic similarity to Lisp: the Lisp lambda expression
(lambda (x) function body)
looks like the lambda calculus expression
λx. f unction body,
and both expressions define functions. However, there are some differences between
Lisp and lambda calculus. For example, lists are the basic data type of Lisp, whereas
functions are the only data type in pure lambda calculus. Another difference is evaluation order, but that topic will not be discussed in detail.
Although the syntax of block-structured languages is farther from lambda calculus than from Lisp, the basic concepts of declarations and function parameterizations
in Algol-like languages are fundamentally the same as in lambda calculus. For example, the C program fragment
int f (int x) {return x};
block body;
4.2 Lambda Calculus
with a function declaration at the beginning of a block is easily translated into lambda
calculus. The translation is easier to understand if we first add declarations to lambda
calculus.
The simple let declaration,
let x = M in N,
which declares that x has value M in the body N, may be regarded as syntactic sugar
for a combination of lambda abstraction and application:
let x = M in N
is sugar for
(λx.N)M.
For those not familiar with the concept of syntactic sugar, this means that let x = M
in N is “sweeter” to write in some cases, but we can just think of the syntax let x = M
in N as standing for (λx.N)M.
Using let declarations, we can write the C program from above as
let f = (λx .x ) in block body
Note that the C form expression and the lambda expression have the same free and
bound variables and a similar structure. One difference between C and lambda calculus is that C has assignment statements and side effects, whereas lambda calculus is
purely functional. However, by introducing stores, mappings from variable locations
to values, we can translate C programs with side effects into lambda terms as well.
The translation preserves the overall structure of programs, but makes programs
look a little more complicated as the dependencies on the state of the machine are
explicit.
Equivalence and Substitution
We have already discussed α equivalence of terms. This is one of the basic axioms
of lambda calculus, meaning that it is one of the properties of lambda terms that
defines the system and that is used in proving other properties of lambda terms.
Stated more carefully than before, the equational proof system of lambda calculus
has the equational axiom
λx.M = λy.[y/x]M,
(α)
where [y/x]M is the result of substituting y for free occurrences of x in M and
y cannot already appear in M. There are three other equational axioms and four
inference rules for proving equations between terms. However, we look at only one
other equational axiom.
The central equational axiom of lambda calculus is used to calculate the value
of a function application (λx.M)N by substitution. Because λx.M is the function we
get by treating M as a function of x, we can write the value of this function at N by
substituting N for x. Again writing [N/x]M for the result of substituting N for free
occurrences of x in M, we have
(λx.M)N = [N/x]M,
(β)
which is called the axiom of β-equivalence. Some important warnings about
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Fundamentals
substitution are discussed in the next subsection. The value of λ f. f x applied to λy.y
may be calculated if the argument λy.y is substituted in for the bound variable f :
(λ f. f x)(λy.y) = (λy.y)x
Of course, (λy.y)x may be simplified by an additional substitution, and so we have
(λ f. f x)(λy.y) = (λy.y)x = x
Any readers old enough to be familiar with the original documentation of Algol 60
will recognize β-equivalence as the copy rule for evaluating function calls. There are
also parallels between (β) and macroexpansion and in-line substitution of functions.
Renaming Bound Variables
Because λ bindings in M can conflict with free variables in N, substitution [N/x]M
is a little more complicated than we might think at first. However, we can avoid
all of the complications by following the variable convention: renaming bound variables in (λx.M)N so that all of the bound variables are different from each other
and different from all of the free variables. For example, let us reduce the term
(λ f.λx. f ( f x))(λy.y + x). If we first rename bound variables and then perform βreduction, we get
(λ f.λz. f ( f z))(λy.y + x) = λz.((λy.y + x)((λy.y + x)z)) = λz.z + x + x
If we were to forget to rename bound variables and substitute blindly, we might
simplify as follows:
(λ f.λx. f ( f x))(λy.y + x) = λx.((λy.y + x)((λy.y + x)x)) = λx.x + x + x
However, we should be suspicious of the second reduction because the variable x is
free in (λy.y + x) but becomes bound in λx.x + x + x. Remember: In working out
[N/x]M, we must rename any bound variables in M that might be the same as free
variables in N. To be precise about renaming bound variables in substitution, we
define the result [N/x]M of substituting N for x in M by induction on the structure
of M:
[N/x]x = N,
[N/x]y = y, where y is any variable different from x,
[N/x](M1 M2 ) = ([N/x]M1 )([N/x]M2 ),
[N/x](λx.M) = λx.M,
[N/x](λy.M) = λy.([N/x]M), where y is not free in N.
Because we are free to rename the bound variable y in λy.M, the final clause
λy.([N/x]M) always makes sense. With this precise definition of substitution, we
now have a precise definition of β-equivalence.
4.2.3 Programming in Lambda Calculus
Lambda calculus may be viewed as a simple functional programming language. We
will see that, even though the language is very simple, we can program fairly naturally
4.2 Lambda Calculus
if we adopt a few abbreviations. Before declarations and recursion are discussed, it
is worth mentioning the problem of multiargument functions.
Functions of Several Arguments
Lambda abstraction lets us treat any expression M as a function of any variable x
by writing λx.M. However, what if we want to treat M as function of two variables,
x and y? Do we need a second kind of lambda abstraction λ2 x, y.M to treat M as
function of the pair of arguments x, y? Although we could add lambda operators
for sequences of formal parameters, we do not need to because ordinary lambda
abstraction will suffice for most purposes.
We may represent a function f of two arguments by a function λx.(λy.M) of a
single argument that, when applied, returns a second function that accepts a second
argument and then computes a result in the same way as f . For example, the function
f (g, x) = g(x)
has two arguments, but can be represented in lambda calculus by ordinary lambda
abstraction. We define fcurry by
fcurry = λg.λx.gx
The difference between f and fcurry is that f takes a pair (g, x) as an argument,
whereas fcurry takes a single argument g. However, fcurry can be used in place of f
because
fcurry g x = gx = f (g, x)
Thus, in the end, fcurry does the same thing as f . This simple idea was discovered by
Schönfinkel, who investigated functionality in the 1920s. However, this technique for
representing multiargument functions in lambda calculus is usually called Currying,
after the lambda calculus pioneer Haskell Curry.
Declarations
We saw in Subsection 4.2.2 that we can regard simple let declarations,
let x = M in N
as lambda terms by adopting the abbreviation
let x = M in N = (λx.N)M
The let construct may be used to define a composition function, as in the expression
let compose = λ f.λg.λx. f (gx) in compose h h
Using β-equivalence, we can simplify this let expression to λx. h(hx), the composition
of h with itself. In programming language parlance, the let construct provides local
declarations.
Recursion and Fixed Points
An amazing fact about pure lambda calculus is that it is possible to write recursive
functions by use of a self-application “trick.” This does not have a lot to do with
comparisons between modern programming languages, but it may interest readers
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Fundamentals
with a technical bent. (Some readers and some instructors may wish to skip this
subsection.)
Many programming languages allow recursive function definitions. The characteristic property of a recursive definition of a function f is that the body of the function contains one or more calls to f. To choose a specific example, let us suppose we
define the factorial function in some programming language by writing a declaration
like
function f(n) {if n=0 then 1 else n∗f(n-1)};
where the body of the function is the expression inside the braces. This definition has
a straightforward computational interpretation: when f is called with argument a, the
function parameter n is set to a and the function body is evaluated. If evaluation of
the body reaches the recursive call, then this process is repeated. As definitions of a
computational procedure, recursive definitions are clearly meaningful and useful.
One way to understand the lambda calculus approach to recursion is to associate
an equation with a recursive definition. For example, we can associate the equation
f(n) = if n=0 then 1 else n∗f(n-1)
with the preceding recursive declaration. This equation states a property of factorial.
Specifically, the value of the expression f(n) is equal to the value of the expression
if n=0 then 1 else n∗f(n-1) when f is the factorial function. The lambda calculus
approach may be viewed as a method of finding solutions to equations in which an
identifier (the name of the recursive function) appears on both sides of the equation.
We can simplify the preceding equation by using lambda abstraction to eliminate
n from the left-hand side. This gives us
f = λn. if n=0 then 1 else n∗f(n-1)
Now consider the function G that we obtain by moving f to the right-hand-side of the
equation:
G = λf. λn. if n=0 then 1 else n∗f(n-1)
Although it might not be clear what sort of “algebra” is involved in this manipulation of equations, we can check, by using lambda calculus reasoning and basic
understanding of function equality, that the factorial function f satisfies the equation
f = G(f)
This shows that recursive declarations involve finding fixed points.
4.2 Lambda Calculus
A fixed point of a function G is a value f such that f = G( f ). In lambda calculus,
fixed points may be defined with the fixed-point operator:
Y = λ f.(λx. f (xx))(λx. f (xx)).
The surprising fact about this perplexing lambda expression is that, for any f , the
application Yf is a fixed point of f . We can see this by calculation. By β-equivalence,
we have
Yf = (λx. f (xx))(λx. f (xx)).
Using β-equivalence again on the right-hand term, we get
Yf = (λx. f (xx))(λx. f (xx)) = f (λx. f (xx))(λx. f (xx)) = f (Yf ).
Thus Yf is a fixed point of f .
Example 4.3
We can define factorial by fact = YG, where lambda terms Y and G are as given
above, and calculate fact 2 = 2! by using the calculation rules of lambda calculus.
Here are the calculation steps representing the first “call” to factorial:
fact 2 = (YG) 2
= G (YG) 2
= (λ f.λn. if n = 0 then 1 else n∗ f (n − 1))(YG) 2
= (λn. if n = 0 then 1 else n∗((YG)(n − 1)) 2
= if 2 = 0 then 1 else 2∗((YG)(2 − 1))
= if 2 = 0 then 1 else 2∗((YG) 1)
= 2∗((YG) 1).
Using similar steps, we can calculate (YG) 1 = 1! = 1 to complete the calculation.
It is worth mentioning that Y is not the only lambda term that finds fixed points
of functions. There are other expressions that would work as well. However, this
particular fixed-point operator played an important role in the history of lambda
calculus.
4.2.4 Reduction, Confluence, and Normal Forms
The computational properties of lambda calculus are usually described with a form
of symbolic evaluation called reduction. In simple terms, reduction is equational
reasoning, but in a certain direction. Specifically, although β-equivalence was written
as an equation, we have generally used it in one direction to “evaluate” function
calls. If we write → instead of = to indicate the direction in which we intend to use
the equation, then we obtain the basic computation step called β-reduction:
(λx.M)N → [N/x]M,
(β)
We say that M β-reduces to N, and write M → N, if N is the result of applying one
β-reduction step somewhere inside M. Most of the examples of calculation in this
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section use β-reduction, i.e., β-equivalence is used from left to right rather than from
right to left. For example,
(λ f.λz. f ( f z))(λy.y + x) → λz.((λy.y + x)((λy.y + x)z)) → λz.z + x + x.
Normal Forms
Intuitively, we think of M → N as meaning that, in one computation step, the expression M can be evaluated to the expression N. Generally, this process can be repeated,
as illustrated in the preceding subsection. However, for many expressions, the process eventually reaches a stopping point. A stopping point, or expression that cannot
be further evaluated, is called a normal form. Here is an example of a reduction
sequence that leads to a normal form:
(λ f.λx. f ( f x))(λy.y + 1) 2
→ (λx.(λy.y + 1)((λy.y + 1)x)) 2
→ (λx.(λy.y + 1)(x + 1)) 2
→ (λx.(x + 1 + 1)) 2
→ (2 + 1 + 1).
This last expression is a normal form if our only computation rule is β-reduction, but
not a normal form if we have a computation rule for addition. Assuming the usual
evaluation rule for expressions with addition, we can continue with
2+1+1
→3 + 1
→ 4.
This example should give a good idea of how reduction in lambda calculus corresponds to computation. Since the 1930s, lambda calculus has been a simple mathematical model of expression evaluation.
Confluence
In our example starting with (λ f.λx. f ( f x)) (λy.y + 1) 2, there were some steps in
which we had to chose from several possible subexpressions to evaluate. For example,
in the second expression,
(λx.(λy.y + 1)((λy.y + 1)x)) 2,
we could have evaluated either of the two function calls beginning with λy. This is
not an artifact of lambda calculus itself, as we also have two choices in evaluating the
purely arithmetic expression
2 + 1 + 1.
An important property of lambda calculus is called confluence. In lambda calculus, as
a result of confluence, evaluation order does not affect the final value of an expression.
Put another way, if an expression M can be reduced to a normal form, then there
is exactly one normal form of M, independent of the order in which we choose to
4.3 Denotational Semantics
evaluate subexpressions. Although the full mathematical statement of confluence is
a bit more complicated than this, the important thing to remember is that, in lambda
calculus, expressions can be evaluated in any order.
4.2.5 Important Properties of Lambda Calculus
In summary, lambda calculus is a mathematical system with some syntactic and computational properties of a programming language. There is a general notation for
functions that includes a way of treating an expression as a function of some variable
that it contains. There is an equational proof system that leads to calculation rules,
and these calculation rules are a simple form of symbolic evaluation. In programming language terminology, these calculation rules are a form of macro expansion
(with renaming of bound variables!) or function in-lining. Because of the relation to
in-lining, some common compiler optimizations may be defined and proved correct
by use of lambda calculus.
Lambda calculus has the following imortant properties:
Every computable function can be represented in pure lambda calculus. In the
terminology of Chapter 2, lambda calculus is Turing complete. (Numbers can be
represented by functions and recursion can be expressed by Y.)
Evaluation in lambda calculus is order independent. Because of confluence, we
can evaluate an expression by choosing any subexpression. Evaluation in pure
functional programming languages (see Section 4.4) is also confluent, but evaluation in languages whose expressions may have side effects is not confluent.
Macro expansion is another setting in which a form of evaluation is confluent. If we
start with a program containing macros and expand all macro calls with the macro
bodies, then the final fully expanded program we obtain will not depend on the order
in which macros are expanded.
4.3 DENOTATIONAL SEMANTICS
In computer science, the phrase denotational semantics refers to a specific style
of mathematical semantics for imperative programs. This approach was developed
in the late 1960s and early 1970s, following the pioneering work of Christopher
Strachey and Dana Scott at Oxford University. The term denotational semantics
suggests that a meaning or denotation is associated with each program or program
phrase (expression, statement, declaration, etc.). The denotation of a program is a
mathematical object, typically a function, as opposed to an algorithm or a sequence
of instructions to execute.
In denotational semantics, the meaning of a simple program like
x := 0; y:=0; while x –< z do y := y+x; x := x+1
is a mathematical function from states to states, in which a state is a mathematical
function representing the values in memory at some point in the execution of a
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program. Specifically, the meaning of this program will be a function that maps any
state in which the value of z is some nonnegative integer n to the state in which
x =n, y is the sum of all numbers up to n, and all other locations in memory are left
unchanged. The function would not be defined on machine states in which the value
of z is not a nonnegative integer.
Associating mathematical functions with programs is good for some purposes
and not so good for others. In many situations, we consider a program correct if we
get the correct output for any possible input. This form of correctness depends on
only the denotational semantics of a program, the mathematical function from input
to output associated with the program. For example, the preceding program was
designed to compute the sum of all the nonnegative integers up to n. If we verify that
the actual denotational semantics of this program is this mathematical function, then
we have proved that the program is correct. Some disadvantages of denotational
semantics are that standard denotational semantics do not tell us anything about the
running time or storage requirements of a program. This is sometimes an advantage in
disguise because, by ignoring these issues, we can sometimes reason more effectively
about the correctness of programs.
Forms of denotational semantics are commonly used for reasoning about program
optimization and static analysis methods. If we are interested in analyzing running
time, then operational semantics might be more useful, or we could use more detailed denotational semantics that also involves functions representing time bounds.
An alternative to denotational semantics is called operational semantics, which involves modeling machine states and (generally) the step-by-step state transitions
associated with a program. Lambda calculus reduction is an example of operational
semantics.
Compositionality
An important principle of denotational semantics is that the meaning of a program
is determined from its text compositionally. This means that the meaning of a program must be defined from the meanings of its parts, not something else, such as
the text of its parts or the meanings of related programs obtained by syntactic
operations. For example, the denotation of a program such as if B then P else Q
must be explained with only the denotations of B , P , and Q ; it should not be defined with programs constructed from B , P , and Q by syntactic operations such as
substitution.
The importance of compositionality, which may seem rather subtle at first, is that
if two program pieces have the same denotation, then either may safely be substituted for the other in any program. More specifically, if B , P , and Q have the
same denotations as B ’, P ’, and Q ’, respectively, then if B then P else Q must have
the same denotation as if B’ then P’ else Q’. Compositionality means that the denotation of an expression or program statement must be detailed enough to capture
everything that is relevant to its behavior in larger programs. This makes denotational semantics useful for understanding and reasoning about such pragmatic issues
as program transformation and optimization, as these operations on programs involve replacing parts of programs without changing the overall meaning of the whole
program.
4.3 Denotational Semantics
4.3.1 Object Language and Metalanguage
One source of confusion in talking (or writing) about the interpretation of syntactic expressions is that everything we write is actually syntactic. When we study a
programming language, we need to distinguish the programming language we study
from the language we use to describe this language and its meaning. The language we
study is traditionally called the object language, as this is the object of our attention,
whereas the second language is called the metalanguage, because it transcends the
object language in some way.
To pick an example, let us consider the mathematical interpretation of a simple
algebraic expression such as 3 + 6 − 4 that might appear in a program written in C,
Java, or ML. The ordinary “mathematical” meaning of this expression is the number
obtained by doing the addition and subtraction, namely 5. Here, the symbols in the
expression 3 + 6 − 4 are in our object language, whereas the number 5 is meant to be
in our metalanguage. One way of making this clearer is to use an outlined number,
such as 1, to mean “the mathematical entity called the natural number 1.” Then we
can say that the meaning of the object language expression 3 + 6 − 4 is the natural
number 5. In this sentence, the symbol 5 is a symbol of the metalanguage, whereas
the expression 3 + 6 − 4 is written with symbols of the object language.
4.3.2 Denotational Semantics of Binary Numbers
The following grammar for binary expressions is similar to the grammar for decimal
expressions discussed in Subsection 4.1.2:
e ::= n | e+e | e-e
n ::= b | nb
b ::= 0 | 1
We can give a mathematical interpretation of these expressions in the style of denotational semantics. In denotational semantics and other studies of programming
languages, it is common to forget about how expressions are converted into parse
trees and just give the meaning of an expression as a function of its parse tree.
We may interpret the expressions previously defined as natural numbers by using
induction on the structure of parse trees. More specifically, we define a function from
parse trees to natural numbers, defining the function on a compound expression by
referring to its value on simpler expressions. A historical convention is to write [[e]]
for any parse tree of the expression e. When we write [[e1 +e2 ]], for example, we mean
a parse tree of the form
e
[[e1]]
+
[[e2]]
with [[e1 ]] and [[e2 ]] as immediate subtrees.
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Using this notation, we may define the meaning E[[e]] of an expression e, according to its parse tree [[e]], as follows:
E[[0]] = 0
E[[1]] = 1
E[[nb]] = E[[n]] ∗ 2 + E[[b]]
E[[e1 +e2 ]] = E[[e1 ]] + E[[e2 ]]
E[[e1 -e2 ]] = E[[e1 ]]–E[[e2 ]]
In words, the value associated with a parse tree of the form [[e1 +e2 ]], for example,
is the sum of the values given to the subtrees [[e1 ]] and [[e2 ]]. This is not a circular
definition because the parse trees [[e1 ]] and [[e2 ]] are smaller than the parse tree
[[e1 +e2 ]].
On the right-hand side of the equal signs, numbers and arithmetic operations ∗,
+, and − are meant to indicate the actual natural numbers and the standard integer operations of multiplication, addition, and subtraction. In contrast, the symbols
+ and − in expressions surrounded by double square brackets on the left-hand
side of the equal signs are symbols of the object language, the language of binary
expressions.
4.3.3 Denotational Semantics of While Programs
Without going into detail about the kinds of mathematical functions that are used,
let us take a quick look at the form of semantics used for a simplified programming
language with assignment and loops.
Expressions with Variables
Program statements contain expressions with variables. Here is a grammar for arithmetic expressions with variables. This is the same as the grammar in Subsection 4.1.2,
except that expressions can contain variables in addition to numbers:
e ::= v | n | e+e | e-e
n ::= d | nd
d ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
v ::= x | y | z | . . .
In the simplified programming language we consider, the value of a variable depends
on the state of the machine. We model the state of the machine by a function from
variables to numbers and write E[[e]](s) for the value of expression e in state s. The
value of an expression in a state is defined as follows.
E[[x]](s) = s(x)
E[[0]](s) = 0
E[[1]](s) = 1
... = …
4.3 Denotational Semantics
E[[9]](s) = 9
E[[nd]](s) = E[[n]](s) ∗ 10 + E[[d]](s)
E[[e1 +e2 ]](s) = E[[e1 ]](s) + E[[e2 ]](s)
E[[e1 − e2 ]](s) = E[[e1 ]](s) − E[[e2 ]](s)
Note that the state matters in the definition in the base case, the value of a variable.
Otherwise, this is essentially the same definition as that in the semantics of variablefree expressions in the preceding subsection.
The syntax and semantics of Boolean expressions can be defined similarly.
While Programs
The language of while programs may be defined over any class of value expressions
and Boolean expressions. Without specifying any particular basic expressions, we
may summarize the structure of while programs by the grammar
P ::= x := e | P; P | if e then P else P | while e do P
where we assume that x has the appropriate type to be assigned the value of e in the
assignment x :=e and that the test e has type bool in if-then-else and while statements.
Because this language does not have explicit input or output, the effect of a program
will be to change the values of variables. Here is a simple example:
x := 0; y:=0; while x –< z do (y := y+x; x := x+1)
We may think of this program as having input z and output y . This program uses an
additional variable x to set y to the sum of all natural numbers up to z.
States and Commands
The meaning of a program is a function from states to states. In a more realistic
programming language, with procedures or pointers, it is necessary to model the
fact that two variable names may refer to the same location. However, in the simple
language of while programs, we assume that each variable is given a different location.
With this simplification in mind, we let the set S tate of mathematical representations
of machine states be
State = Variables → Values
In words, a state is a function from variables to values. This is an idealized view of
machine states in two ways: We do not explicitly model the locations associated with
variables, and we use infinite states that give a value to every possible variable.
The meaning of a program is an element of the mathematical set Command of
commands, defined by
Command = State → State
In words, a command is a function from states to states. Unlike states themselves,
which are total functions, a command may be a partial function. The reason we need
partial functions is that a program might not terminate on an initial state.
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A basic function on states that is used in the semantics of assignment is modify,
which is defined as follows:
modify(s,x,a) = λν ∈ Variables. if ν=x then a else s(ν)
In words, modify(s,x,a) is the state (function from variables to values) that is just like
state s , except that the value of x is a .
Denotational Semantics
The denotational semantics of while programs is given by the definition of a function
C from parsed programs to commands. As with expressions, we write [[P]] for a
parse tree of the program P. The semantics of programs are defined by the following
clauses, one for each syntactic form of program:
C[[x := e]](s) = modify(s,x, E[[ e ]](s))
C[[P1 ;P2 ]](s) = C[[ P2 ]]( C[[P1 ]](s))
C[[if e then P1 else P2 ]](s) = if E[[ e ]](s) then C[[P1 ]](s) else C[[P2 ]](s)
C[[while e do P]](s) = if not E[[ e ]](s) then s
else C[[while e do P]](C[[P]](s))
Because e is an expression, not a statement, we apply the semantic function E to
obtain the value of e in a given state.
In words, we can describe the semantics of programs as follows:
C[[x := e]](s) is the state similar to s , but with x having the value of e.
C[[P1 ;P2 ]](s) is the state we obtain by applying the semantics of P2 the to the state
we obtain by applying the semantics of P1 to state s .
C[[if e then P1 else P2 ]](s) is the state we obtain by applying the semantics of P1 to
s if e is true in s and P2 to s otherwise.
C[[while e do P]](s) is a recursively defined function f, from states to states. In
words, f (s) is either s if e is false or the state we obtain by applying f to the
state resulting from executing P once in s .
The recursive function definition in the while clause is relatively subtle. It also raises
some interesting mathematical problems, as it is not always mathematically reasonable to define functions by arbitrary recursive conditions. However, in the interest of
keeping our discussion simple and straightforward, we just assume that a definition
of this form can be made mathematically rigorous.
Example 4.4
We can calculate the semantics of various programs by using this definition. To begin
with, let us consider a simple loop-free program,
if x>y then x :=y else y :=x
which sets both x and y to the minimum of their initial values. For concreteness, let
us calculate the semantics of this program in the state s0 , where s0 (x) = 1 and s0 (y)=2.
4.3 Denotational Semantics
Because E[[x>y]](s0 ) = false, we have
C[[if x>y then x :=y else y := x ]](s0 )
= if E[[x>y]](s0 ) then C[[ x := y ]](s0 ) else C[[ y := x ]](s0 )
= C[[ y := x ]](s0 )
= modify(s0 , y, E[[ x ]](s0 ))
In words, if the program if x>y then x :=y else y := x is executed in the state s0 , then
the result will be the state that is the same as s0 , but with variable y given the value
that x has in state s0 .
Example 4.5
Although it takes a few more steps than the previous example, it is not too difficult
to work out the semantics of the program
x := 0; y:=0; while x –< z do (y := y+x; x := x+1)
in the state s0 , where s0 (z)=2. A few preliminary definitions will make the calculation
easier. Let s1 and s2 be the states
s1 = C[[ x:= 0 ]](s0 )
s2 = C[[ y := 0 ]](s1 )
Using the semantics of assignment, as above, we have
s1 = modify(s0 , x, 0)
s2 = modify(s1 , y, 0)
Returning to the preceding program, we have
C[[x := 0; y:=0; while x –< z do (y := y+x; x := x+1) ]](s0 )
= C[[ y:=0; while x –< z do (y := y+x; x := x+1) ]]( C [[ x := 0 ]](s0 ) )
= C[[ y:=0; while x –< z do (y := y+x; x := x+1) ]](s1 )
= C[[ while x –< z do (y := y+x; x := x+1) ]]( C [[ y := 0 ]](s1 ) )
= C[[ while x –< z do (y := y+x; x := x+1) ]](s2 )
= if not E[[ x –< z ]](s2 ) then s2
else C[[ while x –< z do (y := y+x; x := x+1) ]](C[[ y := y+x; x := x+1 ]](s2 ))
= C[[ while x –< z do (y := y+x; x := x+1) ]](s3 )
where s3 has y set to 0 and x set to 1. Continuing in the same manner, we have
C[[ while x –< z do (y := y+x; x := x+1) ]](s3 )
= if not E[[x –< z ]](s3 ) then s3
else C[[ while x –< z do (y := y+x; x := x+1) ]](C[[y := y+x; x := x+1 ]](s3 ))
= C[[ while x –< z do (y := y+x; x := x+1) ]](s4 )
= if not E[[x –< z ]](s4 ) then s4
else C[[ while x –< z do (y := y+x; x := x+1) ]](C[[y := y+x; x := x+1 ]](s4 ))
= C[[ while x –< z do (y := y+x; x := x+1) ]](s5 )
= s5
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Fundamentals
where s4 has y set to 1 and x to 2 and s5 has y set to 3 and x to 3. The steps are tedious
to write out, but you can probably see without doing so that if s0 (z) = 5, for example,
then this program will yield a state in which the value of x is 0+1+2+3+4+5.
As these examples illustrate, the denotational semantics of while programs unambiguously associates a partial function from states to states with each program.
One important issue we have not discussed in detail is what happens when a loop
does not terminate. The meaning C[[ while x=x do x := x ]] of a loop that does not
terminate in any state is a partial function that is not defined on any state. In other
words, for any state s, C[[ while x=x do x := x ]](s) is not defined. Similarly, C[[ while
x=y do x := y ]](s) is s if s(x) =/ s(y) and undefined otherwise. If you are interested in
more information, there are many books that cover denotational semantics in detail.
4.3.4 Perspective and Nonstandard Semantics
There are several ways of viewing the standard methods of denotational semantics.
Typically, denotational semantics is given by the association of a function with each
program. As many researchers in denotational semantics have observed, a mapping
from programs to functions must be written in some metalanguage. Because lambda
calculus is a useful notation for functions, it is common to use some form of lambda
calculus as a metalanguage. Thus, most denotational semantics actually have two
parts: a translation of programs into a lambda calculus (with some extra operations
corresponding to basic operations in programs) and a semantic interpretation of the
lambda calculus expressions as mathematical objects. For this reason, denotational
semantics actually provides a general technique for translating imperative programs
into functional programs.
Although the original goal of denotational semantics was to define the meanings
of programs in a mathematical way, the techniques of denotational semantics can
also be used to define useful “nonstandard” semantics of programs.
One useful kind of nonstandard semantics is called abstract interpretation. In abstract interpretation, programs are assigned meaning in some simplified domain. For
example, instead of interpreting integer expressions as integers, integer expressions
could be interpreted elements of the finite set {0, 1, 2, 3, 4, 5, . . . , 100, >100}, where
>100 is a value used for expressions whose value might be greater than 100. This
might be useful if we want to see if two array expressions A[e1 ] and A[e2 ] refer to the
same array location. More specifically, if we assign values to e1 from the preceding
set, and similarly for e2 , we might be able to determine that e1 =e2 . If both are assigned
the value >100, then we would not know that they are the same, but if they are assigned the same ordinary integer between 0 and 100, then we would know that these
expressions have the same value. The importance of using a finite set of values is that
an algorithm could iterate over all possible states. This is important for calculating
properties of programs that hold in all states and also for calculating the semantics
of loops.
Example. Suppose we want to build a program analysis tool that checks programs to
make sure that every variable is initialized before it is used. The basis for this kind of
program analysis can be described by denotational semantics, in which the meaning
of an expression is an element of a finite set.
4.3 Denotational Semantics
Because the halting problem is unsolvable, program analysis algorithms are usually designed to be conservative. Conservative means that there are no false positivies:
An algorithm will output correct only if the program is correct, but may sometimes
output error even if there is no error in the program. We cannot expect a computable
analysis to decide correctly whether a program will ever access a variable before it
is initialized. For example, we cannot decide whether
(complictated error-free program); x := y
executes the assignment x := y without deciding whether complictated error-free program halts. However, it is often possible to develop efficient algorithms for conservative analysis. If you think about it, you will realize that most compiler warnings are
conservative: Some warnings could be a problem in general, but are not a problem
in a specific program because of program properties that the compiler is not “smart”
enough to understand.
We describe initialize-before-use analysis by using an abstract representation of
machine states that keep track only of whether a variable has been initialized or not.
More precisely, a state will either be a special error state called error or a function
from variable names to the set {init, uninit} with two values, one representing any
value of initialized variable and the other an uninitialized one. The set State of
mathematical abstractions of machine states is therefore
State = {error} ∪ (Variables → {init, uninit})
As usual, the meaning of a program will be a function from states to states. Let us
assume that E[[e]](s) = error if e contains any variable y with s(y) = uninit and E[[e]](s) =
Ok otherwise.
The semantics of programs is given by a set of clauses, one for each program
form, as usual. For any program P, we let C[[P]](error) = error. The semantic clause
for assignment in state s =/ error can be written as
C[[x :=e ]](s) = if E[[e]](s)=Ok then modify(s, x, init) else error
In words, if we execute an assignment x:= e in a state s different from error, then the
result is either a state that has x initialized or error if there is some variable in e that
was not initialized in s. For example, let
s0 = λv ∈ Variables. uninit
be the state with every variable uninitialized. Then
C[[ x := 0 ]](s0 ) = modify(s0 , x, init)
is the state with variable x initialized and all other variables uninitialized.
The clauses for sequences P1; P2 are essentially straightforward. For s =/ error,
C[[P1 ;P2 ]](s) = if C[[P1 ]](s) = error then error else C[[ P2 ]]( C[[P1 ]](s) )
The clause for conditional is more complicated because our analysis tool is not going
to try to evaluate a Boolean expression. Instead, we treat conditional as if it were
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Fundamentals
possible to execute either branch. (This is the conservative part of the analysis.)
Therefore, we change a variable to initialized only if it is initialized in both branches.
If s1 and s2 are states different from error, then let s1 + s2 be the state
s1 + s2 = λv ∈ Variables. If s1 (v) = s2 (v) = init then init else uninit
We define the meaning of a conditional statement by
C[[if e then P1 else P2 ]](s)
= if E[[ e ]](s)= error or C[[ P1 ]](s)= error or C[[ P2 ]](s) = error
then error
else C[[ P1 ]](s) + C[[ P2 ]](s)
For example, using s0 as above, we have
C[[ if 0=1 then x := 0 else x := 1; y := 2 ]](s0 ) = modify(s0 , x, init)
as only x is initialized in both branches of the conditional.
For simplicity, we do not consider the clause for while e do P.
4.4 FUNCTIONAL AND IMPERATIVE LANGUAGES
4.4.1 Imperative and Declarative Sentences
The languages that humans speak and write are called natural languages as they
developed naturally, without concern for machine readability. In natural languages,
there are four main kinds of sentences: imperative, declarative, interrogative, and
exclamatory.
In an imperative sentence, the subject of the sentence is implicit. For example,
the subject of the sentence
Pick up that fish
is (implicitly) the person to whom the command is addressed. A declarative sentence
expresses a fact and may consist of a subject and a verb or subject, verb, and object.
For example,
Claude likes bananas
is a declarative sentence.
Interrogatives are questions. An exclamatory sentence may consist of only an
interjection, such as Ugh! or Wow!
In many programming languages, the basic constructs are imperative statements.
For example, an assignment statement such as
x:=5
is a command to the computer (the implied subject of the utterance) to store the value
5 in a certain location. Programming languages also contain declarative constructs
such as the function declaration
function f(int x) { return x+1;}
4.4 Functional and Imperative Languages
that states a fact. One reading of this as a declarative sentence is that the subject is
the name f and the sentence about f is “f is a function whose return value is 1 greater
than its argument.”
In programming, the distinction between imperative and declarative constructs
rests on the distinction between changing an existing value and declaring a new value.
The first is imperative, the latter declarative. For example, consider the following
program fragment:
{ int x = 1;
x = x+1;
{ int y = x+1;
{ int x = y+1;
}}}
/* declares new x */
/* assignment to existing x */
/* declares new y */
/* declares new x */
Here, only the second line is an imperative statement. This is an imperative command
that changes the state of the machine by storing the value 2 in the location associated
with variable x. The other lines contain declarations of new variables.
A subtle point is that the last line in the preceding code declares a new variable
with the same name as that of a previously declared variable. The simplest way to
understand the distinction between declaring a new variable and changing the value
of an old one is by variable renaming. As we saw in lambda calculus, a general
principle of binding is that bound variables can be renamed without changing the
meaning of an expression or program. In particular, we can rename bound variables
in the preceding program fragment to get
{ int x = 1;
x = x+1;
{ int y = x+1;
{ int z = y+1;
}}}
/* declares new x */
/* assignment to existing x */
/* declares new y */
/* declares new z */
(If there were additional occurrences of x inside the inner block, we would rename
them to z also.) After rewriting the program to this equivalent form, we easily see
that the declaration of a new variable z does not change the value of any previously
existing variable.
4.4.2 Functional versus Imperative Programs
The phrase functional language is used to refer to programming languages in which
most computation is done by evaluation of expressions that contain functions. Two
examples are Lisp and ML. Both of these languages contain declarative and imperative constructs. However, it is possible to write a substantial program in either
language without using any imperative constructs.
Some people use the phrase functional language to refer to languages that do
not have expressions with side effects or any other form of imperative construct.
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However, we will use the more emphatic phrase pure functional language for declarative languages that are designed around flexible constructs for defining and using
functions. We learned in Subsection 3.4.9 that pure Lisp, based on atom, eq, car, cdr,
cons, lambda, define, is a pure functional language. If rplaca, which changes the car of
a cell, and rplacd, which changes the cdr of a cell, are added, then the resulting Lisp
is not a pure functional language.
Pure functional languages pass the following test:
Declarative Language Test: Within the scope of specific declarations of x1 , . . . ,
xn , all occurrences of an expression e containing only variables x1 , . . . , xn have
the same value.
As a consequence, pure functional languages have a useful optimization property: If
expression e occurs several places within a specific scope, this expression needs to be
evaluated only once. For example, suppose a program written in pure Lisp contains
two occurrences of (cons a b). An optimizing Lisp compiler could compute (cons a b)
once and use the same value both places. This not only saves time, but also space, as
evaluating cons would ordinarily involve a new cell.
Referential Transparency
In some of the academic literature on programming languages, including some textbooks on programming language semantics, the concept that is used to distinguish
declarative from imperative languages is called referential transparency. Although
it is easy to define this phrase, it is a bit tricky to use it correctly to distinguish one
programming language from another.
In linguistics, a name or noun phrase is considered referentially transparent if it
may be replaced with another noun phrase with the same referent (i.e., referring to
the same thing) without changing the meaning of the sentence that contains it. For
example, consider the sentence
I saw Walter get into his car.
If Walter owns a Maserati Biturbo, say, and no other car, then the sentence
I saw Walter get into his Maserati Biturbo
has the same meaning because the noun phrases (in italics) have the same meaning.
A traditional counterexample to referential transparency, attributed to the language
philosopher Willard van Orman Quine, occurs in the sentence
He was called William Rufus because of his red beard.
The sentence refers to William IV of England and rufus means reddish or orange in
color. If we replace William Rufus with William IV, we get a sentence that makes no
sense:
He was called William IV because of his red beard.
Obviously, the king was called William IV because he was the fourth William, not
because of the color of his beard.
Returning to programming languages, it is traditional to say that a language is
referentially transparent if we may replace one expression with another of equal
value anywhere in a program without changing the meaning of the program. This is
a property of pure functional languages.
The reason referential transparency is subtle is that it depends on the value we
associate with expressions. In imperative programming languages, we can say that
4.4 Functional and Imperative Languages
a variable x refers to its value or to its location. If we say that a variable refers
to its location in memory, then imperative languages are referentially transparent,
as replacing one variable with another that names the same memory location will
not change the meaning of the program. On the other hand, if we say that a variable refers to the value stored in that location, then imperative languages are not
referentially transparent, as the value of a variable may change as the result of
assignment.
Historical Debate
John Backus received the 1977 ACM Turing Award for the development of Fortran.
In his lecture associated with this award, Backus argued that pure functional programming languages are better than imperative ones. The lecture and the accompanying
paper, published by the Association for Cumputing Machinery, helped inspire a number of research projects aimed at developing practical pure functional programming
languages. The main premise of Backus’ argument is that pure functional programs
are easier to reason about because we can understand expressions independently of
the context in which they occur.
Backus asserts that, in the long run, program correctness, readability, and reliability are more important than other factors such as efficiency. This was a controversial position in 1977, when programs were a lot smaller than they are today and
computers were much slower. In the 1990s, computers finally reached the stage at
which commercial organizations began to choose software development methods
that value programmer development time over run-time efficiency. Because of his
belief in the importance of correctness, readability, and reliability, Backus thought
that pure functional languages would be appreciated as superior to languages with
side effects.
To advance his argument, Backus proposed a pure functional programming language called FP, an acronym for functional programming. FP contains a number of
basic functions and a rich set of combining forms with which to build new functions
from old ones. An example from Backus’ paper is a simple program to compute the
inner product of two vectors. In C, the inner product of vectors stored in arrays a and
b could be written as
int i, prod;
prod=0;
for (i=0; i < n; i++) prod = prod + a[i] * b[i];
In contrast, the inner product function would be defined in FP by combining functions
+ and × (multiplication) with vector operations. Specifically, the inner product would
be expressed as
Inner product = (Insert +) ◦ (ApplyToAll x) ◦ Transpose
where
◦
is function composition and Insert, ApplyToAll, and Transpose are vector
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operations. Specifically, the Transpose of a pair of lists produces a list of pairs and
ApplyToAll applies the given operation to every element in a list, like the maplist
function in Subsection 3.4.7. Given a binary operation and a list, Insert has the effect
of inserting the operation between every pair of adjacent list elements and calculating the result. For example, (Insert +) <1, 2, 3, 4> = 1+2+3+4=10, where <1, 2, 3, 4> is
the FP notation for the list with elements 1, 2, 3, 4.
Although the C syntax may seem clearer to most readers, it is worth trying to
imagine how these would compare if you had not seen either before. One facet of
the FP expression is that all of its parts are functions that we can understand without
thinking about how vectors are represented in memory. In contrast, the C program
has extra variables i and prod that are not part of the function we are trying to
compute. In this sense, the FP program is higher level, or more abstract, than the
C code.
A more general point about FP programs is that if one functional expression is
equivalent to another, then we can replace one with the other in any program. This
leads to a set of algebraic laws for FP programs. In addition to algebraic laws, an
advantage of functional programming languages is the possibility of parallelism in
implementations. This is subsequently discussed.
In retrospect, Backus’ argument seems more plausible than his solution. The
importance of program correctness, readability, and reliability has increased compared with that of run-time efficiency. The reason is that these affect the amount of
time that people must spend in developing, debugging, and maintaining code. When
computers become faster, it is acceptable to run less efficient programs – hardware
improvements can compensate for software. However, increases in computer speed
do not make humans more efficient. In this regard, Backus was absolutely right about
the aspects of programming languages that would be important in the future.
Backus’ language FP, on the other hand, was not a success. In the years since his
lecture, there was an effort to develop a FP implementation at IBM, but the language
was not widely used. One problem is that the language has a difficult syntax. Perhaps
more importantly, there are severe limitations in the kind of data structures and
control structures that can be defined. FP languages that allow variable names and
binding (as in Lisp and lambda calculus) have been more successful, as have all
programming languages that support modularity and reuse of library components.
However, Backus raised an important issue that led to useful reflection and debate.
Functional Programming and Concurrency
An appealing aspect of pure functional languages and of programs written in the pure
functional subset of larger languages is that programs can be executed concurrently.
This is a consequence of the declarative language test mentioned at the beginning
of this subsection. We can see how parallelism arises in pure functional languages
by using the example of a function call f(e1 ,...,en ), where function arguments e1 ,...,en
are expressions that may need to be evaluated.
Functional Programming: We can evaluate f(e1 ,...,en ) by evaluating e1 ,...,en
in parallel because values of these expressions are independent.
Imperative Programming: For an expression such as f(g(x), h(x)), the function
g might change the value of x. Hence the arguments of functions in imperative
4.4 Functional and Imperative Languages
languages must be evaluated in a fixed, sequential order. This ordering restricts
the use of concurrency.
Backus used the term von Neumann bottleneck for the fact that in executing an
imperative program, computation must proceed one step at a time. Because each
step in a program may depend on the previous one, we have to pass values one at a
time from memory to the CPU and back. This sequential channel between the CPU
and memory is what he called the von Neumann bottleneck.
Although functional programs provide the opportunity for parallelism, and parallelism is often an effective way of increasing the speed of computation, effectively
taking advantage of inherent parallelism is difficult. One problem that is fairly easy
to understand is that functional programs sometimes provide too much parallelism.
If all possible computations are performed in parallel, many more computation steps
will be executed than necessary. For example, full parallel evaluation of a conditional
if e1 then e2 else e3
will involve evaluating all three expressions. Eventually, when the value of e1 is
found, one of the other computations will turn out to be irrelevant. In this case, the
irrelevant computation can be terminated, but in the meantime, resources will have
been devoted to calculation that does not matter in the end.
In a large program, it is easy to generate so many parallel tasks that the time setting up and switching between parallel processes will detract from the efficiency of
the computation. In general, parallel programming languages need to provide some
way for a programmer to specify where parallelism may be beneficial. Parallel implementations of functional languages often have the drawback that the programmer
has little control over the amount of parallelism used in execution.
Practical Functional Programming
Backus’ Turing Award lecture raises a fundamental question:
Do pure functional programming languages have significant practical advantages
over imperative languages?
Although we have considered many of the potential advantages of pure FP languages
in this section, we do not have a definitive answer. From one theoretical point of
view, FP languages are as good as imperative programming languages. This can be
demonstrated when a translation is made of C programs into FP programs, lambda
calculus expressions, or pure Lisp. Denotational semantics provides one method for
doing this.
To answer the question in practice, however, we would need to carry out large
projects in a functional language and see whether it is possible to produce usable
software in a reasonable amount of time. Some work toward answering this question was done at IBM on the FP project (which was canceled soon after Backus
retired). Additional efforts with other languages such as Haskell and Lazy ML are
still being carried out at other research laboratories and universities. Although most
programming is done in imperative languages, it is certainly possible that, at some
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future time, pure or mostly pure FP languages will become more popular. Whether or
not that happens, FP projects have generated many interesting language design ideas
and implementation techniques that have been influential beyond pure functional
programming.
4.5 CHAPTER SUMMARY
In this chapter, we studied the following topics:
the outline of a simple compiler and parsing issues,
lambda calculus,
denotational semantics,
the difference between functional and imperative languages.
A standard compiler transforms an input program, written in a source language,
into an output program, written in a target language. This process is organized into
a series of six phases, each involving more complex properties of programs. The
first three phases, lexical analysis, syntax analysis, and semantic analysis, organize
the input symbols into meaningful tokens, construct a parse tree, and determine
context-dependent properties of the program such as type agreement of operators
and operands. (The name semantic analysis is commonly used in compiler books, but
is somewhat misleading as it is still analysis of the parse tree for context-sensitive
syntactic conditions.) The last three phases, intermediate code generation, optimization, and target code generation, are aimed at producing efficient target code through
language transformations and optimizations.
Lambda calculus provides a notation and symbolic evaluation mechanism that
is useful for studying some properties of programming languages. In the section on
lambda calculus, we discussed binding and α conversion. Binding operators arise in
many programming languages in the form of declarations and in parameter lists of
functions, modules, and templates. Lambda expressions are symbolically evaluated
by use of β-reduction, with the function argument substituted in place of the formal parameter. This process resembles macro expansion and function in-lining, two
transformations that are commonly done by compilers. Although lambda calculus is
a very simple system, it is theoretically possible to write every computable function in
the lambda calculus. Untyped lambda calculus, which we discussed, can be extended
with type systems to produce various forms of typed lambda calculus.
Denotational semantics is a way of defining the meanings of programs by specifying the mathematical value, function, or function on functions that each construct
denotes. Denotational semantics is an abstract way of analyzing programs because
it does not consider issues such as running time and memory requirements. However, denotational semantics is useful for reasoning about correctness and has been
used to develop and study program analysis methods that are used in compilers and
programming environments. Some of the exercises at the end of the chapter present
applications for type checking, initialization-before-use analysis, and simplified security analysis. From a theoretical point of view, denotational semantics shows that
every imperative program can be transformed into an equivalent functional program.
In pure functional programs, syntactically identical expressions within the same
scope have identical values. This property allows certain optimizations and makes
Exercises
it possible to execute independent subprograms in parallel. Because functional
languages are theoretically as powerful as imperative languages, we discussed some
of the pragmatic differences between functional and imperative languages. Although
functional languages may be simpler to reason about in certain ways, imperative languages often make it easier to write efficient programs. Although Backus argues that
functional programs can eliminate the von Neumann bottleneck, practical parallel
execution of functional programs has not proven as successful as he anticipated in
his Turing Award lecture.
EXERCISES
4.1 Parse Tree
Draw the parse tree for the derivation of the expression 25 given in Subsection
4.1.2. Is there another derivation for 25? Is there another parse tree?
4.2 Parsing and Precedence
Draw parse tress for the following expressions, assuming the grammar and precedence described in Example 4.2:
(a) 1 - 1 ∗ 1.
(b) 1 - 1 + 1.
(c) 1 - 1 + 1 - 1 + 1, if we give + higher precedence than −.
4.3 Lambda Calculus Reduction
Use lambda calculus reduction to find a shorter expression for (λx.λy.xy)(λx.xy).
Begin by renaming bound variables. You should do all possible reductions to get
the shortest possible expression. What goes wrong if you do not rename bound
variables?
4.4 Symbolic Evaluation
The Algol-like program fragment
function f(x)
return x+4
end;
function g(y)
return 3-y
end;
f(g(1));
can be written as the following lambda expression:




 (λ f.λg. f (g 1)) (λx.x + 4)  (λy.3 − y) .
main
f
g
Reduce the expression to a normal form in two different ways, as described below.
(a) Reduce the expression by choosing, at each step, the reduction that eliminates
a λ as far to the left as possible.
(b) Reduce the expression by choosing, at each step, the reduction that eliminates
a λ as far to the right as possible.
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4.5 Lambda Reduction with Sugar
Here is a “sugared” lambda expression that uses let declarations:
let compose = λ f. λg. λx. f (g x) in
let h = λx. x + x in
compose h h 3
The “desugared” lambda expression, obtained when each let z = U in V is replaced with (λz. V) U is
(λcompose.
(λh. compose h h 3) λx. x + x)
λ f. λg. λx. f (g x).
This is written with the same variable names as those of the let form to make it
easier to read the expression.
Simplify the desugared lambda expression by using reduction. Write one or two
sentences explaining why the simplified expression is the answer you expected.
4.6 Translation into Lambda Calculus
A programmer is having difficulty debugging the following C program. In theory,
on an “ideal” machine with infinite memory, this program would run forever. (In
practice, this program crashes because it runs out of memory, as extra space is
required every time a function call is made.)
int f(int (*g)(. . .)){ /* g points to a function that returns an int */
return g(g);
}
int main(){
int x;
x = f(f);
printf(”Value of x = %d\n”, x);
return 0;
}
Explain the behavior of the program by translating the definition of f into lambda
calculus and then reducing the application f(f). This program assumes that the type
checker does not check the types of arguments to functions.
4.7 Order of Evaluation
In pure lambda calculus, the order of evaluation of subexpressions does not effect
the value of an expression. The same is true for pure Lisp: if a pure Lisp expression has a value under the ordinary Lisp interpreter, then changing the order of
evaluation of subterms cannot produce a different value.
To give a concrete example, consider the following section of Lisp code:
(define a ( . . . ))
(define b ( . . . ))
...
(define f (lambda (x y z) (cons (car x) (cons (car y) (cdr z)))))
...
(f e1 e2 e3)
Exercises
The ordinary evaluation order for the function call
(f e1 e2 e3)
is to evaluate the arguments e1 e2 e3 from left to right and then pass this list of
values to the function f.
Give an example of Lisp expressions e1 e2 e3, possibly by using functions rplaca
or rplacd with side effects, so that evaluating expressions from left to right gives a
different result from that obtained by evaluating them from right to left. You may
fill in specific declarations for a or b if you like and refer to a or b in your expressions.
Explain briefly, in one or two sentences, why one order of evaluation is different
from the other.
4.8 Denotational Semantics
The text describes a denotational semantics for the simple imperative language
given by the grammar
P ::= x := e | P 1 ;P 2 | if e then P 1 else P 2 | while e do P.
Each program denotes a function from states to states, in which a state is a function
from variables to values.
(a) Calculate the meaning C[[ x := 1; x := x + 1;]](s0 ) in approximately the same detail
as that of the examples given in the text, where s0 = λv ∈ variables. 0, giving
every variable the value 0.
(b) Denotational semantics is sometimes used to justify ways of reasoning about
programs. Write a few sentences, referring to your calculation in part (a),
explaining why
C[[x := 1; x := x + 1;]](s) = C[[x := 2;]](s)
for every state s.
4.9 Semantics of Initialize-Before-Use
A nonstandard denotational semantics describing initialize-before-use analysis is
presented in the text.
(a) What is the meaning of
C[[x := 0; y := 0; if x = y then z := 0 else w := 1 ]](s0 )
in the state s0 = λy ∈ variables.uninit? Show how to calculate your answer.
(b) Calculate the meaning
C[[if x = y then z := y else z := w]](s)
in state s with s(x) = init, s(y) = init, and s(v) = uninit or every other variable v.
4.10 Semantics of Type Checking
This problem asks about a nonstandard semantics that characterizes a form of type
analysis for expressions given by the following grammar:
e ::= 0 | 1 | true | false | x | e + e | if e then e else e | let x:τ =e in e
In the let expresion, which is a form of local declaration, the type τ may be either
int or bool. The meaning V[[e]](η) of an expression e depends on an environment.
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Fundamentals
An environment η is a mapping from variables to values. In type analysis, we use
three values,
Values = {integer, boolean, type error}.
Intuitively, V[[e]](η) = integer means that the value of expression e is an integer (in
environment η) and V[[e]](η) = type error means that evaluation of e may involve
a type error.
Here are the semantic clauses for the first few expression forms.
V[[0]]η
V[[1]]η
V[[true]]η
V[[false]]η
=
=
=
=
integer,
integer,
boolean,
boolean.
The value of a variable in some environment is simply the value the environment
gives to that variable:
V[[x]]η = η(x)
For addition, the value will be type error unless both operands are integers:
integer
if V[[e1 ]]η = integer and V[[e2 ]]η = integer
.
V[[e1 + e2 ]]η =
type error otherwise
Because declarations involve setting the types of variables, and the types of variables are recorded in an environment, the semantics of declarations involves changing the environment. Before giving the clause for declarations, we define the
notation η[x → σ ] for an environment that is similar to η, except that it must
map the variable x to type σ . More precisely,
η[x → σ ] = λy ∈ Variables. if y = x then σ else η(y).
Using this notation, we can define the semantics of declarations by
V[[let x :τ = e1 in e2 ]]η

 V[[e2 ]](η[x → integer])
= V[[e2 ]](η[x → boolean])

type error
if V[[e1 ]]η = integer and τ = int
if V[[e1 ]]η = boolean and τ = bool.
otherwise
The clause for conditional is
V[[i f e1 then e2 else e3 ]]η

if V[[e1 ]]η = V[[e2 ]]η = V[[e3 ]]η = boolean
 boolean
= integer
if V[[e1 ]]η = boolean and V[[e2 ]]η = V[[e3 ]]η = integer.

type error otherwise
For example, with η0 = λy ∈ Var. type error,
V[[if true then 0 + 1 else x + 1]]η0 = type error
as the expression x + 1 produces a type error if evaluating x produces a type error.
On the other hand,
V[[let x :int = (1+1) in (if true then 0 else x )]]η0 = integer
as the let expression declares that x is an integer variable.
Questions:
(a) Show how to calculate the meaning of the expression if false then 0 else 1 in
the environment η0 = λy ∈ Var. type error.
Exercises
(b) Suppose e1 and e2 are expressions with V[[e1 ]]η = integer and V[[e2 ]]η = boolean
in every environment. Show how to calculate the meaning of the expression
let x :int=e1 in (if e2 then e1 else x ) in environment η0 = λy ∈ Var. type error.
(c) In declaration let x :τ =e in e, the type of x is given explicitly. It is also possible
to leave the type out of the declaration and infer it from context. Write a
semantic clause for the alternative form, let x =e1 in e2 by using the following
partial solution (i.e., fill in the missing parts of this definition):
V[[let x =e1 in e2 ]]η =
V[[e2 ]](η[x → σ ])
if
type error
otherwise
.
4.11 Lazy Evaluation and Parallelism
In a “lazy” language, we evaluate a function call f(e) by passing the unevaluated
argument to the function body. If the value of the argument is needed, then it is
evaluated as part of the evaluation of the body of f. For example, consider the
function g defined by
fun g(x,y) = if = x=0
then 1
else if x+y=0
then 2
else 3;
In a lazy language, we evaluate the call g(3,4+2) by passing some representation of
the expressions 3 and 4+2 to g. We evaluate the test x=0 by using the argument 3. If it
were true, the function would return 1 without ever computing 4+2. Because the test
is false, the function must evaluate x+y, which now causes the actual parameter 4+2
to be evaluated. Some examples of lazy functional languages are Miranda, Haskell,
and Lazy ML; these languages do not have assignment or other imperative features
with side effects.
If we are working in a pure functional language without side effects, then for any
function call f(e1 , e2 ), we can evaluate e1 before e2 or e2 before e1 . Because neither
can have side effects, neither can affect the value of the other. However, if the
language is lazy, we might not need to evaluate both of these expression. Therefore,
something can go wrong if we evaluate both expressions and one of them does not
terminate.
As Backus argues in his Turing Award lecture, an advantage of pure functional
languages is the possibility of parallel evaluation. For example, in evaluating a
function call f(e1 , e2 ) we can evaluate both e1 and e2 in parallel. In fact, we could
even start evaluating the body of f in parallel as well.
(a) Assume we evaluate g(e1 , e2 ) by starting to evaluate g, e1 , and e2 in parallel,
where g is the function defined above. Is it possible that one process will have
to wait for another to complete? How can this happen?
(b) Now, suppose the value of e1 is zero and evaluation of e2 terminates with an
error. In the normal (i.e., eager) evaluation order that is used in C and other
common languages, evaluation of the expression g(e1 , e2 ) will terminate in
error. What will happen with lazy evaluation? Parallel evaluation?
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Fundamentals
(c) Suppose you want the same value, for every expression, as lazy evaluation,
but you want to evaluate expressions in parallel to take advantage of your
new pocket-sized multiprocessor. What actions should happen, if you evaluate
g(e1 , e2 ) by starting g, e1 , and e2 in parallel, if the value of e1 is zero and evaluation
of e2 terminates in an error?
(d) Suppose now that the language contains side effects. What if e1 is z and e2
contains an assignment to z; can you still evaluate the arguments of g(e1 , e2 ) in
parallel? How? Or why not?
4.12 Single-Assignment Languages
A number of so-called single-assignment languages have been developed over the
years, many designed for parallel scientific computing. Single-assignment conditions are also used in program optimization and in hardware description languages.
Single-assignment conditions arise in hardware as only one assignment to each variable may occur per clock cycle.
One example of a single-assignment language is SISAL, which stands for
streams and iteration in a single-assignment language. Another is SAC, or singleassignment C. Programs in single-assignment languages must satisfy the following
condition:
Single-Assignment Condition: During any run of the program, each variable
may be assigned a value only once, within the scope of the variable.
The following program fragment satisfies this condition,
if (y>3) then x = 42+29/3 else x = 13.39;
because only one branch of the if-then-else will be executed on any run of the
program. The program x=2; loop forever; x=3 also satisfies the condition because no
execution will complete both assignments.
Single-assignment languages often have specialized loop constructs, as otherwise it would be impossible to execute an assignment inside a loop body that gets
executed more than once. Here is one form, from SISAL:
for range
body
returns returns clause
end for
An example illustrating this form is the following loop, which computes the dot (or
inner) product of two vectors:
for i in 1, size
elt prod := x[i]∗y[i]
returns value of sum elt prod
end for
This loop is parallelizable because different products x[i]∗y[i] can be computed in
parallel. A typical SISAL program has a sequential outer loop containing a set of
parallel loops.
Suppose you have the job of building a parallelizing compiler for a singleassignment language. Assume that the programs you compile satisfy the singleassignment condition and do not contain any explicit process fork or other parallelizing instructions. Your implementation must find parts of programs that can
Exercises
be safely executed in parallel, producing the same output values as if the program
were executed sequentially on a single processor.
Assume for simplicity that every variable is assigned a value before the value of
the variable is used an in expression. Also assume that there is no potential source
of side effects in the language other than assignment.
(a) Explain how you might execute parts of the sample program
x = 5;
y = f(g(x),h(x));
if y==5 then z=g(x) else z=h(x);
in parallel. More specifically, assume that your implementation will schedule
the following processes in some way:
process 1 - set x to 5
process 2 - call g(x)
process 3 - call h(x)
process 4 - call f(g(x),h(x)) and set y to this value
process 5 - test y==5
process 6 - call g(x) and then set z=g(x)
process 7 - call h(x) and then set z=h(x)
For each process, list the processes that this process must wait for and list the
processes that can be executed in parallel with it. For simplicity, assume that a
call cannot be executed until the parameters have been evaluated and assume
that processes 6 and 7 are not divided into smaller processes that execute the
calls but do not assign to z. Assume that parameter passing in the example
code is by value.
(b) If you further divide process 6 into two processes, one that calls g(x) and one that
assigns to z, and similarly divide process 7 into two processes, can you execute
the calls g(x) and h(x) in parallel? Could your compiler correctly eliminate
these calls from processes 6 and 7? Explain briefly.
(c) Would the parallel execution of processes you describe in parts (a) and (b), if
any, be correct if the program does not satisfy the single-assignment condition?
Explain briefly.
(d) Is the single-assignment condition decidable? Specifically, given an program
written in a subset of C, for concreteness, is it possible for a compiler to decide
whether this program satisfies the single-assignment condition? Explain why
or why not. If not, can you think of a decidable condition that implies the singleassignment condition and allows many useful single-assignment programs to
be recognized?
(e) Suppose a single-assignment language has no side-effect operations other than
assignment. Does this language pass the declarative language test? Explain why
or why not.
4.13 Functional and Imperative Programs
Many more lines of code are written in imperative languages than in functional
ones. This question asks you to speculate about reasons for this. First, however, an
explanation of what the reasons are not is given:
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It is not because imperative languages can express programs that are not possible
in functional languages. Both classes can be made Turing complete, and indeed a
denotational sematics of an imperative language can be used to translate it into
a functional language.
It is not because of syntax. There is no reason why a functional language could
not have a syntax similar to that of C, for example.
It is not because imperative languages are always compiled whereas functional
languages are always interpreted. Basic is imperative, but is usually interpreted,
whereas Haskell is functional and usually compiled.
For this problem, consider general properties of imperative and functional languages. Assume that a functional language supports higher-order functions and
garbage collection, but not assignment. For the purpose of this question, an imperative language is a language that supports assignment, but not higher-order
functions or garbage collection. Use only these assumptions about imperative and
functional languages in your answer.
(a) Are there inherent reasons why imperative languages are superior to functional
ones for the majority of programming tasks? Why?
(b) Which variety (imperative or functional) is easier to implement on machines
with limited disk and memory sizes? Why?
(c) Which variety (imperative or functional) would require bigger executables
when compiled? Why?
(d) What consequence might these facts have had in the early days of computing?
(e) Are these concerns still valid today?
4.14 Functional Languages and Concurrency
It can be difficult to write programs that run on several processors concurrently
because a task must be decomposed into independent subtasks and the results of
subtasks often must be combined at certain points in the computation. Over the
past 20 years, many researchers have tried to develop programming languages that
would make it easier to write concurrent programs. In his Turing Lecture, Backus
advocated functional programming because he believed functional programs would
be easier to reason about and because he believed that functional programs could
be executed efficiently in parallel.
Explain why functional programming languages do not provide a complete
solution to the problem of writing programs that can be executed efficiently in
parallel. Include two specific reasons in your answer.
PART 2
Procedures, Types, Memory
Management, and Control
5
The Algol Family and ML
The Algol-like programming languages evolved in parallel with the Lisp family of
languages, beginning with Algol 58 and Algol 60 in the late 1950s. The most prominent
Algol-like programming languages are Pascal and C, although C differs from most
of the Algol-like languages in some significant ways.
In this chapter, we look at some of the historically important languages from the
Algol family, including Algol 60, Pascal, and C. Because many of the central features
of the Algol family are used in ML, we then use the ML programming language to
discuss some important concepts in more detail. The ML section of this chapter is
also a useful reference for later chapters that use ML examples to illustrate concepts
that are not found in C.
There are many Algol-related languages that we do not have time to cover, such
as Algol 58, Algol W, Euclid, EL1, Mesa, Modula-2, Oberon, and Modula-3. We will
discuss Modula and modules in Chapter 9.
5.1 THE ALGOL FAMILY OF PROGRAMMING LANGUAGES
A number of important language ideas were developed in the Algol family, which
began with work on Algol 58 and Algol 60 in the late 1950s. The Algol family developed in parallel with Lisp languages and led to the late development of ML and
Modula.
The main characteristics of the Algol family are the familiar colon-separated
sequence of statements used in most languages today, block structure, functions and
procedures, and static typing.
5.1.1 Algol 60
Algol 60 was designed between 1958 and 1963 by a committee that included many
important computer pioneers, such as John Backus (designer of Fortran), John
McCarthy (designer of Lisp), and Alan Perlis. Algol 60 was intended to be a generalpurpose language, which at the time meant there was emphasis on scientific and
numerical applications. Compared with Fortran, Algol 60 provided better ways to
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ROBIN MILNER
A thoughtful, engaging, and optimistic person, Robin Milner has had a
profound effect on several areas of computer science and on computing
in the United Kingdom in general. Always looking for new insight and
open to new ideas, Robin is an unassuming but forceful presence at any
discussion over coffee after a conference talk or workshop presentation.
Milner was awarded the 1991 ACM Turing Award for “several distinct and complete achievements: LCF, probably the first theoretically
based yet practical tool for machine-assisted proof construction; ML,
the first language to include polymorphic type inference and a type-safe
exception-handling mechanism; CCS, a general theory of concurrency;
and full abstraction, the study of the relationship between operational
and denotational semantics.”
After approximately 20 years in Edinburgh, Robin returned to
Cambridge University, where he held a Chair in Computer Science and
was Head of the Computer Laboratory until his retirement in 1999.
represent data structures and, like LISP, allowed functions to be called recursively.
Until the development of Pascal, Algol 60 was the academic standard for describing
complex algorithms in scientific and engineering publications.
The following characteristics are some important features of Algol 60:
simple statement-oriented syntax, involving colon-separated sequences of statements
blocks indicated by begin . . . end (corresponding to curly braces {. . .} in C)
recursive functions and stack storage allocation
5.1 The Algol Family of Programming Languages
Lisp
Algol 60
Algol 68
Pascal
ML
Modula
fewer ad hoc restrictions than previous languages. For example,
general expressions inside array indices
procedures that could be called with procedure parameters
a primitive static type system, later improved on in Algol 68 and Pascal.
Here is an example program (subsequently explained) that will give you some
feel for Algol 60 syntax:
real procedure average(A,n);
real array A; integer n;
begin
real sum; sum := 0;
for i = 1 step 1 until n do
sum := sum + A[i];
average := sum/n
end;
In Algol, the two-character sequence := is used for assignment, and the single character = for test for equality. In this program, note that the types of the parameters
to the procedure (function) are declared in the line following the procedure name.
Although A is declared to be a real array, no array bounds are given as part of the
declaration. The return value of the procedure is given when a value is assigned to
the name of the procedure. (This is the assignment to average in the last line.) An
irritating syntactic peculiarity of Algol 60 is the way that semicolons must be (and
must not be) used between statements. In particular, it would be a syntactic error to
place a semicolon after the last assignment and before the keyword end. There is a
systematic explanation for why semicolons are needed some places and not others
in Algol 60, but the systematic reason is hard for programmers to learn, remember,
and apply when programs are edited.
There were a number of trouble spots in Algol 60 that motivated computer scientists to develop better programming languages:
The Algol 60 type discipline had some shortcomings. There are two examples
that are illustrated in more detail in the exercises:
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The type of a procedure parameter to a procedure does not include the types
of parameters.
An array parameter to a procedure is given type array, without array bounds.
Algol 60 was designed around two parameter-passing mechanisms, pass-by-value
and pass-by-name:
Pass-by-name interacts badly with side effects.
Pass-by-value is expensive for arrays.
There are some awkward issues related to control flow, such as memory management, when a program jumps out of a nested block.
Pass-by-Name. Perhaps the strangest feature of Algol 60, in retrospect, is the use
of pass-by-name. In pass-by-name, the result of a procedure call is the same as if
the formal parameter were substituted into the body of the procedure. This rule for
defining the result of a procedure call by copying the procedure and substituting for
the formal parameters is called the Algol 60 copy rule. Although the copy rule works
well for pure functional programs, as illustrated by β reduction in lambda calculus,
the interaction with side effects to the formal parameter are a bit strange. Here is
an example program showing a technique referred to as Jensen’s device: passing an
expression and a variable it contains to a procedure so that the procedure can use
one parameter to change the location referred to by the other:
begin integer i;
integer procedure sum(i, j);
integer i, j;
comment parameters passed by name;
begin integer sm; sm := 0;
for i := 1 step 1 until 100 do sm := sm + j;
sum := sm
end;
print(sum(i, i*10 ))
end
In this program, the procedure sum(i,j) adds up the values of j as i goes from 1 to
100. If you look at the code, you will realize that the procedure makes no sense
unless changes to i cause some change in the value of j; otherwise, the procedure
just computes 100∗j. In the call sum(i, i10) shown here, the for loop in the body of
procedure sum adds up the value of i10 as i goes from 1 to 100.
BNF. An important by-product of the Algol 60 design effort was the invention
of Backus Normal Form (or BNF), which was used in the Algol 60 report to define
the well-formed programs of the language. BNF, summarized in Subsection 4.1.2,
remains the standard notation for describing the syntax of programming languages.
Although Algol 60 was very influential and commonly used in academic circles and
in Europe, it was not a commercial success in the United States.
5.1 The Algol Family of Programming Languages
5.1.2 Algol 68
Algol 68 was intended to remove some of the difficulties found in Algol 60 and
to improve the expressiveness of the language. However, in the end, the Algol 68
committee produced a design that was more problematic than that of Algol 60.
One main problem is that, although programming in Algol 68 appears no more
difficult than that of Algol 60, some features of Algol 68 made it difficult to compile
efficiently. One source of difficulty was the combination of procedure parameters
and procedure return values, which was not well understood at the time. (We will
discuss the implementation consequences of higher-order functions in Section 7.4.)
Another reason that Algol 68 was not entirely successful was that the authors chose
to define entirely new terminology for the language and its documentation. This
made it difficult for programmers to move from Algol 60 to Algol 68.
One contribution of Algol 68 was its regular, systematic type system. For some
reason, the Algol 68 designers chose to call types modes. The modes of Algol 68 are
either primitive or compound modes. The primitive modes included int, real, char,
bool, string, complex, bits, bytes, semaphore, format (for input and output), and file.
The compound modes include modes formed with the forms array, structure,
procedure, set, and pointer.
These type constructions can be combined without restriction, so that a programmer can build an array of pointers to procedures, for example. The decision to allow
unrestricted combinations of the mode constructors made the type system seem more
systematic than previous languages.
Some other advances in Algol 68 were in the areas of memory management
and parameter passing. (The general concepts of memory management, parameter passing, and related issues are covered in Chapter 7.) Algol 68 memory management involves a stack for local variables and heap storage for data that are
intended to live beyond the current function call. As in C, Algol 68 data on the
heap are explicitly allocated, but unlike C, heap data are reclaimed by garbage collection. This combination of explicit allocation and garbage collection carried over
into Pascal. Algol 68 parameter passing is pass-by-value, with pass-by-reference accomplished by pointer types. This is essentially the same design as that adopted in
C some years later. The decision to allow independent constructs to be combined
without restriction also led to some complex features, such as assignable procedure
variables.
5.1.3 Pascal
Pascal was design in the 1970s by Niklaus Wirth, who used the data-structuring
ideas advanced by C.A.R. (Tony) Hoare. Wirth first designed and implemented a
language called Algol W and then refined the design of Algol W to produce Pascal.
Wirth designed Pascal around a series of teaching exercises, enumerated in his book
Algorithms + Data Structures = Programs (Prentice-Hall, 1975). He also designed
the language so that it could have a simple one-pass compiler.
Pascal was significantly simpler than Algol 68 and achieved more widespread
acceptance than either Algol 60 or Algol 68. Although the use of Pascal declined
in the 1990s, Pascal was one of the most widely used programming languages over
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The Algol Family and ML
approximately a 20-year period. Pascal was very successful as a programming language for teaching, in part because it was designed explicitly for this purpose. Pascal
was also used for a significant number of production programming projects, including
operating systems and applications for the Apple Macintosh.
The Pascal type system is more expressive than the Algol 60 type system. The type
system also repairs some of the Algol 60 type loopholes, such as Algol 60 procedure
types that do not include the types of parameters. The Pascal type system is also
simpler and more limited than the Algol 68 type system, eliminating some of the
compilation difficulties of Algol 68.
An important contribution of the Pascal type system is the rich set of datastructuring concepts. These include records (similar to C structs), variant records
(a form of union type), and subranges. For example, the subrange [1 .. 10] of integers
between 1 and 10 became a type for the first time in Pascal. A restriction that made
Pascal simpler than Algol 68 was that procedure parameters could not be procedures with procedure parameters. More specifically, in Pascal syntax, procedures of
the form
procedure DoSomething( j, k : integer );
procedure DoSomething( procedure P(i:integer); j,k, : integer);
are allowed. The first is a procedure with integer parameters, and the second is a
procedure whose parameter is a procedure with integer parameters. However, a
procedure of the form
procedure NotAllowed( procedure MyProc( procedure P(i:integer)));
with a procedure parameter that has a procedure parameter, is not allowed.
One place where Wirth’s focus on teaching and on systematic language design
caused a small problem was in the typing of array parameters. In Pascal, the type of
an array has the form
array indexType of entryType
where indexType is often a subrange type. The important detail here is that the index
type is part of the type of an array, and two array types are equal only if they have
the same index type and entry type. This definition of array type allows a procedure
declaration such as
procedure p(a : array [1..10] of integer)
5.2 The Development of C
where the argument to the procedure is an array of some array type, but does not
allow
procedure p(n: integer, a : array [1..n] of integer)
as array [1..n] of integer is not considered a legal type in Pascal. A procedure of the
form
procedure p(a : array [1..10] of integer)
may be called only with an array of length 10 as an actual parameter. This is an
unfortunate limitation. If we want to write a sort procedure, for example, then we
will have to write the procedure out for sorting arrays of some fixed length. If we
want to sort arrays of several different lengths, then Pascal (as originally designed)
would make it necessary to copy over the procedure several times, changing the array
length in each copy.
Not only is this awkward, it is also unnecessary because the memory associated
with an array is already allocated before it is passed to any procedure. Therefore there
is no reason, other than type checking, for a procedure to allow array parameters
of only a fixed length. Because this aspect of the Pascal design was awkward and
unnecessary, later versions of Pascal have this limitation removed.
5.1.4 Modula
The Modula programming language is a descendent of Pascal, developed by Pascal
designer Niklaus Wirth in Switzerland in the late 1970s. The main innovation of
Modula over Pascal is a module system, used for grouping sets of related declarations into program units. Some examples of Modula-2, a successful version
of the language, appear in Subsection 9.3.1 as part of the discussion of program
modules.
5.2 THE DEVELOPMENT OF C
Although Pascal was a successful academic and teaching language, C eventually
eclipsed Pascal as a production programming language. There are several reasons
for the success of C. One reason, which is unrelated to the design of the language
itself, is the popularity of the Unix operating system, which was written in C. When
programs are written to run under Unix, all of the basic system calls are immediately
available in C. Therefore, it is easier to write many Unix applications in C than in
other programming languages. Another reason for the popularity of C is that it
has a distinctive memory model that is close to the underlying hardware. Although
C has many of the same concepts as Pascal, C is less rigid in its enforcement of
basic principles and restrictions. Many C programmers like the resulting flexibility
of C.
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The Algol Family and ML
C was originally designed and implemented from 1969 to 1973, as part of the Unix
operating system project at Bell Laboratories. C was designed by Dennis Ritchie, one
of the original designers of Unix, so that he and Ken Thompson could build Unix
in a language that they liked. The design evolved from Ritchie’s and Thompson’s B
language (hence the name C, the next letter in the alphabet), which was in turn based
on a language called BCPL. Significant changes in C occurred from 1977 to 1979, as
part of a push to achieve portability of the Unix system, and in the mid-1980s when
committee of the American National Standards Institute (ANSI) standardized the
language. BCPL was a systems programming language designed in the 1960s and
used by Bell Laboratories members of the Multics operating system project. B was
a pared-down version of BCPL, designed to run on the small (PDP) computer used
by the Unix project. The main difference between B and C is that B was untyped
whereas the C language has types and type-checking rules.
One characteristic that distinguishes C from other popular languages is the treatment of memory locations, arrays, and pointers. This part of C is inherited from
BCPL and B. The only data type in B is the “word,” or “cell,” a fixed-length bit
pattern. Memory in BCPL and B is presented to the programmer as a linear array
of words. Pointers are treated as integer indices into this array, and programmerdeclared arrays are treated as contiguous words drawn from the larger array of all
memory words. This view has several consequences. One is that arrays and pointers
are largely equivalent, as described below. Another consequence is that, because
pointers are integer indices in the memory array, pointer arithmetic is considered
meaningful: If p is the address of a memory location, then p+1 is the address of the
next location.
C Arrays and Pointers
In C, pointers and arrays are declared differently, as if pointers and arrays are different types of values. For example, the following code declares a pointer p to an integer
location and an array A of integers:
int * p;
int A[5];
In most languages, there is some operation for dereferencing a pointer. Dereferencing
is the operation that returns the location pointed to by the pointer. In most languages
with arrays, there is an indexing operation that can be used to find one of the locations
within the array. There are some similarities between these two operations, but most
typed languages (including others in the Algol family) would consider dereferencing
an array name or indexing a pointer illegal. In C, however, arrays are effectively
treated as pointers. To quote Dennis Ritchie’s 1975 C Reference Manual,
Every time an identifier of array type appears in an expression, it is converted into
a pointer to the first member of the array. . . . By definition, the subscript operator
[ ] is interpreted in such a way that “E1[E2]” is identical to “∗ ((E1)+(E2)).” Because
of the conversion rules which apply to +, if E1 is an array and E2 is an integer, then
E1[E2] refers to the E2-th member of E1.
5.3 The LCF System and ML
There is no other programming language in widespread use that allows pointer arithmetic in this way.
Critique
An important feature of C has been the tolerance of C compilers to type errors. This
is partly because C evolved from typeless languages. Ritchie had to adapt existing
programs as the language developed and to make an allowance for existing code in
a typeless language. As C evolved further and was later standardized by an ANSI
committee, backward compatibility with the then-existing C code also prevented
strong typing restrictions. Although some C programmers have liked the ability to
write and compile programs with type errors, most C programmers have eventually
come to consider the weak type checking of many C compilers to be a disadvantage.
In fact, one of the most commonly cited advantages of C++ over C is the fact that
C++ provides better type checking.
As Dennis Ritchie says in The Development of the C Language (ACM Second
History of Programming Languages conference, ACM 1993), “C is quirky, flawed,
and an enormous success.” Although accidents of history surely helped, C evidently
satisfied a need for a system implementation language efficient enough to displace
assembly language, yet sufficiently abstract and fluent to describe algorithms and
interactions in a wide variety of environments.
An interesting discussion may be found in Kernighan’s “Why Pascal is not my
favorite programming language,” Bell Labs CSTR 100, July 1981. If you are interested
in doing extra reading, you can find this document on the web.
5.3 THE LCF SYSTEM AND ML
ML might be called a mostly functional language with imperative features or perhaps
a function-oriented imperative language. ML has very flexible function features, similar to Lisp, allowing functions to be created in-line as parts of expressions, passed as
arguments to functions, and returned as function results. At the same time, it is possible to write imperative Algol-like programs in a syntax that resembles that of the
Algol family with approximately the same degree of ease as for modern descendants
of Algol. ML also has concurrent extensions, making it suitable for developing concurrent systems, and an object-oriented extension. However, our main use of ML in
this part of the book is to examine concepts common to Algol-like languages, Lisplike languages, and concurrent and object-oriented extensions of these languages.
Therefore, we focus primarily on the core fragment of ML.
The following list enumerates our main reasons for looking at ML in some detail:
ML illustrates most of the important concepts of the Lisp/Algol families of
languages.
Type systems have been an important part of programming language design from
1960 to the present day, and the ML type system is often considered the cleanest
and most expressive type system to date.
Because most readers are familiar with C and many have not written a lot of
programs in significantly different languages, it is useful to have a language other
than C to use for examples in the following chapters.
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ML allows higher-order functions and other constructions that are discussed in
the following chapters.
One distinguishing feature of ML is its type system, which extends the successful
Pascal type system in a number of ways. Unlike C, which has numerous loopholes,
the ML type system is sound in a precise mathematical sense. Specifically, if the
type checker determines that an expression has a certain type, then any terminating
evaluation of that expression is guaranteed to produce a legitimate value of that type.
For example, if an expression has a type such as “pointer to string,” then the value
of that expression is guaranteed to be a pointer to allocated memory that contains
a string. It cannot be a dangling pointer to a location that has been deallocated or
used to store some value other than a string.
Before ML, programming languages with sound type systems were generally considered unpleasantly restrictive. Many C programmers have considered it important
to “break” the type system in various ways (confusing integers and pointers, for example), and Lisp fans have valued their freedom from static typing. However, the
ML type system is unobtrusive, as many type declarations are automatically deduced
by the compiler, and flexible, as the type system allows an expression to have many
possible types. We will explore these aspects of the ML type system in more detail in
Chapter 6.
Most successful programming languages were originally designed for a single
application or a set of closely related programming tasks. The ML programming
language was designed by Robin Milner and his associates as part of the LCF project.
The LCF project, aimed at developing a Logic for Computable Functions, drew
inspiration from a set of logical principles outlined by Dana Scott. Robin Milner’s
goal was to build a system that would make it practical to prove interesting properties
of functional programs in an automated or semiautomated manner. His LCF project
started at Stanford in 1970 and continued at Edinburgh through the 1980s, making
substantial progress toward this goal and stimulating a number of related efforts in
the process.
ML was designed as the Meta-Language (hence its name) of the LCF System. Its
original purpose was for writing programs that would attempt to construct mathematical proofs. As any reader who has developed mathematical proofs will know, this
can be a very difficult task. In many cases, it is necessary to try a number of methods
for finding proofs. A fundamental concept in the LCF system is that of proof tactic.
A proof tactic is a function that, given a formula making some assertion, tries to
find a proof of the formula. Because a tactic may search indefinitely or reach some
situation in which it is clear that no further search is likely to produce a proof, there
are three possible results of applying a tactic to a formula:

 succeed and return proof
tactic(formula) = search forever
.

fail
We may use the concept of tactic to understand some basic properties of the ML
programming language. Because the goal of LCF is to find correct proofs, a programming language mechanism that ensures correctness, in whole or in part, might
improve the LCF system. An idea that was adopted in LCF was to try to use a type
5.4 The ML Programming Language
system to distinguish successful proofs from unsuccessful ones. In particular, there
was a type proof, with the intent that values of type proof are correct proofs, not
incorrect ones.
Once we have a type of correct proofs, a problem arises. If a tactic fails to find a
proof, what should the function do? More specifically, because a tactic is a function
from formulas to proofs, the type of a tactic would be a function type:
tactic : formula → proof
However, this type seems to say that if a tactic is applied to a function, the result will
be a proof. However, what if the formula is not a correct statement and therefore has
no proof? The solution was to develop an exception mechanism and allow a tactic to
raise an exception if the computation determines that no proof will be found. From
this inspiration, Milner developed the first type-safe exception mechanism, one of the
accomplishments that led to his Turing Award in 1991. Allowing for the possibility
of exceptions, a function f that maps A to B, written as
f:A→B
in ML, means that, for all x in A, if f (x) terminates normally without raising an
exception, then f (x) is in B.
Thus type correctness and exceptions, two basic concepts in ML, arose naturally
as the result of the intended application of ML.
Another emphasis of ML is the use of higher-order functions. This can also be
attributed to the interest in defining complex proof-search tactics: Because a tactic
is a function, a method for combining tactics into a proof-search strategy is a function from functions to functions. For example, here is the outline of a function that
combines two tactics according to an if-then-else strategy:
f(tactic1 , tactic2 ) =
λformula. try tactic1 (formula)
else tactic2 (formula)
Given two tactics tactic1 and tactic2 , the function f returns a tactic that, given a
formula, first tries to prove the formula by using tactic1 and then uses tactic2 if tactic1
fails.
5.4 THE ML PROGRAMMING LANGUAGE
Because ML is used in the next few chapters of the book to illustrate properties
of programming languages, we will study ML in a little more detail than we will
study some other languages. The version of ML that we will use is called Standard
ML 97 (SML97). Compilers for SML97 are available on the Internet without charge.
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Several books and manuals covering the language are available. In addition to online sources easily located by web search, Ullman’s Elements of ML Programming
(Prentice-Hall, 1994) is a good reference.
5.4.1 Interactive Sessions and the Run-Time System
Most ML compilers are based on the same kind of read-eval-print loop as many Lisp
implementations. The standard way of interacting with the ML system is to enter
expressions and declarations one at a time. As each is entered, the source code is
type checked, compiled, and executed. Once an identifier has been given a value by
a declaration, that identifier can be used in subsequent expressions.
Expressions
For expressions, user interaction with the ML compiler has the form
---- <expression>;
val it = <print value> : <type>
where “—” is the prompt for user input and the line below is output from the
ML compiler and run-time system. The preceding lines show that if you type in
an expression, the compiler will compile the expression and evaluate it. The output is a bit cryptic: it is a special identifier bound to the value of the last expression entered, so it = <print value> : <type> means that the value of the expression is
<print value> and this is a value of type <type>. It is probably easier to understand
the idea from a few examples. Here are four lines of input and the resulting compiler
output:
---- (5+3) -2;
val it = 6 : int
---- it + 3;
val it = 9 : int
---- if true then 1 else 5;
val it = 1 : int
---- (5 = 4);
val it = false : bool
In words, the value of the first expression is the integer 6. The second expression adds 3
to the value it of the previous expression, giving integer value 9. The third expression
is an if-then-else, which evaluates to the integer 1, and the fourth expression is a
Boolean-valued expression (comparison for equality) with value false.
Each expression is parsed, type checked, compiled, and executed before the next
input is read. If an expression does not parse correctly or does not pass the typechecking phase of the compiler, no code is generated and no code is executed. The
5.4 The ML Programming Language
ill-typed expression
if true then 3 else false;
for example, parses correctly because this has the correct form for an if-then-else.
However, the type checker rejects this expression because the ML type checker
requires the then and else parts of an if-then-else expression to have the same types,
as described in the next subsection. The compiler output for this expression includes
the error message
stdIn:1.1-29.10 Error: types of rules don’t agree [literal]
indicating a type mismatch. A full discussion of ML type checking appears in Chapter 6.
Declarations
User input can be an expression or a declaration. The standard form for ML declarations, followed by compiler output, is
---- val <identifier> = <expression>;
val <identifier> = <print value> : <type>
The keyword val stands for value. When a declaration is given to the compiler, the
value associated with the identifier is computed and bound to that identifier. Because
the value of the expression used in a declaration has a name, the compiler output
uses this name instead of it for the value of the expression. Here are some examples:
---- val x=7+2;
val x = 9 : int
---- val y = x+3;
val y = 12 : int
---- val z = x*y - (x div y);
val z = 108 : int
In words, the first declaration binds the integer value 9 to the identifier x. The second
declaration refers to the value of x from the first declaration and binds the value 12
to the identifier y. The third declaration refers to both of the previous declarations
and binds the integer value 108 to the identifier z.
You might note that integer division in ML is written as div instead of /. There are
a few syntactic peculiarities of ML like this, especially when it comes to integer and
real-number arithmetic. As you will see from the discussion of ML type inference in
Chapter 6, it is useful for the compiler to distinguish operations of different types.
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In ML, / is used for real-number (floating-point) division, and there is no automatic
conversion from integer to real. Therefore x/y would not have been syntactically
well formed if we had written this instead of x div y in the declaration of z.
Functions can be declared with the keyword fun (which stands for function)
instead of val. The general form of user input and compiler output is
---- fun <identifier> <arguments> = <expression>;
val <identifier> = fn <arg type> → <result type>
This declares a function whose name is <identifier>. The argument type is determined by the form of <arguments> and the result type is determined by the form of
<expression>.
Here is an example:
---- fun f(x) = x + 5;
val f = fn : int → int
This declaration binds a function value to the identifier f. The value of f is a function
from integers to integers. The same function can be declared by a val declaration,
written as
---- val f = fn x => x+5;
val f = fn : int → int
In this declaration, the identifier f is given the value of expression fn x => x+5, which
is a function expression like (lambda (x) (+ x 5)) in Lisp or λx. x + 5 in lambda
calculus. We will discuss function declarations and function expressions further in
Subsection 5.4.3.
The system prints fn for the value of f because the value of a function is not
printable. One reason why function values are not printed is that most functions are
infinite values in principle – an integer function has infinitely many possible results,
one for each possible integer argument. After a function is declared, the compiler
stores compiled code for the function. It would be possible to print the compiled
code, but this is not in ML. It is in some other target low-level language, and printing
it would not usually be useful to a programmer.
Identifiers vs. Variables. An important aspect of ML is that the value of an identifier cannot be changed by assignment. More specifically, if an identifier x is declared
by val x = 3, for example, then the value of x will always be 3. It is not possible
to change the value of x by assignment. In other words, ML declarations introduce
constants, not variables. The way to declare an assignable variable in ML is to define
a reference cell, which is similar to a cons cell in Lisp, except that reference cells do
not come in pairs. References and assignment are explained in Subsection 5.4.5.
Although most readers will initially think otherwise, the ML treatment of identifiers and variables is more uniform than the treatment of identifiers and variables in
5.4 The ML Programming Language
languages such as Pascal and C. If an integer identifier is declared in C or Pascal, it
is treated as an assignable variable. On the other hand, if a function is declared and
given a name in either of these languages, the name of the function is a constant, not
a variable. It is not possible to assign to the function name and change it to a different
function. Thus, Pascal and C choose between variables and constants according to
the type of the value given to the identifier. In ML, a val declaration works the same
way for all types of values.
5.4.2 Basic Types and Type Constructors
The core expression, declaration, and statement parts of ML are best summarized
by a list of the basic types along with the expression forms associated with each type.
Unit
The type unit has only one element, written as empty parentheses:
( ) : unit
Like void in C, unit is used as the result type for functions that are executed only
for side effects. The type unit is also used as the type of argument for functions
that have no arguments. C programmers may be confused by the fact that unit suggests one element, whereas void seems to mean no elements. From a mathematical
point of view, “one element” is correct. In particular, if a function is supposed to
return an element of an empty type, then that function cannot return because the
empty set (or empty type) has no elements. On the other hand, a function that
returns an element of a one-element type can return. However, we do not need
to keep track of what value such a function returns, as there is only one thing
that it could possibly return. The ML type system is based on years of theoretical
study of types; most of the typing concepts in ML have been considered with great
care.
Bool
There are two values of type bool, true and false:
true : bool
false : bool
The most common expression form associated with Booleans is conditional, with
if e1 then e2 else e3
having the same type as e2 and e3 if these have the same type and e1 has type bool.
There is no if-then without else, as a conditional expression must have a value whether
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the test is true or false. For example, an expression
---- val nonsense = if a then 3;
is not legal ML because there is no value for nonsense if the expression a is false.
More specifically, the input if a then 3 does not even parse correctly; there is no parse
tree for this string in the syntax of ML.
There are also ML Boolean operations for and, or, not, and so on. These are similar
to and, or, and not in Pascal or &&, ||, and ! in C, with some minor differences.
Negation is written as not, conjunction (and) is written as andalso and disjunction (or)
is written as orelse.
For example, here is a function that determines whether its two arguments have
the same Boolean value, followed by an expression that calls this function:
---- fun equiv(x,y) = (x andalso y) orelse ((not x) andalso (not y));
val equiv = fn : bool * bool -> bool
---- equiv(true,false);
val it = false : bool
In words, Boolean arguments x and y are the same Boolean value if they are either
both true or both false. The first subexpression, (x andalso y), is true if x and y are
both true and the second subexpression, ((not x) andalso (not y)), is true if they are
both false.
The reason for the long names andalso and orelse is to emphasize evaluation order.
In the expression (a andalso b), where a and b are both expressions, a is evaluated first.
If a is true, then b is evaluated. Otherwise the value of the expression (a andalso b)
is determined to be false without evaluating b. Similarly, b in (a orelse b) is evaluated
only if the value of a is false.
Integers
Many ML integer expressions are written in the usual way, with number constants
and standard arithmetic operations:
0,1,2, . . . ,-1,-2, . . . : int
+, -, *, div : int * int → int
The operator div is a binary infix operator on integers, used as follows:
---- fun quotient(x,y) = x div y;
val quotient = fn : int * int → int
The identifier div by itself is not an expression, though. Similarly, +, -, and ∗ are infix
binary operators.
5.4 The ML Programming Language
Strings
Strings are written as a sequence of symbols between double quotes:
“William Jefferson Clinton” : string
“Boris Yeltsin” : string
String concatenation is written as ∧ , so we have
---- “Chelsey” ∧ “ ” ∧ “Clinton”;
val it = “Chelsey Clinton” : string
Real
The ML type for floating-point numbers is real. For reasons that will be easier to
understand when we come to type inference, ML requires a decimal point in real
constants:
1.0, 2.0, 3.14159, 4.44444, . . . : real
The arithmetic operators +, -, and ∗ may be applied to either integers or real numbers.
Here are some example expressions and the resulting compiler output:
---- 3+4;
val it = 7 : int
---- 4.0 + 5.1;
val it = 9.1 : real
Note that when + has two integer arguments the result is an integer, and when + has
two real arguments the result is a real number. However, it is a type error to combine
integer and real arguments. Here is part of the compiler output for an expression
that adds an integer to a real number:
---- 4+5.1;
stdIn:1.1-1.6 Error: operator and operand don’t agree [literal]
operator domain: int * int
operand:
int * real
This error message is telling us that, because the first argument is an integer, the +
symbol is considered to be integer addition. Therefore, the operator + has domain
int * int, which is ML notation for the type of pairs of integers. However, the operand
is applied to a pair of type int * real, which is ML notation for the type of pairs with
one integer and one real number.
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Conversion from integer to real is done by the explicit conversion function real.
For example, the value of the expression real(3) is 3.0. Conversion from real to
integer can be done with functions floor (round down), ceil (round up), round, and
trunc.
Although arithmetic expressions in ML are a little more cumbersome than in
some other languages, the language is generally usable for most purposes. In part,
explicit typing of numeric constants and explicit conversion is the price to pay for
automatic type inference, a useful feature of ML that is described in Chapter 6.
Tuples
A tuple may be a pair, triple, quadruple, and so on. In ML, tuples may be formed
of any types of values. Tuple values are written with parentheses and tuple types are
written with *. For example, here is the compiler output for a pair, a triple, and a
quadruple:
---- (3,4);
val it = (3,4) : int * int
---- (4,5,true);
val it = (4,5,true) : int * int * bool
---- (“Bob”, “Carol”, “Ted”, “Alice”);
val it = (“Bob”,“Carol”,“Ted”,“Alice”) : string * string * string * string
For all types τ1 and τ2 , the type τ1 ∗ τ2 is the type of pairs whose first component has
type τ1 and whose second component has type τ2 . The type τ1 ∗ τ2 ∗ τ3 is a type of triples,
the type τ1 ∗ τ2 ∗ τ3 ∗ τ4 a type of quadruples, and so on.
Components of a tuple are accessed by functions that name the position of the
desired component. For example, #1, selects the first component of any tuple, #2
selects the second component of any tuple with at least two components, and so on.
Here are some examples:
---- #2(3,4);
val it = 4 : int
---- #3(“John”, “Paul”, “George”, “Ringo”);
val it = “George” : string
Records
Like Pascal records and C structs, ML records are similar to tuples, but with named
components. Record values and record types are written with curly braces, as follows:
— { First name = “Donald”, Last name = “Knuth” };
val it = {First name=“Donald”,Last name=“Knuth”}
: {First name:string, Last name:string}
5.4 The ML Programming Language
The expression here has two components, one called First name and the other called
Last name. The type of this record tells us the type of each component. Record components can be accessed with # functions like tuples, but are named according to the
component names instead of position. Here is one example:
---- #First name({First name=“Donald”, Last name=“Knuth”});
val it = “Donald” : string
Another way of selecting components of tuples and records is by pattern matching,
which is described in Subsection 5.4.3.
Lists
ML lists can have any length, but all elements of a list must have the same type.
We can write lists by listing their elements, separated by commas, between brackets.
Here are some example lists of different types:
---- [1,2,3,4];
val it = [1,2,3,4] : int list
---- [true, false];
val it = [true,false] : bool list
---- [“red”, “yellow”, “blue”];
val it = [“red”,“yellow”,“blue”] : string list
---- [ fn x => x+1, fn x => x+2];
val it = [fn,fn] : (int -> int) list
For short lists, the compiler prints the elements of the list when showing that a list
expression has been evaluated. For longer lists, the last elements are replaced with
an ellipsis (three dots; . . .). As the last list example above shows, it is possible to write
a list of functions.
In general, a τ list is the type of all lists whose elements have type τ .
As in Lisp, the empty list is written nil in ML. List cons is an infix operator written
as a pair of colons:
---- 3 :: nil;
val it = [3] : int list
---- 4 :: 5 :: it;
val it = [4,5,3] : int list
In the first list expression, 3 is “consed” onto the front of the empty list. The result
is a list containing the single element 3. In the second expression, 4 and 5 are consed
onto this list. In both cases, the result is an int list.
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5.4.3 Patterns, Declarations, and Function Expressions
The declarations we have seen so far bind a value to a single identifier. One very
convenient syntactic feature of ML is that declarations can also bind values to a set
of identifiers by using patterns.
Value Declarations
The general form of value declaration associates a value with a pattern. A pattern is
an expression containing variables (such as x, y, z . . .) and constants (such as true,
false, 1, 2, 3 . . .), combined by certain forms such as tupling, record expressions, and
a form of operation called a constructor. The general form of value declaration is
val <pattern> = <exp> ;
where the common forms of patterns are summarized by the following grammar:
<pattern> ::= <id> | <tuple> | <cons> | <record> | <constr>
<tuple> ::= (<pattern>, . . . , <pattern>)
<cons> ::= <pattern>::pattern
<record> ::= {<id>=<pattern>, . . . , <id>=<pattern>}
<constr> ::= <id>(<pattern>, . . . , <pattern>)
In words, a pattern can be an identifier, a tuple pattern, a list cons pattern, a record
pattern, or a declared data-type constructor pattern. A tuple pattern is a sequence of
patterns between parentheses, a list cons pattern is two patterns separated by double
colons, a record pattern is a recordlike expression with each field in the form of a
pattern, and a constructor pattern is an identifier (a declared constructor) applied to
the right number of pattern arguments. This BNF does not define the set of patterns
exactly, as some conditions on patterns are not context free and therefore cannot
be expressed by BNF. For example, the conditions that in a constructor pattern the
identifier must be a declared constructor and that the constructor must be applied
to the right number of pattern arguments are not context free conditions. An additional condition on patterns, subsequently discussed in connection with function
declarations, is that no variable can occur twice in any pattern.
Because a variable is a pattern, a value declaration can simply associate a value
with a variable. For example, here is a declaration that binds a tuple to one identifier,
followed by a declaration that uses a tuple pattern to bind components of the tuple:
---- val t = (1,2,3);
val t = (1,2,3) : int * int * int
---- val (x,y,z) = t;
val x = 1 : int
val y = 2 : int
val z = 3 : int
5.4 The ML Programming Language
Note that there are two lines of input in this example and four lines of compiler
output. In the first declaration, the identifier t is bound to a tuple. In the second
declaration, the tuple pattern (x,y,z) is given the value of t. When the pattern (x,y,z)
is matched against the triple t, identifier x gets value 1, identifier y gets value 2, and
identifier z gets value 3.
Function Declarations
The general form of a function declaration uses patterns. A single-clause definition
has the form
fun f( <pattern> ) = <exp>
and a multiple-clause definition has the form
fun f( <pattern1> ) = <exp1> | . . . | f( <patternn> ) = <expn>
For example, a function adding its arguments can be written as
fun f(x,y) = x + y;
Technically, the formal parameter of this function is a pattern (x, y) that must match
the actual parameter on a call to f. The formal parameter to f is a tuple, which is
broken down by pattern matching into its first and second components. You may
think you are calling a function of two arguments. In reality, you are calling a function
of one argument. That argument happens to be a pair of values. Pattern matching
takes the tuple apart, binding x to what you might think is the first parameter and y
to the second.
An example in which more than one clause is used is the following function, which
computes the length of a list:
---- fun length(nil) = 0
| length( x :: xs ) = 1 + length(xs);
val length = fn : ’a list → int
This code is subsequently explained. The first two lines here are input (the declaration
of function length) and the last line is the compiler output giving the type of this
function. Here is an example application of length and the resulting value:
---- length [“a”, “b”, “c”, “d”];
val it = 4 : int
When the function length is applied to an argument, the clauses are matched in the
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order they are written. If the argument matches the constant nil (i.e., the argument is
the empty list), then the function returns the value 0, as specified by the first clause.
Otherwise the argument is matched against the pattern given in the second clause
(x::xs), and then the code for the second branch is executed. Because type checking
guarantees that length will be applied only to a list, these two clauses cover all values
that could possibly be passed to this function. The type of length, ’a list → int, will
be explained in the next chapter.
In addition to declarations, ML has syntax for anonymous functions. We have
already seen some simple examples. The general form allows the argument to be
given by a pattern:
fn <pattern> => <exp>,
As mentioned briefly in passing in an earlier example, fn <pattern> => <exp> is
like (lambda (<parameters>) (<exp>)) in Lisp. Here is an example, with compiler
output:
---- fn (x,y) => x+y;
val it = fn : int * int → int
The function expressed here takes a pair and adds its two components. The type of
this function is int * int → int, meaning a function that maps a pair of integers to a
single integer.
Here are some more examples illustrating other forms of patterns, each shown
with an associated compiler output:
---- fun f(x, (y,z)) = y;
val f = fn : ’a * (’b * ’c) -> ’b
---- fun g(x::y::z) = x::z;
val g = fn : ’a list -> ’a list
---- fun h {a=x, b=y, c=z} = {d=y, e=z};
val h = fn : {a:’a, b:’b, c:’c} -> {d:’b, e:’c}
The first is a function on nested tuples, the second a function on lists that have at least
two elements, and the third a function on records. The second declaration produces
a compiler warning, as the function g is not defined for lists that have fewer than two
elements.
Pattern matching is applied in order. For example, when the function
fun f (x,0) = x
| f (0,y) = y
| f (x,y) = x+y;
5.4 The ML Programming Language
is applied to an argument (a,b), the first clause is used if b=0, the second clause if b =/ 0
and a=0, and the third clause if b =/ 0 and a =/ 0. The ML type system will keep f from
being applied to any argument that is not a pair (a,b).
An important condition on patterns is that no variable can occur twice in any
pattern. For example, the following function declaration is not syntactically correct
because the identifier x occurs twice in the pattern:
---- fun eq(x,x) = true
| eq(x,y) = false;
stdIn:24.5-25.20 Error: duplicate variable in pattern(s):
This function is not allowed because multiple occurrences of variables express equality, and equality must be written explicitly into the body of a function.
5.4.4 ML Data-Type Declaration
The ML data-type declaration is a special form of type declaration that declares a
type name and operations for building and making use of elements of the type. The
ML data-type declaration has the syntactic form
datatype <type name> = <constructor clause> | . . . | <constructor clause>
where a constructor clause has the form
<constructor clause> ::= <constructor> | <constructor> of <arg types>
The idea is that each constructor clause tells one way to construct elements of the
type. Elements of the type may be “deconstructed” into their constituent parts by
pattern matching. This is illustrated by three examples that show some common ways
of using data-type declarations in ML programs.
Example. An Enumerated Data Type: Types consisting of a finite set of tokens can
be declared as ML data types. Here is a type consisting of three tokens, named to
indicate three specific colors:
---- datatype color = Red | Blue | Green;
datatype color = Blue | Green | Red
The compiler output, which looks just like the ML input code, indicates that the three
elements of type color are Blue, Green, and Red. Technically, values Blue, Green, and
Red are called constructors. They are called constructors because they are the ways
of constructing values with type color.
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Example. A Tagged Union Data Type: ML constructors can be declared so that they
must be applied to arguments when constructing elements of the data type. Constructors do not actually do anything to their arguments, other than to “tag” their
arguments so that values constructed in different ways can be distinguished by pattern matching.
Suppose we are keeping student records, with names of B.S. students, names and
undergraduate institutions of M.S. students, and names and faculty supervisors of
Ph.D. students. Then we could define a type student that allows these three forms of
tuples as follows:
---- datatype student = BS of name | MS of name*school | PhD of name*faculty;
In this data-type declaration, BS, MS, and PhD are each constructors. However, unlike
in the color example, each student constructor must be applied to arguments to
construct a value of type student. We must apply BS to a name, MS to a pair consisting
of a name and a school, and PhD to a pair consisting of a name and a faculty name in
order to produce a value of type student.
In effect, the type student is the union of three types,
student ≈ union {name, name*school, name*faculty }
except that in ML “unions” (which are defined by datatype), each value of the union
is tagged by a constructor that tells which of the constituent types the value comes
from. This is illustrated in the following function, which returns the name of a student:
---- fun name(BS(n)) = n
| name(MS(n,s)) = n
| name(PhD(n,f)) = n;
val name = fn : student → name
The first three lines are the declaration of the function name, and the last line is
the compiler output indicating that name is a function from students to names. The
function has three clauses, one for each form of student.
Example. A Recursive Type: Data-type declaration may be recursive in that the type
name may appear in one or more of the constructor argument types. Because of the
way type recursion is implemented, ML data type provides a convenient, high-level
language construct that hides a common form of routine pointer manipulation.
The set of trees with integer labels at the leaves may be defined mathematically
as follows:
A tree is either
a leaf, with an associated integer label, or
a compound tree, consisting of a left subtree and a right subtree.
5.4 The ML Programming Language
This definition can be expressed as an ML data-type declaration, with each part of
the definition corresponding to a clause of the data-type declaration:
datatype tree = LEAF of int | NODE of (tree * tree);
The identifiers LEAF and NODE are constructors, and the elements of the data type
are all values that can be produced by the application of constructors to legal (typecorrect) arguments. In words, a tree is either the result of applying the constructor
LEAF to an integer (signifying a leaf with that integer label) or the result of applying
the constructor NODE to two trees. These two trees, of course, must be produced
similarly with constructors LEAF and NODE.
The following function shows how the constructors may be used to define a function on trees:
---- fun inTree(x, LEAF(y)) = x = y
|
inTree(x, NODE(y,z)) = inTree(x, y) orelse inTree(x, z);
val inTree = fn : int * tree → bool
This function looks for a specific integer value x in a tree. If the tree has the form
LEAF(y), then x is in the tree only if x=y. If the tree has the form NODE(y,z), with
subtrees y and z, then x is in the tree only if x is in the subtree y or the subtree z. The
type output by the compiler shows that inTree is a function that, given an integer and
a tree, returns a Boolean value.
An example of a polymorphic data-type declaration appears in Subsection 6.5.3,
after the discussion of polymorphism in Section 6.4.
5.4.5 ML Reference Cells and Assignment
None of the ML constructs discussed in earlier sections of this chapter have side
effects. Each expression has a value, but evaluating an expression does not have the
side effect of changing the value of any other expression. Although most large ML
programs are written in a style that avoids side effects when possible, most large ML
programs do use assignment occasionally to change the value of a variable.
The way that assignable variables are presented in ML is different from the way
that assignable variables appear in other programming languages. The main reasons
for this are to preserve the uniformity of ML as a programming language and to
separate side effects from pure expressions as much as possible.
ML assignment is restricted to reference cells. In ML, a reference cell has a different type than immutable values such as integers, strings, lists, and so on. Because
reference cells have specific reference types, restrictions on ML assignment are enforced as part of the type system. This is part of the elegance of ML: Almost all
restrictions on the structure of programs are part of the type system, and the type
system has a systematic, uniform definition.
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L-values and R-values
Before looking at assignment in ML, let us think about the difference between memory locations and their contents. This distinction is part of machine architectures
(memory locations contain data) and relevant to many programming languages. The
following pseudocode fragment illustrates the idea:
x : int;
y : int;
x := y + 3;
In the assignment, the value stored in variable y is added to 3 and the result stored
in the location for x. The central point is that the two variables are used differently.
The command uses only the value stored in y and does not depend on the location of
y. In contrast, the command uses the location of x, but does not depend on the value
stored in x before the assignment occurs.
The location of a variable is called its L-value, and the value stored in this location
is called the R-value of the variable. This is standard terminology that you will see in
many books on programming languages. The two values are called L and R to stand
for left and right, as typically we use L-values on the left-hand sides of an assignment
statement and R-values on the right-hand side.
ML Reference Cells
In ML, L-values and R-values have different types. In other words, an assignable
region of memory has a different type than a value that cannot be changed. In ML,
an L-value, or assignable region of memory, is called a reference cell. The type of a
reference cell indicates that it is a reference cell and specifies the type of value that
it contains. For example, a reference cell that contains an integer has type int ref,
meaning an integer reference cell.
When a reference cell is created, it must be initialized to a value of the correct
type. Therefore ML does not have uninitialized variables or dangling pointers. When
an assignment changes the value stored in a reference cell, the assignment must
be consistent with the type of the reference cell: An integer reference cell will always contain an integer, a list reference cell will always contain (or refer to) a list,
and so on.
Operations on Reference Cells
ML has operations to create reference cells, to access their contents, and to change
their contents. These are ref, !, and :=, which behave as follows:
ref v ---- creates a reference cell containing value v
!r
---- returns the value contained in reference cell r
r := v ---- places value v in reference cell r
5.4 The ML Programming Language
Here are some examples:
---- val x = ref 0;
val x = ref 0 : int ref
---- x := 3*(!x) + 5;
val it = () : unit
---- !x;
val it = 5 : int
The first input line binds identifier x to a new reference cell with contents 0. As
the compiler output indicates, the value of x is this reference cell, which has type
int ref. The next input line multiplies 3 times the contents of x, adds 5, and stores
the resulting integer value in cell x. Because ML is expression oriented, this “statement” is an ML expression. The type of this expression is unit, which, as described
earlier, is the type used for expressions that are evaluated for side effect. The last
input line is an expression reading the contents of x, which is the integer 5.
Because ML does not have any operations for computing the address of a value,
there is no way to observe whether assignment is by value or by pointer. As a result,
it is a convenient and accurate abstraction to regard a reference cell as a box holding
a value of any size and to regard assignment as an operation that places a value inside
the box. For example, the preceding code that creates a reference cell named x and
changes its contents can be visualized as follows, with the double arrow indicating
changes as a result of assignment:
x := 5:
x
0
x
5
Because reference cells can be created for any type of value, we can define a string
reference cell and change its contents by assignment:
---- val y = ref “Apple”;
val y = ref “Apple” : string ref
---- y := “Fried green tomatoes”;
val it = () : unit
---- !y;
val it = “Fried green tomatoes” : string
As in the integer example, the associated reference cell can be visualized as a box
that can contain any string:
y
y
As you know, different strings may require different amounts of memory. Therefore, it
does not seem likely that the memory cell bound to y can hold any string of any length.
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In fact, when the declaration val y = ref “Apple” is processed, storage is allocated to
contain the string “Apple” and the reference cell y is initialized to a pointer to this
location. When the assignment y := “Fried green tomatoes” is executed, the contents
of the cell y are changed to a pointer to “Fried green tomatoes”. Comparing the integer
and string examples, ML assignment is implemented as ordinary value assignment
for some types of cells and pointer assignment for others. However, because ML
has no way of finding the address of an expression, this implementation difference is
completely hidden from the programmer. If a compiler writer wanted to implement
integer assignment as pointer assignment, all programs would behave in exactly the
same way.
Here is one last simple code example to show how ML reference cells may be
used in an iterative loop. This loop sums the numbers between 1 and 10:
val i = ref 0;
val j = ref 0;
while !i < 10 do (i := !i + 1;j := !j + !i);
!j;
In the first two lines, the identifiers i and j are bound to new reference cells initialized
to value 0. The while loop increments i until !i, the contents of i, is not less than 10.
The final expression reveals the final value of j, because the compiler prints the value
of !j. Some important details are that a test i < 10 would not be legal, as this compares
a reference cell to an integer. Similarly, i := i+1 is not legal as a reference cell cannot
be added to 1; only integers or real numbers can be added.
As illustrated in this example, two imperative expressions can be combined with
a semicolon. Parentheses are used to keep the preceding while loop from parsing
as a loop followed by j := !j+i. In fact, a semicolon can be used to combine any two
expressions. The expression
e1; e2
is equivalent to
(fn x => e2) e1
where x is chosen not to appear in e2. As a result, the value of e1; e2 is the value of
e2 after e1 has been evaluated.
Typing Imperative Operations. As previously mentioned, reference cells have a
different type than the values they contain. Here is the typing rule:
If expression e has type τ , then the expression ref e has type τ ref.
The function ! can be applied to any argument of type τ ref, and assignment x := e is
5.5 Chapter Summary
type correct only if x has type τ ref and e has type τ for some type τ . In summary,
x : int
---- not assignable (like a constant in other languages)
y : int ref ---- assignable reference cell
5.4.6 ML Summary
ML is a programming language that encourages programming with functions. It
is easy to define functions with function arguments and function return results. In
addition, most data structures in ML programs are not assignable. Although it is
possible to construct reference cells for any type of value and modify reference cells
by assignment, side effects occur only when reference cells are used. Although most
large ML programs do use reference cells and have side effects, the pure parts of ML
are expressive enough that reference cells are used sparingly.
ML has an expressive type system. There are basic types for many common kinds
of computable values, such as Booleans, integers, strings, and reals. There are also
type constructors, which are type operators that can be applied to any type. The type
constructors include tuples, records, and lists. In ML, it is possible to define tuples of
lists of functions, for example. There is no restriction on the types of values that can
be placed in data structures.
The ML type system is often called a strong type system, as every expression
has a type and there are no mechanisms for subverting the type system. When the
ML type checker determines that an expression has type int, for example, then
any successful evaluation of that expression is guaranteed to produce an integer.
There are no dangling pointers that refer to unallocated locations in memory and no
casts that allow values of one type to be treated as values of another type without
conversion.
ML has several forms that allow programmers to define their own types and
type constructors. In this chapter, we looked at data-type declarations, which can be
used to define ML versions of enumerated types (types consisting of a finite list of
values), disjoint unions (types whose elements are drawn from the union of two or
more types), and recursively defined types. Another important aspect of the ML type
system is polymorphism, which we will study in the next chapter, along with other
aspects of the ML type system. We will discuss additional type definition and module
forms in Chapter 9.
5.5 CHAPTER SUMMARY
In this chapter, we discussed some of the basic properties of Algol-like languages
and examined some of the advances and problem areas in Algol 60, Algol 68, Pascal,
and C. The Algol family of languages established the command-oriented syntax,
with blocks, local declarations, and recursive functions, that are used in most current
programming languages. The Algol family of languages is all statically typed, as each
expression has a type that is determined by its syntactic form and the compiler checks
before running the program to make sure that the types of operations and operands
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agree. In looking at the improvements from Algol 60 to Algol 68 to Pascal, we saw
improvements in the static type systems.
The C programming language is similar to Algol 60, Algol 68, and Pascal in
some respects: command-oriented syntax, blocks, local declarations, and recursive
functions. However, C also shares some features with its untyped precursor BCPL,
such as pointer arithmetic. C is also more restricted than most Algol-based languages
in that functions cannot be declared inside nested blocks: All functions are declared
outside the main program. This simplifies storage management for C, as we will see
in Chapter 7.
In the second half of this chapter, we looked at the ML programming language
in more detail than we did the Algol family of languages. One reason to study ML is
that this language combines many if the important features of the Algol family with
features of Lisp; this language provides a good summary of the important language
features that developed before 1980.
The part of ML that we covered in this chapter comes from what is called core
ML. This is ML without the module features that were added in the 1980s. Core ML
has the following types,
unit, Booleans, integers, strings, reals, tuples, lists, records
and the following constructs
patterns, declarations, functions, polymorphism, overloading, type declarations,
reference cells, exceptions
We discussed most of these in this chapter, except polymorphism, which is
covered in Chapter 6, and exceptions, which are studied in Chapter 8. The
study of ML was summarized in this chapter in Subsection 5.4.6.
EXERCISES
5.1 Algol 60 Procedure Types
In Algol 60, the type of each formal parameter of a procedure must be given. However, proc is considered a type (the type of procedures). This is much simpler than
the ML types of function arguments. However, this is really a type loophole; because
calls to procedure parameters are not fully type checked, Algol 60 programs may
produce run-time type errors.
Write a procedure declaration for Q that causes the following program fragment
to produce a run-time type error:
proc P (proc Q)
begin Q(true) end;
P(Q);
where true is a Boolean value. Explain why the procedure is statically type correct,
but produces a run-time type error. (You may assume that adding a Boolean to an
integer is a run-time type error.)
5.2 Algol 60 Pass-By-Name
The following Algol 60 code declares a procedure P with one pass-by-name integer
parameter. Explain how the procedure call P(A[i]) changes the values of i and A by
substituting the actual parameters for the formal parameters, according to the Algol
Exercises
60 copy rule. What integer values are printed by tprogram? By using pass-by-name
parameter passing?
The line integer x does not declare local variables – this is just Algol 60 syntax
declaring the type of the procedure parameter:
begin
integer i;
integer array A[1:2];
procedure P(x);
integer x;
begin
i := x;
x := i
end
i := 1;
A[1] := 2; A[2] := 3;
P (A[i]);
print (i, A[1], A[2])
end
5.3 Nonlinear Pattern Matching
ML patterns cannot contain repeated variables. This exercise explores this language
design decision.
A declaration with a single pattern is equivalent to a sequence of declarations
using destructors. For example,
val p = (5,2);
val (x,y) = p;
is equivalent to
val p = (5,2);
val x = #1(p);
val y = #2(p);
where #1(p) is the ML expression for the first component of pair p and #2 similarly returns the second component of a pair. The operations #1 and #2 are called destructors
for pairs.
A function declaration with more than one pattern is equivalent to a function
declaration that uses standard if-then-else and destructors. For example,
fun f nil = 0
| f (x::y) = x;
is equivalent to
fun f(z) = if z=nil then 0 else hd(z);
where hd is the ML function that returns the first element of a list.
Questions:
(a) Write a function declaration that does not use pattern matching and that is
equivalent to
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The Algol Family and ML
fun f (x,0) = x
| f (0,y) = y
| f (x,y) = x+y;
ML pattern matching is applied in order, so that when this function is applied to
an argument (a, b), the first clause is used if b = 0, the second clause if b =/ 0 and
a=0, and the third clause if b =/ 0 and a =/ 0.
(b) Does the method you used in part (a), combining destructors and if-then-else,
work for this function?
fun eq(x,x) = true
| eq(x,y) = false;
(c) How would you translate ML functions that contain patterns with repeated
variables into functions without patterns? Give a brief explanation of a general
method and show the result for the function eq in part (b).
(d) Why do you think the designers of ML prohibited repeated variables in patterns?
(Hint: If f, g : int → int, then the expression f = g is not type-correct ML as the
test for equality is not defined on function types.)
5.4 ML Map for Trees
(a) The binary tree data type
datatype ’a tree = LEAF of ’a |
NODE of ’a tree ∗ ’a tree;
describes a binary tree for any type, but does not include the empty tree (i.e.,
each tree of this type must have at least a root node).
Write a function maptree that takes a function as an argument and returns
a function that maps trees to trees by mapping the values at the leaves to new
values, using the function passed in as a parameter. In more detail, if f is a
function that can be applied to the leaves of tree t and t is the tree on the left,
then maptree f t should result in the tree on the right:
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For example, if f is the function fun f(x)=x+1 then
maptree f (NODE(NODE(LEAF 1,LEAF 2),LEAF 3))
should evaluate to NODE(NODE(LEAF 2,LEAF 3),LEAF 4). Explain your definition in
one or two sentences.
(b) What is the type ML gives to your function? Why is it not the expected type
(’a → ’a) → ’a tree → ’a tree?
Exercises
5.5 ML Reduce for Trees
Assume that the data type tree is defined as in problem 4. Write a function
reduce : ( ’a ∗ ’a → ’a) → ’a tree → ’a
that combines all the values of the leaves by using the binary operation passed as a
parameter. In more detail, if oper : ’a ∗ ’a → ’a and t is the nonempty tree on the left
in this picture,
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then reduce oper t should be the result we obtain by evaluating the tree on the right.
For example, if f is the function,
fun f(x : int, y : int) = x + y;
then reduce f (NODE(NODE(LEAF 1, LEAF 2), LEAF 3)) = (1 + 2) + 3 = 6. Explain your
definition of reduce in one or two sentences.
5.6 Currying
This problem asks you to show that the ML types ’a → (’b → ’c) and (’a ∗ ’b) → ’c
are essentially equivalent.
(a) Define higher-order ML functions
Curry: ((’a ∗ ’b) → ’c)→ (’a → (’b→ ’c))
and
UnCurry: (’a → (’b → ’c))→ ((’a ∗ ’b) → ’c)
(b) For all functions f : (’a ∗ ’b) → ’c and g : ’a → (’b → ’c), the following two equalities
should hold (if you wrote the right functions):
UnCurry(Curry(f)) = f
Curry(UnCurry(g)) = g
Explain why each is true for the functions you have written. Your answer can
be three or four sentences long. Try to give the main idea in a clear, succinct
way. (We are more interested in insight than in number of words.) Be sure to
consider termination behavior as well.
5.7 Disjoint Unions
A union type is a type that allows the values from two different types to be combined
in a single type. For example, an expression of type union(A, B) might have a value
of type A or a value of type B. The languages C and ML both have forms of union
types.
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The Algol Family and ML
(a) Here is a C program fragment written with a union type:
...
union IntString {
int i;
char *s;
} x;
int y;
if ( . . . ) x.i = 3 else x.s = “here, fido”;
...
y = (x.i) + 5;
...
A C compiler will consider this program to be well typed. Despite the fact that
the program type checks, the addition may not work as intended. Why not? Will
the run-time system catch the problem?
(b) In ML, a union type union(A,B) would be written in the form datatype UnionAB =
tag a of A | tag b of B and the preceding if statement could be written as
datatype IntString = tag int of int | tag str of string;
...
val x = if . . . then tag int(3) else tag str(“here, fido”);
...
let val tag int (m) = x in m + 5 end;
Can the same bug occur in this program? Will the run-time system catch the
problem? The use of tags enables the compiler to give a useful warning message
to the programmer, thereby helping the programmer to avoid the bug, even
before running the program. What message is given and how does it help?
5.8 Lazy Evaluation and Functions
It is possible to evaluate function arguments at the time of the call (eager evaluation) or at the time they are used (lazy evaluation). Most programming languages
(including ML) use eager evaluation, but we can simulate lazy evaluation in an eager
language such as ML by using higher-order functions.
Consider a sequence data structure that starts with a known value and continues
with a function (known as a thunk) to compute the rest of the sequence:
---- datatype ’a Seq = Nil
| Cons of ’a * (unit -> ’a Seq);
---- fun head (Cons (x, )) = x;
val head = fn : ’a Seq -> ’a
---- fun tail (Cons ( , xs)) = xs();
val tail = fn : ’a Seq -> ’a Seq
---- fun BadCons (x, xs) = Cons (x, fn() =>xs);
val BadCons = fn : ’a * ’a Seq -> ’a Seq
Note that BadCons does not actually work, as xs was already evaluated on entering the
function. Instead of calling BadCons(x, xs), you would need to use Cons(x, fn() =>xs)
for lazy evaluation.
Exercises
This lazy sequence data type provides a way to create infinite sequences, with
each infinite sequence represented by a function that computes the next element in
the sequence. For example, here is the sequence of infinitely many 1s:
---- val ones = let fun f() = Cons(1,f) in f() end;
We can see how this works by defining a function that gets the nth element of a
sequence and by looking at some elements of our infinite sequence:
---- fun get(n,s) = if n=0 then head(s) else get(n-1,tail(s));
val get = fn : int * ’a Seq -> ’a
---- get(0,ones);
val it = 1 : int
---- get(5,ones);
val it = 1 : int
---- get(245, ones);
val it = 1 : int
We can define the infinite sequence of all natural numbers by
---- val natseq = let fun f(n)() = Cons(n,f(n+1)) in f(0) () end;
Using sequences, we can represent a function as a potentially infinite sequence of
ordered pairs. Here are two examples, written as infinite lists instead of as ML code
(note that ∼ is a negative sign in ML):
add1 = (0, 1) :: (∼ 1, 0) :: (1, 2) :: (∼ 2, ∼ 1) :: (2, 3) :: . . .
double = (0, 0) :: (∼ 1, ∼ 2) :: (1, 2) :: (∼ 2, ∼ 4) :: (2, 4) :: . . .
Here is ML code that constructs the infinite sequences and tests this representation of
functions by applying the sequences of ordered pairs to sample function arguments.
---- fun make ints(f)=
let
fun make pos (n) = Cons( (n, f(n)), fn() =>make pos(n + 1))
fun make neg (n) = Cons( (n, f(n)), fn() =>make neg(n - 1))
in
merge (make pos (0), make neg(∼ 1))
end;
val make ints = fn : (int -> ’a) -> (int * ’a) Seq
---- val add1 = make ints (fn(x) => x+1);
val add1 = Cons ((0,1),fn) : (int * int) Seq
---- val double = make ints (fn(x) => 2*x);
val double = Cons ((0,0),fn) : (int * int) Seq
---- fun apply (Cons( (x1,fx1), xs) , x2) =
if (x1=x2) then fx1
else apply(xs(), x2);
val apply = fn : (”a * ’b) Seq * ”a -> ’b
---- apply(add1, ∼ 4);
val it = ∼ 3 : int
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---- apply(double, 7);
val it = 14 : int
(a) Write merge in ML. Merge should take two sequences and return a sequence
containing the values in the original sequences, as used in the make ints function.
(b) Using the representation of functions as a potentially infinite sequence of ordered pairs, write compose in ML. Compose should take a function f and a
function g and return a function h such that h(x) = f (g(x)).
(c) It is possible to represent a partial function whose domain is not the entire set
of integers as a sequence. Under what conditions will your compose function
not halt? Is this acceptable?
6
Type Systems and Type Inference
Programming involves a wide range of computational constructs, such as data structures, functions, objects, communication channels, and threads of control. Because
programming languages are designed to help programmers organize computational
constructs and use them correctly, many programming languages organize data and
computations into collections called types. In this chapter, we look at the reasons for
using types in programming languages, methods for type checking, and some typing
issues such as polymorphism, overloading, and type equality. A large section of this
chapter is devoted to type inference, the process of determining the types of expressions based on the known types of some symbols that appear in them. Type inference
is a generalization of type checking, with many characteristics in common, and a
representative example of the kind of algorithms that are used in compilers and programming environments to determine properties of programs. Type inference also
provides an introduction to polymorphism, which allows a single expression to have
many types.
6.1 TYPES IN PROGRAMMING
In general, a type is a collection of computational entities that share some common
property. Some examples of types are the type int of integers, the type int→int of
functions from integers to integers, and the Pascal subrange type [1 .. 100] of integers
between 1 and 100. In concurrent ML there is the type int channel of communication
channels carrying integer values and, in Java, a hierarchy of types of exceptions.
There are three main uses of types in programming languages:
naming and organizing concepts,
making sure that bit sequences in computer memory are interpreted consistently,
providing information to the compiler about data manipulated by the program.
These ideas are elaborated in the following subsections.
Although some programming language descriptions will say things like, “Lisp
is an untyped language,” there is really no such thing as an untyped programming
language. In Lisp, for example, lists and atoms are two different types: list operations
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can be applied to lists but not to atoms. Programming languages do vary a great deal,
however, in the ways that types are used in the syntax and semantics (implementation)
of the language.
6.1.1 Program Organization and Documentation
A well-designed program uses concepts related to the problem being solved. For
example, a banking program will be organized around concepts common to banks,
such as accounts, customers, deposits, withdrawals, and transfers. In modern programming languages, customers and accounts, for example, can be represented as
separate types. Type checking can then check to make sure that accounts and customers are treated separately, with account operations applied to accounts but not
used to manipulate customers. Using types to organize a program makes it easier for
someone to read, understand, and maintain the program. Types therefore serve an
important purpose in documenting the design and intent of the program.
An important advantage of type information, in comparison with comments written by a programmer, is that types may be checked by the programming language
compiler. Type checking guarantees that the types written into a program are correct.
In contrast, many programs contain incorrect comments, either because the person
writing the explanation was careless or because the program was later changed but
the comments were not.
6.1.2 Type Errors
A type error occurs when a computational entity, such as a function or a data value,
is used in a manner that is inconsistent with the concept it represents. For example, if
an integer value is used as a function, this is a type error. A common type error is to
apply an operation to an operand of the wrong type. For example, it is a type error
to use integer addition to add a string to an integer. Although most programmers
have a general understanding of type errors, there are some subtleties that are worth
exploring.
Hardware Errors. The simplest kind of type error to understand is a machine
instruction that results in a hardware error. For example, executing a “function call”
x()
is a type error if x is not a function. If x is an integer variable with value 256, for
example, then executing x() will cause the machine to jump to location 256 and begin
executing the instructions stored at that place in memory. If location 256 contains data
that do not represent a valid machine instruction, this will cause a hardware interrupt.
Another example of a hardware type error occurs in executing an operation
float add(3, 4.5)
6.1 Types in Programming
where the hardware floating-point unit is invoked on an integer argument 3. Because
the bit pattern used to represent 3 does not represent a floating-point number in the
form expected by the floating-point hardware, this instruction will cause a hardware
interrupt.
Unintended Semantics. Some type errors do not cause a hardware fault or interrupt because compiled code does not contain the same information as the program
source code does. For example, an operation
int add(3, 4.5)
is a type error, as int add is an integer operation and is applied here to a floating-point
number. Most hardware would perform this operation. Because the bits used to represent the floating-point number 4.5 represent an integer that is not mathematically
related to 4.5, the operation it is not meaningful. More specifically, int add is intended
to perform addition, but the result of int add(3, 4.5) is not the arithmetic sum of the
two operands.
The error associated with int add(3, 4.5) may become clearer if we think about
how a program might apply integer addition to a floating-point argument. To be
concrete, suppose a program defines a function f that adds three to its argument,
fun f(x) = 3+x;
and someplace within the scope of this definition we also declare a floating-point
value z:
float z = 4.5;
If the programming language compiler or interpreter allows the call f(z) and the language does not automatically convert floating-point numbers to integers in this situation, then the function call f(z) will cause a run-time type error because int add(3, 4.5)
will be executed. This is a type error because integer addition is applied to a noninteger argument.
The reason why many people find the concept of type error confusing is that
type errors generally depend on the concepts defined in a program or programming
language, not the way that programs are executed on the underlying hardware. To be
specific, it is just as much of a type error to apply an integer operation to a floatingpoint argument as it is to apply a floating-point operation to an integer argument. It
does not matter which causes a hardware interrupt on any particular computer.
Inside a computer, all values are stored as sequences of bytes of bits. Because
integers and floating-point numbers are stored as four bytes on many machines,
some integers and floating-point numbers overlap; a single bit pattern may represent
an integer when it is used one way and a floating-point number when it is used another.
Nonetheless, a type error occurs when a pattern that is stored in the computer for the
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purpose of representing one type of value is used as the representation of another
type of value.
6.1.3 Types and Optimization
Type information in programs can be used for many kinds of optimizations. One
example is finding components of records (as they are called in Pascal and ML)
or structs (as they are called in C). The component-finding problem also arises in
object-oriented languages. A record consists of a set of entries of different types. For
example, a student record may contain a student name of type string and a student
number of type integer. In a program that also has records for undergraduate students,
these might be represented as related type that also contains a field for the year in
school of the student. Both types are written here as ML-style type expressions:
Student = {name : string, number : int}
Undergrad = {name : string, number : int, year : int}
In a program that manipulates records, there might be an expression of the form
r.name, meaning the name field of the record r. A compiler must generate machine
code that, given the location of record r in memory at run time, finds the location
of the field name of this record at run time. If the compiler can compute the type
of the record at compile time, then this type information can be used to generate efficient code. More specifically, the type of r makes it is possible to compute
the location of r.name relative to the location r, at compile time. For example, if the
type of r is Student, then the compiler can build a little table storing the information
that name occurs before number in each Student record. Using this table, the compiler
can determine that name is in the first location allocated to the record r. In this case,
the expression r.name is compiled to code that reads the value stored in location r+1
(if location r is used for something else besides the first field). However, for records
of a different type, the name field might appear second or third. Therefore, if the type
of r is not known at compile time, the compiler must generate code to compute the
location of name from the location of r at run time. This will make the program run
more slowly. To summarize: Some operations can be computed more efficiently if
the type of the operand is known at compile time.
In some object-oriented programming languages, the type of an object may be
used to find the relative location of parts of the object. In other languages, however,
the type system does not give this kind of information and run-time search must be
used.
6.2 TYPE SAFETY AND TYPE CHECKING
6.2.1 Type Safety
A programming language is type safe if no program is allowed to violate its type
distinctions. Sometimes it is not completely clear what the type distinctions are in a
specific programming language. However, there are some type distinctions that are
meaningful and important in all languages. For example, a function has a different
6.2 Type Safety and Type Checking
type from an integer. Therefore, any language that allows integers to be used as
functions is not type safe. Another action that we always consider a type error is to
access memory that is not allocated to the program.
The following table characterizes the type safety of some common programming
languages. We will discuss each form of type error listed in the table in turn.
Safety
Example languages
Explanation
Not safe
Almost safe
Safe
C and C++
Pascal
Lisp, ML, Smalltalk, Java
Type casts, pointer arithmetic
Explicit deallocation; dangling pointers
Complete type checking
Type Casts. Type casts allow a value of one type to be used as another type. In C
in particular, an integer can be cast to a function, allowing a jump to a location that
does not contain the correct form of instructions to be a C function.
Pointer Arithmetic. C pointer arithmetic is not type safe. The expression *(p+i)
has type A if p is defined to have type A*. Because the value stored in location p+i
might have any type, an assignment like x = *(p+i) may store a value of one type into
a variable of another type and therefore may cause a type error.
Explicit Deallocation and Dangling Pointers. In Pascal, C, and some other languages, the location reached through a pointer may be deallocated (freed) by the
programmer. This creates a dangling pointer, a pointer that points to a location that
is not allocated to the program. If p is a pointer to an integer, for example, then after
we deallocate the memory referenced by p, the program can allocate new memory
to store another type of value. This new memory may be reachable through the old
pointer p, as the storage allocation algorithm may reuse space that has been freed.
The old pointer p allows us to treat the new memory as an integer value, as p still
has type int. This violates type safety. Pascal is considered “mostly safe” because this
is the only violation of type safety (after the variant record and other original type
problems are repaired).
6.2.2 Compile-Time and Run-Time Checking
In many languages, type checking is used to prevent some or all type errors. Some
languages use type constraints in the definition of legal program. Implementations
of these languages check types at compile time, before a program is started. In these
languages, a program that violates a type constraint is not compiled and cannot be
run. In other languages, checks for type errors are made while the program is running.
Run-Time Checking. In programming languages with run-time type checking, the
compiler generates code so that, when an operation is performed, the code checks to
make sure that the operands have the correct type. For example, the Lisp language
operation car returns the first element of a cons cell. Because it is a type error to
apply car to something that is not a cons cell, Lisp programs are implemented so that,
before (car x) is evaluated, a check is made to make sure that x is a cons cell. An
advantage of run-time type checking is that it catches type errors. A disadvantage is
the run-time cost associated with making these checks.
Compile-Time Checking. Many modern programming languages are designed
so that it is possible to check expressions for potential type errors. In these
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languages, it is common to reject programs that do not pass the compile-time
type checks. An advantage of compile-time type checking is that it catches errors
earlier than run-time checking does: A program developer is warned about the error
before the program is given to other users or shipped as a product. Because compiletime checks may eliminate the need to check for certain errors at run time, compiletime checking can make it possible to produce more efficient code. For a specific
example, compiled ML code is two to four times faster than Lisp code. The primary
reason for this speed increase is that static type checking of ML programs greatly
reduces the need for run-time tests.
Conservativity of Compile-Time Checking. A property of compile-time type checking is that the compiler must be conservative. This mean that compile-time type
checking will find all statements and expressions that produce run-time type errors,
but also may flag statements or expressions as errors even if they do not produce
run-time errors. To be more specific about it, most checkers are both sound and conservative. A type checker is sound if no programs with errors are considered correct.
A type checker is conservative if some programs without errors are still considered
to have errors.
There is a reason why most type checkers are conservative: For any Turingcomplete programming language, the set of programs that may produce a run-time
type error is undecidable. This follows from the undecidability of the halting problem.
To see why, consider the following form of program expression:
if (complicated-expression-that-could-run-forever)
then (expression-with-type-error)
else (expression-with-type-error)
It is undecidable whether this expression causes a run-time type error, as the only way
for expression-with-type-error to be evaluated is for complicated-expression-that-couldrun-forever to halt. Therefore, deciding whether this expression causes a run-time type
error involves deciding whether complicated-expression-that-could-run-forever halts.
Because the set of programs that have run-time type errors is undecidable, no
compile-time type checker can find type errors exactly. Because the purpose of type
checking is to prevent errors, type checkers for type-safe languages are conservative.
It is useful that type checkers find type errors, and a consequence of the undecidability
of the halting problem is that some programs that could execute without run-time
error will fail the compile-time type-checking tests.
The main trade-offs between compile-time and run-time checking are summarized in the following table.
Form of Type Checking
Advantages
Disadvantages
Run-time
Compile-time
Prevents type errors
Prevents type errors
Eliminates run-time tests
Finds type errors before
execution and run-time
tests
Slows program execution
May restrict programming
because tests are
conservative.
6.3 Type Inference
Combining Compile-Time and Run-Time Checking. Most programming languages
actually use some combination of compile-time and run-time type checking. In
Java, for example, static type checking is used to distinguish arrays from integers, but array bounds errors (which are a form of type error) are checked at run
time.
6.3 TYPE INFERENCE
Type inference is the process of determining the types of expressions based on the
known types of some symbols that appear in them. The difference between type inference and compile-time type checking is really a matter of degree. A type-checking
algorithm goes through the program to check that the types declared by the programmer agree with the language requirements. In type inference, the idea is that
some information is not specified, and some form of logical inference is required
for determining the types of identifiers from the way they are used. For example,
identifiers in ML are not usually declared to have a specific type. The type system infers the types of ML identifiers and expressions that contain them from the
operations that are used. Type inference was invented by Robin Milner (see the
biographical sketch) for the ML programming language. Similar ideas were developed independently by Curry and Hindley in connection with the study of lambda
calculus.
Although practical type inference was developed for ML, type inference can be
applied to a variety of programming languages. For example, type inference could, in
principle, be applied to C or other programming languages. We study type inference
in some detail because it illustrates the central issues in type checking and because
type inference illustrates some of the central issues in algorithms that find any kind
of program errors.
In addition to providing a flexible form of compile-time type checking, ML type
inference supports polymorphism. As we will see when we subsequently look at the
type-inference algorithm, the type-inference algorithm uses type variables as placeholders for types that are not known. In some cases, the type-inference algorithm
resolves all type variables and determines that they must be equal to specific types
such as int, bool, or string. In other cases, the type of a function may contain type
variables that are not constrained by the way the function is defined. In these cases,
the function may be applied to any arguments whose types match the form given by
a type expression containing type variables.
Although type inference and polymorphism are independent concepts, we discuss polymorphism in the context of type inference because polymorphism arises
naturally from the way type variables are used in type inference.
6.3.1 First Examples of Type Inference
Here are two ML type-inference examples to give you some feel for how ML type
inference works. The behavior of the type-inference algorithm is explained only
superficially in these examples, just to give some of the main ideas. We will go through
the type inference process in detail in Subsection 6.3.2.
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Example 6.1
– fun f1(x) = x + 2;
val f1 = fn : int → int
The function f1 adds 2 to its argument. In ML, 2 is an integer constant; the real
number 2 would be written as 2.0. The operator + is overloaded; it can be either
integer addition or real addition. In this function, however, it must be integer addition
because 2 is an integer. Therefore, the function argument x must be an integer. Putting
these observations together, we can see that f1 must have type int → int.
Example 6.2
– fun f2(g,h) = g(h(0));
val f2 = fn : (’a → ’b) * (int → ’a) → ’b
The type (’a → ’b) * (int → ’a) → ’b inferred by the compiler is parsed as ((’a → ’b)
* (int → ’a)) → ’b.
The type-inference algorithm figures out that, because h is applied to an integer
argument, h must be a function from int to something. The algorithm represents
“something” by introducing a type variable, which is written as ’a. (This is unrelated
to Lisp ’a, which would be syntax for a Lisp atom, not a variable.) The type-inference
algorithm then deduces that g must be a function that takes whatever h returns
(something of type ’a) and then returns something else. Because g is not constrained
to return the same type of value as h, the algorithm represents this second something
by a new type variable, ’b. Putting the types of h and g together, we can see that the
first argument to f2 has type (’a → ’b) and the second has type (int → ’a). Function
f2 takes the pair of these two functions as an argument and returns the same type of
value as g returns. Therefore, the type of f2 is ((’a → ’b) * (int → ’a)) → ’b, as shown
in the preceding compiler output.
6.3.2 Type-Inference Algorithm
The ML type-inference algorithm uses the following three steps:
1. A assign a type to the expression and each subexpression. For any compound
expression or variable, use a type variable. For known operations or constants,
such as + or 3, use the type that is known for this symbol.
2. Generates a set of constraints on types, using the parse tree of the expression.
These constraints reflect the fact that if a function is applied to an argument,
for example, then the type of the argument must equal the type of the domain
of the function.
3. Solve these constraints by means of unification, which is a substitution-based
algorithm for solving systems of equations. (More information on unification
appears in the chapter on logic programming.)
6.3 Type Inference
The type-inference algorithm is explained by a series of examples. These examples
present the following issues:
explanation of the algorithm
a polymorphic function definition
application of a polymorphic function
a recursive function
a function with multiple clauses
type inference indicates a program error
Altogether, these six examples should give you a good understanding of the typeinference algorithm, except for the interaction between type inference and overloading. The interaction between overloading and type inference is not covered in this
book.
Example 6.3 Explanation of the Algorithm
We can see how the type-inference algorithm works by considering this example
function:
– fun g(x) = 5 + x;
val g = fn : int → int
The easiest way to see how the algorithm works is by drawing the parse tree of the
expression. We use an abbreviated form of parse tree that lists only the symbols that
occur in the expression, together with the symbol @ for an application of a function
to an argument. For the preceding expression we use the following graph. This is a
form of parse tree, together with a special edge indicating the binding lambda for
each bound variable. Here, the link from x to λ indicates that x is lambda bound at
the beginning of the expression:
λ
@
@
+
x
5
We use this graph to follow our type-inference steps:
1. We assign a type to the expression and each subexpression:
We illustrate this step by redrawing the graph, writing a type next to each node.
To simplify notation, we use single letters r, s, t, u, v, . . . , for type variables instead of
ML syntax ’a, ’b, and so on:
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Type Systems and Type Inference
λ
:r
@
@
:t
+ :
int → int→ int
5 :
:s
x
:u
int
Recall that each node in a parse tree represents a subexpression. Repeating the
information in the picture, the following table lists the subexpressions and their types:
Subexpression
λx. ((+ 5) x)
((+ 5) x)
(+ 5)
+
5
x
Type
r
s
t
int → (int → int)
int
u
Here we have written addition (+) as a Curried function and have chosen type int
→ int → int for this operation. The prefix notation for addition is not ML syntax,
of course, but it is used here to make the pictures simpler. As we saw earlier, + can
either be integer addition or real-number addition. Here we can see from context that
integer addition is needed. The actual ML type-inference algorithm will require a few
steps to figure this out, but we are not concerned with the mechanics of overloading
resolution here.
2. We generate a set of constraints on types, using the parse tree of the expression.
Constraints are equations between type expressions that must be solved. The way
we generate them depends on the form of each subexpression. For each function
application, constraints are generated according to the following rule.
Function Application: If the type of f is a, the type of e is b, and the type of
fe is c, then we must have a = b → c.
This typing rule for function application can be used twice in our expression.
Subexpression (+5),
Constraint int → (int → int) = int → t,
Subexpression (+5) x, Constraint t = u → s.
In the subexpression (+ 5), the type of the function + is int → (int → int), the
type of the argument 5 is int and the type of the application is t. Therefore, we
must have int → (int → int), = int → t. The reasoning for subexpression (+ 5) x is
similar: In the subexpression (+ 5) x, the type of the function (+ 5) is t,the type of
the argument x is u, and the type of the application is s. Therefore, we must have
t = u → s.
Lambda Abstraction (Function Expression): If the type of x is a and the type
of e is b, then the type of λx.e must equal a → b.
6.3 Type Inference
For our example expression, we have one lambda abstraction. This gives us the
following constraint:
Subexpression
λx.((+5)x),
Constraint r = u → s.
In words, the type of the whole expression is r , the type of x is u, and the type of the
subexpression ((+ 5) x) is s. This gives us the equation r = u → s.
3. We solve these constraints by means of unification.
Unification is a standard algorithm for solving systems of equations by substitution.
The general properties of this algorithm are not discussed here. Instead, the process
is shown by example in enough detail that you should be able to figure out the types
of simple expressions on your own.
For our example expression, we have the following constraints.
int → (int → int) = int → t,
t = u → s,
r = u → s.
If there is a way of associating type expression to type variables that makes all of
these equations true, then the expression is well typed. In this case, the type of the
expression will be the type expression equal to the type variable r . If there is no way
of associating type expression to type variables that makes all of these equations true,
then there is no type for this expression. In this case, the type-inference algorithm
will fail, resulting in an error message that says the expression is not well typed.
We can process these equations one at a time. The order is not very important, although it is convenient to put the equation involving the type of the entire expression
last, as this is the output of the type-inference algorithm.
The first equation is true if t = int → int. Because we need t = int → int, we
substitute int → int for t in the remaining equations. This gives us two equations to
solve:
int → int = u → s,
r = u → s.
The only way to have int → int = u → s is if u = s = int. Proceeding as before, we
substitute int for both u and s in the remaining equation. This gives us
r = int → int,
which tells us that the type r of the whole expression is int → int. Because every
constraint is solved, we know that the expression is typeable and we have computed
a type for the expression.
Example 6.4 A Polymorphic Function Definition
– fun apply(f,x) = f(x);
val apply = fn : (’a → ’b) * ’a → ’b
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Type Systems and Type Inference
This is an example of a function whose type involves type variables. The typeinference algorithm begins by assigning a type to each subexpression. Because this
makes it easiest to understand the algorithm, we write the function as a lambda expression with a pair f,x instead of a variable as a formal parameter: apply is defined
by the lambda expression λf,x. f x that maps a pair f,x to the result f x of applying
f to x.
Here is the parse graph of λf,x. f x, in which a pairing node is used on the left
to indicate that the argument f,x of the function is a pair, with links to f and x.
λ
@
f
x
The first step of the algorithm is to assign types to each node in the graph, as
shown here:
λ
:r
t×u
@
f
:t
:s
x:
u
The same information is repeated in the following table, showing the subexpressions
represented by each node and their types.
Subexpression
Type
λf,x. fx
f,x.
fx
f
x
r
t×u
s
t
u
The second step of the algorithm is to generate a set of constraints. For this example,
there is one constraint for the application and one for the lambda abstraction. The
application gives us
t = u → s.
In words, the type of the application f x has type s, provided that the type of the
function f is equal to type of argument → s . Because the type of the argument is u,
this gives us the constraint t = u → s.
The second constraint, from the lambda abstraction, is
r = t ∗ u → s.
6.3 Type Inference
In words, the type of λf,x. fx is r , where r must be equal to type of argument → s ,
as s is the type of the subtree representing the function result. Because the argument
is the pair f,x, the type of the argument is t ∗ u.
The constraints can be solved in order. The first requires t = u → s, which we
solve by substituting u → s for t in the remaining constraint. This gives us
r = (u → s)∗ u → s.
This tells us the type of the function. If we rewrite (u → s) ∗ u → s with ’a and ’b in
place of u and s, then we get the compiler output previously shown. Because there
are type variables in the type of the expression, the function may be used for many
types of arguments. This is illustrated in the following example, which uses the type
we have just computed for apply.
Example 6.5 Application of a Polymorphic Function
In the last example, we calculated the type of apply. The type of apply is (’a → ’b)
* ’a → ’b, which contains type variables. The type variables in this type mean that
apply is a polymorphic function, a function that may be applied to different types
of arguments. In the case of apply, the type (’a → ’b) * ’a → ’b means that apply
may be applied to a pair of arguments of type (’a → ’b) * ’a for any types ’a and
’b. In particular, recall that function fun g(x) = 5 + x from Example 6.3 has type
int → int. Therefore, the pair (g,3) has type int → int.* int, which matches the
form (’a → ’b) * ’a for function apply. In this example, we calculate the type of the
application
apply(g,3);
Following the three steps of the type inference algorithm, we begin by assigning types
to the nodes in the parse tree of this expression:
:r
@
:t
apply :
(u→s)*u →s
g : int→int
3 : int
In this illustration, the smaller unlabeled circle is a pairing node. This node and the
two below it represent the pair (g,3). In the previous example, we associated a product
type with the pairing node. Here, to show that it is equivalent to use a type variable
and constraint, we associate a type variable t with the pairing node and generate the
constraint t = (int → int) ∗ int.
There are two constraints, one for the pairing node and one for the application
node. For pairing, we have
t = (int → int) ∗ int.
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Type Systems and Type Inference
For the application, we have
(u → s) ∗ u → s = t → r.
In words, the type (u → s)∗ u → s of the function must equal type of argument → r,
where r is the type of the application.
Now we must solve the constraints, The first constraint gives a type expression
for t, which we can substitute for t in the second constraint. This gives us
(u → s) ∗ u → s = (int → int) ∗ int → r.
This constraint has an expression on each side of the equal sign. To solve this constraint, corresponding parts of each expression must be equal. In other words, we
can solve this constraint precisely by solving the following four constraints:
u = int, s = int, u = int, s = r.
Because these require u = s = int, we have r = int. Because all of the constraints
are solved, the expression apply(g,3) is typeable in the ML type system. The type of
apply(g,3) is the solution for type variable r , namely int.
We can also apply apply to other types of arguments. If not : bool → bool, then
apply(not, false)
is a well-typed expression with type bool by exactly the same type-inference processes
as those for apply(g,3). This illustrates the polymorphism of apply: Because the type
(’a → ’b) * ’a → ’b of apply contains type variables, the function may be applied to
any type of arguments that can be obtained if the type variables in (’a → ’b) * ’a → ’b
are replaced with type names or type expressions.
Example 6.6 A Recursive Function
When a function is defined recursively, we must determine the type of the function
body without knowing the type of recursive function calls. To see how this works,
consider this simple recursive function that sums the integers up to a given integer.
This function does not terminate, but it does type check:
– fun sum(x) = x+sum(x-1);
val sum = fn : int -> int
Here is a parse graph of the function, with type variables associated with each of the
nodes except for +, −, and 1, as we ignore overloading and treat these as integer
operations and integer constant. Because we are trying to determine the type of sum,
we associate a type variable with sum and proceed:
6.3 Type Inference
λ
:z
:y
@
:t
@
+
@
x:r
sum
:v
:s
@
@
−
:w
:u
1
x:r
Starting with the applications of + and − and proceeding from the lower right
up, the constraints associated with the function applications and lambda abstraction
in this expression are
int → (int → int) = r → t,
int → (int → int) = r → u,
u = int → v,
s = v → w,
t = w → y,
z = r → y.
To this list we add one more because the type of sum must also be the type of the
entire expression:
s = z.
The constraint s = z is the one additional constraint associated with the fact that this
is a recursive declaration of a function. Solving these in order, we have
r = int,
t = int → int,
u = int → int,
v = int,
s = int → w,
t = w → y,
z = r → y,
w = int,
y = int,
z = int → int,
s = int → int.
Because the constraints can be solved, the function is typeable. In the process of
solving the constraints, we have calculated that the type of sum is int→int.
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Example 6.7 A Function with Multiple Clauses
Type inference for functions with several clauses may be done by a type check of
each clause separately. Then, because all clauses define the same function, we impose
the constraint that the types of all clauses must be equal. For example, consider the
append function on lists, defined as follows:
– fun append(nil, l) = l
| append(x::xs, l) = x :: append(xs, l);
val append = fn : ’a list * ’a list → ’a list.
As the type : ’a list * ’a list → ’a list indicates, append can be applied to any pair
of lists, as long as both lists contain the same type of list elements. Thus, append is a
polymorphic function on lists.
We begin type inference for append by following the three-step algorithm for the
first clause of the definition, then repeating the steps for the second clause. This gives
us two types:
append : ’a list * ’b → ’b
append : ’a list * ’b → ’a list
Intuitively, the first clause has type ’a list * ’b → ’b because the first argument must
match nil, but the second argument may be anything. The second clause has type ’a
list * ’b → ’a list because the return result is a list containing one element from the
list passed as the first argument.
If we impose the constraint
’a list * ’b → ’b = ’a list * ’b -> ’a list
then we must have ’b = a list. This gives us the final type for append:
append : ’a list * ’a list → ’a list
Example 6.8 Type Inference Indicates a Program Error
Here is an example that shows how type inference may produce output that indicates
a programming error, even though the program may type correctly. Here is a sample
(incorrect) declaration of a reverse function on lists:
– fun reverse (nil) = nil
| reverse (x::lst) = reverse (lst);
val reverse = fn : ’a list → ’b list
6.4 Polymorphism and Overloading
As the compiler output shows, this function is typeable; there is no type error in this
declaration. However, look carefully at the type of reverse. The type ’a list → ’b list
means that we can apply reverse to any type of list and obtain any type of list as a
result. However, the type of the “reversed” list is not the same as the type of the list
we started with!
Because it does not make sense for reverse to return a list that is a different type
from its argument, there must be something wrong with this code. The problem is
that, in the second clause, the first element x of the input list is not used as part of
the output. Therefore, reverse always returns the empty list.
As this example illustrates, the type-inference algorithm may sometimes return
a type that is more general than the one we expect. This does not indicate a type
error. In this example, the faulty reverse can be used anywhere that a correct reverse
function could be used. However, the type of reverse is useful because it tells the
programmer that there is an error in the program.
6.4 POLYMORPHISM AND OVERLOADING
Polymorphism, which literally means “having multiple forms,” refers to constructs
that can take on different types as needed. For example, a function that can compute
the length of any type of list is polymorphic because it has type ’a list → int for every
type ’a.
There are three forms of polymorphism in contemporary programming languages:
parametric polymorphism, in which a function may be applied to any arguments
whose types match a type expression involving type variables;
ad hoc polymorphism, another term for overloading, in which two or more implementations with different types are referred to by the same name;
subtype polymorphism, in which the subtype relation between types allows an
expression to have many possible types.
Parametric and ad hoc polymorphism (overloading) are discussed in this section.
Subtype polymorphism is considered in later chapters in connection with objectoriented programming.
6.4.1 Parametric Polymorphism
The main characteristic of parametric polymorphism is that the set of types associated with a function or other value is given by a type expression that contains type
variables. For example, an ML function that sorts lists might have the ML type
sort : (’a * ’a → bool) * ’a list → ’a list
In words, sort can be applied to any pair consisting of a function and a list, as long
as the function has a type of the form ’a * ’a -> bool, in which the type ’a must also be
the type of the elements of the list. The function argument is a less-than operation
used to determine the order of elements in the sorted list.
In parametric polymorphism, a function may have infinitely many types, as there
are infinitely many ways of replacing type variables with actual types. The sort
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Type Systems and Type Inference
function, for example, may be used to sort lists of integers, lists of lists of integers,
lists of lists of lists of integers, and so on.
Parametric polymorphism may be implicit or explicit. In explicit parametric polymorphism, the program text contains type variables that determine the way that a
function or other value may be treated polymorphically. In addition, explicit polymorphism often involves explicit instantiation or type application to indicate how
type variables are replaced with specific types in the use of a polymorphic value. C++
templates are a well-known example of explicit polymorphism. ML polymorphism
is called implicit parametric polymorphism because programs that declare and use
polymorphic functions do not need to contain types – the type-inference algorithm
computes when a function is polymorphic and computes the instantiation of type
variables as needed.
C++ Function Templates
For many readers, the most familiar type parameterization mechanism is the C++
template mechanism. Although some C++ programmers associate templates with
classes and object-oriented programming, function templates are also useful for programs that do not declare any classes.
As an illustrative example, suppose you write a simple function to swap the values
of two integer variables:
void swap(int& x, int& y){
int tmp = x; x = y; y = tmp;
}
Although this code is useful for exchanging values of integer variables, the sequence
of instructions also works for other types of variables. If we wish to swap values
of variables of other types, then we can define a function template that uses a type
variable T in place of the type name int:
template <typename T>
void swap(T& x, T& y){
T tmp = x; x = y; y = tmp;
}
For those who are not familiar with templates, the main idea is to think of the type
name T as a parameter to a function from types to functions. When applied to, or
instantiated to, a specific type, the result is a version of swap that has int replaced with
another type. In other words, swap is a general function that would work perfectly
well for many types of arguments. Templates allow us to treat swap as a function with
a type argument.
In C++, function templates are instantiated automatically as needed, with the
types of the function arguments used to determine which instantiation is needed.
This is illustrated in the following example lines of code.
6.4 Polymorphism and Overloading
int i,j; . . .
float a,b; . . .
String s,t; . . .
swap(i,j); // Use swap with T replaced with int
swap(a,b); // Use swap with T replaced with float
swap(s,t); // Use swap with T replaced with String
Comparison with ML Polymorphism
In ML polymorphism, the type-inference algorithm infers the type of a function and
the type of a function application (as explained in Section 6.3). When a function
is polymorphic, the actions of the type-inference algorithm can be understood as
automatically inserting “template declarations” and “template instantiation” into
the program. We can see how this works by considering an ML sorting function that
is analogous to the C++ sort function previously declared:
fun insert(less, x, nil) = [x]
| insert(less, x, y::ys) = if less(x,y) then x::y::ys
else y::insert(less,x,ys);
fun sort(less, nil) = nil
| sort(less, x::xs) = insert(less, x, sort(less,xs));
For sort to be polymorphic, a less-than operation must be passed as a function argument to sort.
The types of insert and sort, as inferred by the type-inference algorithm, are
val insert = fn : (’a * ’a -> bool) * ’a * ’a list -> ’a list
val sort = fn : (’a * ’a -> bool) * ’a list -> ’a list
In these types, the type variable ’a can be instantiated to any type, as needed. In effect,
the functions are treated as if they were “templates.” By use of a combination of C++
template, ML function, and type syntax, the functions previously defined could also
be written as
template <type ’a>
fun insert(less : ’a * ’a -> bool, x : ’a, nil : ’a list) = [x]
| insert(less, x, y::ys) = if less(x,y) then x::y::ys
else y::insert(less,x,ys);
template <type ’a>
fun sort(less : ’a * ’a -> bool, nil : ’a list) = nil
| sort(less, x::xs) = insert(less, x, sort(less,xs));
These declarations are the explicitly typed versions of the implicitly polymorphic
ML functions. In other words, the ML type-inference algorithm may be understood
as a program preprocessor that converts ML expressions without type information
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into expressions in some explicitly typed intermediate language with templates. From
this point of view, the difference between explicit and implicit polymorphism is that
a programming language processor (such as the ML compiler) takes the simpler
implicit syntax and automatically inserts explicit type information, converting from
implicit to explicit form, before programs are compiled and executed.
Finishing this example, suppose we declare a less-than function on integers:
– fun less(x,y) = x < y;
val less = fn : int * int -> bool
In the following application of the polymorphic sort function, the sort template is
automatically instantiated to type int, so sort can be used to sort an integer list:
– sort (less, [1,4,5,3,2]);
val it = [1,2,3,4,5] : int list
6.4.2 Implementation of Parametric Polymorphism
C++ templates and ML polymorphic functions are implemented differently. The reason for the difference is not related to the difference between explicitly polymorphic
syntax and implicitly polymorphic syntax. The need for different implementation
techniques arises from the difference between data representation in C and data
representation in ML.
C++ Implementation
C++ templates are instantiated at program link time. More specifically, suppose that
the swap function template is stored in one file and compiled and a program calling
swap is stored in another file and compiled separately. The so-called relocatable
object files produced by compilation of the calling program will include information
indicating that the compiled code calls a function swap of a certain type. The program
linker is designed to combine the two program parts by linking the calls to swap in the
calling program to the definition of swap in a separate compilation unit. It does so by
instantiating the compiled code for swap in a form that produces code appropriate
for the calls to swap.
If a program calls swap with several different types, then several different instantiated copies of swap will be produced. One reason that a different copy is needed for
each type of call is that function swap declares a local variable tmp of type T. Space
for tmp must be allocated in the activation record for swap. Therefore the compiled
code for swap must be modified according to the size of a variable of type T. If T is
a structure or object, for example, then the size might be fairly large. On the other
hand, if T is int, the size will be small. In either case, the compiled code for swap must
“know” the size of the datum so that addressing into the activation record can be
done properly.
The linking process for C++ is relatively complex. We will not study it in detail.
However, it is worth noting that if < is an overloaded operator, then the correct
6.4 Polymorphism and Overloading
version of < must be identified when the compiled code for sort is linked with a
calling program. For example, consider the following generic sort function:
template <typename T>
void sort( int count, T * A[count] ) {
for (int i=0; i <count-1; i++)
for (int j=i+1; j<count-1; j++)
if (A[j] < A[i]) swap(A[i],A[j]);
}
If A is an array of type T, then sort(n, A) will work only if operator < is defined on
type T. This requirement of sort is not declared anywhere in the C++ code. However,
when the function template is instantiated, the actual type T must have an operator
< defined or a link-time error will be reported and no executable object code will be
produced.
ML Implementation
In ML, there is one sequence of compiled instructions for each polymorphic function. There is no need to produce different copies of the code for different types
of arguments because related types of data are represented in similar ways. More
specifically, pointers are used in parameter passing and in the representation of data
structures such as lists so that when a function is polymorphic, it can access all necessary data in the same way, regardless of its type. This property of ML is called
uniform data representation.
The simplest example of uniform data representation is the ML version of the
polymorphic swap function:
– fun swap(x,y) = let val tmp = x in x := !y; y := !tmp end;
val swap = fn : ’a ref * ’a ref -> unit
As the type indicates, this swap function can be applied to any two references of
the same type. Although ML references generally work like assignable variables in
other languages, the value of a reference is a pointer to a cell that contains a value.
Therefore, when a pair of references is passed to the swap function, the swap function
receives a pair of pointers. The two pointers are the same size (typically 32 bits),
regardless of what type of value is contained in the locations to which they point. In
fact, we can complete the entire computation by doing only pointer assignment. As
a result, none of the compiled code for swap depends on the size of the data referred
to by arguments x and y.
Uniform data representation has its advantages and disadvantages. Because there
is no need to duplicate code for different argument types, uniform data representation leads to smaller code size and avoids complications associated with C++-style
linking. On the other hand, the resulting code can be less efficient, as uniform data
representation often involves using pointers to data instead of storing data directly
in structures.
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For polymorphic list functions to work properly, all lists must be represented
in exactly the same way. Because of this uniformity requirement, small values that
would fit directly into the car part of a list cons cell cannot be placed there because
large values do not fit. Hence we must store pointers to small values in lists, just as we
store pointers to large values. ML programmers and compiler writers call the process
of making all data look the same by means of pointers boxing.
Comparison
Two important points of comparison are efficiency and reporting of error messages.
As far as efficiency, the C++ implementation requires more effort at link time and
produces a larger code, as instantiating a template several times will result in several
copies of the code. The ML implementation will run more slowly unless special
optimizations are applied; uniform data representation involves more extensive use
of pointers and these pointers must be stored and followed.
As a general programming principle, it is more convenient to have program errors
reported at compile time than at link time. One reason is that separate program
modules are compiled independently, but are linked together only when the entire
system is assembled. Therefore, compilation is a “local” process that can be carried
out by the designer or implementer of a single component. In contrast, link-time
errors represent global system properties that are not known until the entire system
is assembled. For this reason, C++ link-time errors associated with operations in
templates can be irritating and a source of frustration.
Somewhat better error reporting for C++ templates could be achieved if the
template syntax included a description of the operations needed on type parameters.
However, this is relatively complicated in C++, because of overloading and other
properties of the language. In contrast, the ML has simpler overloading and includes
more information in parameterized constructs, allowing all type errors to be reported
as a program unit is compiled.
6.4.3 Overloading
Parametric polymorphism can be contrasted with overloading. A symbol is overloaded if it has two (or more) meanings, distinguished by type, and resolved at
compile time. In an influential historical paper, Christopher Strachey referred to
ML-style polymorphism as parametric polymorphism (although ML had not been
invented yet) and overloading as ad hoc polymorphism.
Example. In standard ML, as in many other languages, the operator + has two distinct
implementations associated with it, one of type int * int → int, the other of type
real*real → real. The reason that both of these operations are given the name + is
that both compute numeric addition. However, at the implementation level, the two
operations are really very different. Because integers are represented in one way (as
binary numbers) and real numbers in another (as exponent and mantissa, following
scientific notation), the way that integer addition combines the bits of its arguments to
produce the bits of its result is very different from the way this is done in real addition.
An important difference between parametric polymorphism and overloading is
that parameter polymorphic functions use one algorithm to operate on arguments
6.5 Type Declarations and Type Equality
of many different types, whereas overloaded functions may use a different algorithm
for each type of argument.
A characteristic of overloading is that overloading is resolved at compile time.
If a function is overloaded, then the compiler must choose between the possible
algorithms at compile time. Choosing one algorithm from among the possible algorithms associated with an overloaded function is called resolving the overloading. In
many languages, if a function is overloaded, then only the function arguments are
used to resolve overloading. For example, consider the following two expressions:
3 + 2;
3.0 + 2.0;
/* add two integers */
/* add two real (floating point) numbers */
Here is how the compiler will produce code for evaluating each expression:
3 + 2:
The parsing phase of the compiler will build the parse tree of this expression,
and the type-checking phase will compute a type for each symbol. Because the
type-checking phase will determine that + must have type int * int → int, the
code-generation phase of the compiler will produce machine instructions that
perform integer addition.
3.0 + 2.0: The parsing phase of the compiler will build the parse tree of this
expression, and the type-checking phase will compute a type for each symbol.
Because the type-checking phase will determine that + must have type real *
real → real, the code-generation phase of the compiler will produce machine
instructions that perform integer addition.
Automatic conversion is a separate mechanism that may be combined with overloading. However, it is possible for a language to have overloading and not to have
automatic conversion. ML, for example, does not do automatic conversion.
6.5 TYPE DECLARATIONS AND TYPE EQUALITY
Many kinds of type declarations and many kinds of type equality have appeared
in programming languages of the past. The reason why type declarations and type
equality are related is that, when a type name is declared, it is important to decide
whether this is a “new” type that is different from all other types or a new name whose
meaning is equal to some other type that may be used elsewhere in the program.
Some programming languages have used fairly complicated forms of type equality, leading to confusing forms of type declarations. Instead of discussing many of the
historical forms, we simply look at a few rational possibilities.
6.5.1 Transparent Type Declarations
There are two basic forms of type declaration:
transparent, meaning an alternative name is given to a type that can also be
expressed without this name,
opaque, meaning a new type is introduced into the program that is not equal to
any other type.
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In the ML form of transparent type declaration,
type <identifier> = <type expression>
the identifier becomes a synonym for the type expression. For example,
type Celsius = real;
type Fahrenheit = real;
declare two type names, Celsius and Fahrenheit, whose meaning is the type real, just
the way that the two value declarations
val x = 3;
val y = 3;
declare two identifiers whose value is 3. (Remember that ML identifiers are not
assignable variables; the identifier x will have value 3 wherever it is used.) If we
declare an ML function to convert Fahrenheit to Celsius, this function will have real
→ real:
– fun toCelsius(x) = ((x-32.0)* 0.555556);
val toCelsius = fn : real → real
This should not be surprising because there is no indication that the function argument or return value has any type other than real. If we want to specify the types of
the argument and result, we can add them to the function declaration. This produces
a function of type Fahrenheit → Celsius:
– fun toCelsius(x : Fahrenheit) = ((x-32.0)* 0.555556) : Celsius;
val toCelsius = fn : Fahrenheit → Celsius
This version of the toCelsius function is more informative to read, as the types indicate
the intended purpose of the function. However, because Fahrenheit and Celsius are
synonyms for real, this function can be applied to a real argument:
– toCelsius(74.5);
val it = 23.61113 : Celsius
6.5 Type Declarations and Type Equality
The ML type checker gives the result type Celsius, but because Celsius=real, the result
can be used in real expressions.
ML abstract types are an example of an opaque type declaration. These are
discussed in Chapter 9. ML data type is another form of opaque type declaration;
the opacity of ML data type declarations is discussed in Subsection 6.5.3.
Two historical names for two forms of type equality are name type equality and
structural type equality. Intuitively, name equality means that two type names are
considered equal in type checking only if they are the same name. Structural equality
means that two type names are the same if the types they name are the same (i.e., have
the same structure). Although these terms may seem simple and innocuous, there
are lots of confusing phenomena associated with the use of name and structural type
equivalence in programming languages.
6.5.2 C Declarations and Structs
The basic type declaration construct in C is typedef. Here are some simple examples:
typedef char byte;
typedef byte ten bytes[10];
the first declaring a type byte that is equal to char and the second an array type
ten bytes that is equal to arrays of 10 bytes. Generally speaking, C typedef works
similarly to the transparent ML type declaration discussed in the preceding subsection. However, when structs are involved, the C type checker considers separately
declared type names to be unequal, even if they are declared to name the same struct
type. Here is a short program example showing how this works:
typedef struct {int m;} A;
typedef struct {int m;} B;
A x;
B y;
x=y; /* incompatible types in assignment */
Here, although the two struct types used in the two declarations are the same, the
C type checker does not treat A and B as equal types. However, if we replace the two
declarations with typedef int A; typedef int B;, using int in place of structs, then the
assignment is considered type correct.
6.5.3 ML Data-Type Declaration
The ML data-type declaration, discussed in Subsection 5.4.4, is a form of type declaration that simultaneously defines a new type name and operations for building
and making use of elements of the type. Because all of the examples in Subsection
5.4.4 were monomorphic, we take a quick look at a polymorphic declaration before
discussing type equaliy.
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Here is an example of a polymorphic data type of trees. You may wish to compare
this with the monomorphic (nonpolymorphic) example in Subsection 5.4.4:
datatype ’a tree = LEAF of ’a | NODE of (’a tree * ’a tree);
This declaration defines a polymorphic type ’a tree, with instances int tree, string tree,
and so on, together with polymorphic constructors LEAF and NODE:
– LEAF;
val it = fn : ’a -> ’a tree
– NODE;
val it = fn : ’a tree * ’a tree -> ’a tree
The following function checks to see if an element appears in a tree. The function
uses an exception, discussed in Section 8.2, when the element cannot be found. ML
requires an exception to be declared before it is used:
– exception NotFound;
exception NotFound
– fun inTree(x, EMPTY) = raise NotFound
|
inTree(x, LEAF(y)) = x = y
|
inTree(x, NODE(y,z)) = inTree(x, y) orelse inTree(x, z);
val inTree = fn : ”a * ”a tree → bool
Each ML data-type declaration is considered to define a new type different from all
other types. Even if two data types have the same structure, they are not considered
equivalent. The design of ML makes it hard to declare similar data types, as each
constructor has only one type. For example, the two declarations
datatype A = C of int;
datatype B = C of int;
declare distinct types A and B. Because the second declaration follows the first and
ML considers each declaration to start a new scope, the constructor C has type int
→ B after both declarations have been processed. However, we can see that A and
B are considered different by writing a function that attempts to treat a value of one
type as the other,
fun f(x:A) = x : B;
which leads to the message Error: expression doesn’t match constraint [tycon mismatch].
6.6 Chapter Summary
6.6 CHAPTER SUMMARY
In this chapter, we studied reasons for using types in programming languages, methods for type checking, and some typing issues such as polymorphism, overloading,
and type equality.
Reasons for Using Types
There are three main uses of types in programming languages:
Naming and organizing concepts: Functions and data structures can be given
types that reflect the way these computational constructs are used in a program.
This helps the programmers and anyone else reading a program figure out how
the program works and why it is written a certain way.
Making sure that bit sequences in computer memory are interpreted consistently:
Type checking keeps operations from being applied to operands in incorrect ways.
This prevents a floating-point operation from being applied to a sequence of bits
that represents a string, for example.
Providing information to the compiler about data manipulated by the program: In
languages in which the compiler can determine the type of a data structure, for
example, the type information can be used to determine the relative location of a
part of this structure. This compile-time type information can be used to generate
efficient code for indexing into the data structure at run time.
Type Inference
Type inference is the process of determining the types of expressions based on the
known types of some of the symbols that appear in them. For example, we saw how
to infer that the function g declared by
fun g(x) = 5+x;
has type int → int. The difference between type inference and compile-time type
checking is a matter of degree. A type-checking algorithm goes through the program
to check that the types declared by the programmer agree with the language requirements. In type inference, the idea is that some information is not specified and some
form of logical inference is required for determining the types of identifiers from the
way they are used.
The following steps are used for type inference:
1. Assign a type to the expression and each subexpression by using the known
type of a symbol of a type variable.
2. Generate a set of constraints on types by using the parse tree of the expression.
3. Solve these constraints by using unification, which is a substitution-based algorithm for solving systems of equations.
In a series of examples, we saw how to apply this algorithm to a variety of expressions.
Type inference has many characteristics in common with the kind of algorithms that
are used in compilers and programming environments to determine properties of
programs. For example, some useful alias analysis algorithms that try to determine
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whether two pointers might point to the same location have the same general outline
as that of type inference.
Polymorphism and Overloading
There are three forms of polymorphism: parametric polymorphism, ad hoc polymorphism (another term for overloading), and subtype polymorphism. The first two
were examined in this chapter, with subtype polymorphism left for later chapters
on object-oriented languages. Parametric polymorphism can be either implicit, as in
ML, or explicit, as with C++ templates. There are also two ways of implementing
parametric polymorphism, one in which the same data representation is used for
many types of data and the other in which explicit instantiation of parametric code
is used to match each different data representation.
The difference between parametric polymorphism and overloading is that parametric polymorphism allows one algorithm to be given many types, whereas overloading involves different algorithms. For example, the function + is overloaded in
many languages. In an expression adding two integers, the integer addition algorithm
is used. In adding two floating-point numbers, a completely different algorithm is used
for computing the sum.
Type Declarations and Type Equality
We discussed opaque and transparent type declarations. In opaque type declarations,
the type name stands for a distinct type different from all other types. In transparent
type declarations, the declared name is a synonym for another type. Both forms are
used in many programming languages.
EXERCISES
6.1 ML Types
Explain the ML type for each of the following declarations:
(a) fun a(x,y) = x+2*y;
(b) fun b(x,y) = x+y/2.0;
(c) fun c(f) = fn y => f(y);
(d) fun d(f,x) = f(f(x)));
(e) fun e(x,y,b) = if b(y) then x else y;
Because you can simply type these expressions into an ML interpreter to determine
the type, be sure to write a short explanation to show that you understand why the
function has the type you give.
6.2 Polymorphic Sorting
This function performing insertion sort on a list takes as arguments a comparison
function less and a list l of elements to be sorted. The code compiles and runs
correctly:
fun sort(less, nil) = nil |
sort(less, a : : l) =
let
Exercises
fun insert(a, nil) = a : : nil |
insert(a, b : : l) = if less(a,b) then a : : (b : : l)
else b : : insert(a, l)
in
insert(a, sort(less, l))
end;
What is the type of this sort function? Explain briefly, including the type of the
subsidiary function insert. You do not have to run the ML algorithm on this code;
just explain why an ordinary ML programmer would expect the code to have this
type.
6.3 Types and Garbage Collection
Language D allows a form of “cast” in which an expression of one type can be
treated as an expression of any other. For example, if x is a variable of type integer,
then (string)x is an expression of type string. No conversion is done. Explain how
this might affect garbage collection for language D.
For simplicity, assume that D is a conventional imperative language with integers, reals (floating-point numbers), pairs, and pointers. You do not need to consider
other language features.
6.4 Polymorphic Fixed Point
A fixed point of a function f is some value x such that x = f (x). There is a connection between recursion and fixed points that is illustrated by this ML definition
of the factorial function factorial : int → int:
fun Y f x = f (Y f) x;
fun F f x = if x=0 then 1 else x*f(x-1);
val factorial = Y F;
The first function, Y, is a fixed-point operator. The second function, F, is a function
on functions whose fixed point is factorial. Both of these are curried functions; using
the ML syntax fn x ⇒ . . . for λx . . . , we could also write the function F as
fun F(f) = fn x =>
if x=0 then 1 else x*f(x-1)
This F is a function that, when applied to argument f, returns a function that, when
applied to argument x, has the value given by the expression if x=0 then 1 else
x*f(x-1).
(a) What type will the ML compiler deduce for F?
(b) What type will the ML compiler deduce for Y?
6.5 Parse Graph
Use the following parse graph to calculate the ML type for the function
fun f(g,h) = g(h) + 2;
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λ
✡
✡
✡
✡
✡
✡
✡
✡
✡
✡
✡
✡
❡
❡
❡
❡
❡
❡
@
❡
❡
❡
❡
❡
@
2
❅
❅
❅
+
@
int → int → int
✪
✪
✪
❅
❅
❅
h
g
6.6 Parse Graph
Use the following parse graph to follow the steps of the ML type-inference algorithm on the function declaration
fun f(g) = g(g) + 2;
What is the output of the type checker?
λ
✡
✡
✡
✡
✡
✡
✡
✡
✡
✡
✡
✡
❡
❡
❡
❡
❡
❡
@
❡
❡
❡
❡
❡
@
❅
❅
❅
+
int → int → int
✪
✪
✪
g
@
2
❅
❅
❅
g
Exercises
6.7 Type Inference and Bugs
What is the type of the following ML function?
fun append(nil, l) = l
| append(x : : l, m) = append(l, m);
Write one or two sentences to explain succinctly and informally why append has the
type you give. This function is intended to append one list onto another. However,
it has a bug. How might knowing the type of this function help the programmer to
find the bug?
6.8 Type Inference and Debugging
The reduce function takes a binary operation, in the form of a function f, and a list,
and produces the result of combining all elements in the list by using the binary
operation. For example;
reduce plus [1,2,3] = 1 + 2 + 3 = 6
if plus is defined by
fun plus (x, y : int) = x+y
A friend of yours is trying to learn ML and tries to write a reduce function. Here is
his incorrect definition:
fun reduce(f, x) = x
| reduce(f, (x : : y)) = f(x, reduce(f, y));
He tells you that he does not know what to return for an empty list, but this should
work for a nonempty list: If the list has one element, then the first clause returns
it. If the list has more than one element, then the second clause of the definition
uses the function f. This sounds like a reasonable explanation, but the type checker
gives you the following output:
val reduce = fn : (((’a * ’a list) -> ’a list) * ’a list) -> ’a list
How can you use this type to explain to your friend that his code is wrong?
6.9 Polymorphism in C
In the following C min function, the type void is used in the types of two arguments.
However, the function makes sense and can be applied to a list of arguments in
which void has been replaced with another type. In other words, although the C type
of this function is not polymorphic, the function could be given a polymorphic type
if C had a polymorphic type system. Using ML notation for types, write a type for
this min function that captures the way that min could be meaningfully applied to
arguments of various types. Explain why you believe the function has the type you
have written.
int min (
void *a[ ],
/* a is an array of pointers to data of unknown type */
int n,
/* n is the length of the array */
int (*less)(void*, void*)
/* parameter less is a pointer to function */
)
/* that is used to compare array elements */
{
int i;
int m;
m=0;
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Type Systems and Type Inference
for (i=1; i < n; i++)
if (less(a[i], a[m])) m=i;
return(m);
}
6.10 Typing and Run-Time Behavior
The following ML functions have essentially identical computational behavior,
fun f(x) = not f(x);
fun g(y) = g(y) * 2;
because except for typing differences, we could replace one function with the other
in any program without changing the observable behavior of the program. In more
detail, suppose we turn off the ML type checker and compile a program of the form
P[fun f(x) = not f(x)]. Whatever this program does, the program P[fun f(y) = f(y) * 2]
we obtain by replacing one function definition with the other will do exactly the
same thing. In particular, if the first does not lead to a run-time type error such as
adding an integer to a string, neither will the second.
(a) What is the ML type for f?
(b) What is the ML type for g?
(c) Give an informal explanation of why these two functions have the same runtime behavior.
(d) Because the two functions are equivalent, it might be better to give them the
same type. Why do you think the designers of the ML typing algorithm did not
work harder to make it do this? Do you think they made a mistake?
6.11 Dynamic Typing in ML
Many programmers believe that a run-time typed programming language like Lisp
or Scheme is more expressive than a compile-time typed language like ML, as there
is no type system to “get in your way.” Although there are some situations in which
the flexibility of Lisp or Scheme is a tremendous advantage, we can also make
the opposite argument. Specifically, ML is more expressive than Lisp or Scheme
because we can define an ML data type for Lisp or Scheme expressions.
Here is a type declaration for pure historical Lisp:
datatype LISP = Nil
| Symbol of string
| Number of int
| Cons of LISP * LISP
| Function of (LISP -> LISP)
Although we could have used (Symbol ”nil”) instead of a primitive Nil, it seems
convenient to treat nil separately.
(a) Write an ML declaration for the Lisp function atom that tests whether its
argument is an atom. (Everything except a cons cell is an atom – The word atom
comes from the Greek word atomos, meaning indivisible. In Lisp, symbols,
numbers, nil, and functions cannot be divided into smaller pieces, so they are
considered to be atoms.) Your function should have type LISP → LISP, returning
atoms Symbol(”T”) or Nil.
Exercises
(b) Write an ML declaration for the Lisp function islist that tests whether its argument is a proper list. A proper list is either nil or a cons cell whose cdr
is a proper list. Note that not all listlike structures built from cons cells are
proper lists. For instance, (Cons (Symbol(”A”), Symbol(”B”))) is not a proper list
(it is instead what is known as a dotted list), and so (islist (Cons (Symbol(”A”),
Symbol(”B”)))) should evaluate to Nil. On the other hand, (Cons (Symbol(”A”),
(Cons (Symbol(”B”), Nil)))) is a proper list, and so your function should evaluate
to Symbol(”T”). Your function should have type LISP → LISP, as before.
(c) Write an ML declaration for Lisp car function and explain briefly. The function
should have type LISP → LISP.
(d) Write Lisp expression (lambda (x) (cons x ’A)) as an ML expression of type LISP
→ LISP. Note that ’A means something completely different in Lisp and ML.
The ’A here is part of a Lisp expression, not an ML expression. Explain briefly.
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7
Scope, Functions, and Storage
Management
In this chapter storage management for block-structured languages is described by
the run-time data structures that are used in a simple, reference implementation.
The programming language features that make the association between program
names and memory locations interesting are scope, which allows two syntactically
identical names to refer to different locations, and function calls, which each require
a new memory area in which to store function parameters and local variables. Some
important topics in this chapter are parameter passing, access to global variables, and
a storage optimization associated with a particular kind of function call called a tail
call. We will see that storage management becomes more complicated in languages
with nested function declarations that allow functions to be passed as arguments or
returned as the result of function calls.
7.1 BLOCK-STRUCTURED LANGUAGES
Most modern programming languages provide some form of block. A block is a
region of program text, identified by begin and end markers, that may contain declarations local to this region. Here are a few lines of C code to illustrate the idea:




{ int x = 2;



 { int y = 3; inner
outer
x = y+2; block
block 



}


}
In this section of code, there are two blocks. Each block begins with a left brace, {,
and ends with a right brace, }. The outer block begins with the first left brace and
ends with the last right brace. The inner block is nested inside the outer block. It
begins with the second left brace and ends with the first right brace. The variable
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7.1 Block-Structured Languages
x is declared in the outer block and the variable y is declared in the inner block.
A variable declared within a block is said to be local to that block. A variable declared in an enclosing block is said to be global to the block. In this example, x is
local to the outer block, y is local to the inner block, and x is global to the inner
block.
C, Pascal, and ML are all block-structured languages. In-line blocks are delineated
by { . . . } in C, begin...end in Pascal, and let...in...end in ML. The body of a procedure
or function is also a block in each of these languages.
Storage management mechanisms associated with block structure allow functions
to be called recursively.
The versions of Fortran in widespread use during the 1960s and 1970s were not
block structured. In historical Fortran, every variable, including every parameter
of each procedure (called a subroutine in Fortran) was assigned a fixed-memory
location. This made it impossible to call a procedure recursively, either directly or
indirectly. If Fortran procedure P calls Q, Q calls R, and then R attempts to call P, the
second call to P is not allowed. If P were called a second time in this call chain, the
second call would write over the parameters and return address for the first call. This
would make it impossible for the call to return properly.
Block-structured languages are characterized by the following properties:
New variables may be declared at various points in a program.
Each declaration is visible within a certain region of program text, called a block.
Blocks may be nested, but cannot partially overlap. In other words, if two blocks
contain any expressions or statements in common, then one block must be entirely
contained within the other.
When a program begins executing the instructions contained in a block at run
time, memory is allocated for the variables declared in that block.
When a program exits a block, some or all of the memory allocated to variables
declared in that block will be deallocated.
An identifier that is not declared in the current block is considered global to
the block and refers to the entity with this name that is declared in the closest
enclosing block.
Although most modern general-purpose programming languages are block structured, many important languages do not provide full support for all combinations of
block-structured features. Most notably, standard C and C++ do not allow local function declarations within nested blocks and therefore do not address implementation
issues associated with the return of functions from nested blocks.
In this chapter, we look at the memory management and access mechanisms for
three classes of variables:
local variables, which are stored on the stack in the activation record associated
with the block
parameters to function or procedure blocks, which are also stored in the activation
record associated with the block
global variables, which are declared in some enclosing block and therefore must
be accessed from an activation record that was placed on the run-time stack
before activation of the current block.
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Scope, Functions, and Storage Management
Registers
Code
Data
Stack
Program
Counter
Environment
Pointer
Heap
Figure 7.1. Program stack.
It may seem surprising that most complications arise in connection with access to
global variables. However, this is really a consequence of stack-based memory management: The stack is used to make it easy to allocate and access local variables. In
placing local variables close at hand, a global variable may be buried on the stack
under any number of activation records.
Simplified Machine Model
We use the simplified machine model in Figure 7.1 to look at the memory management
in block-structured languages.
The machine model in Figure 7.1 separates code memory from data memory.
The program counter stores the address of the current program instruction and is
normally incremented after each instruction. When the program enters a new block,
an activation record containing space for local variables declared in the block is added
to the run-time stack (drawn here at the top of data memory), and the environment
pointer is set to point to the new activation record. When the program exits the block,
the activation record is removed from the stack and the environment pointer is reset
to its previous location. The program may store data that will exist longer than the
execution of the current block on the heap. The fact that the most recently allocated
activation record is the first to be deallocated is sometimes called the stack discipline.
Although most block-structured languages are implemented by a stack, higher-order
functions may cause the stack discipline to fail.
Although Figure 7.1 includes some number of registers, generally used for shortterm storage of addresses and data, we will not be concerned with registers or the
instructions that may be stored in the code segment of memory.
Reference Implementation. A reference implementation is an implementation of
a language that is designed to define the behavior of the language. It need not
be an efficient implementation. The goal in this chapter is to give you enough information about how blocks are implemented in most programming languages so
that you can understand when storage needs to be allocated, what kinds of data
are stored on the run-time stack, and how an executing program accesses the data
locations it needs. We do this by describing a reference implementation. Because our
goal is to understand programming languages, not build a compiler, this reference
7.2 In-Line Blocks
implementation will be simple and direct. More efficient methods for doing many of
the things described in this chapter, tailored for specific languages, may be found in
compiler books.
A Note about C
The C programming language is designed to make C easy to compile and execute,
avoiding several of the general scoping and memory management techniques described in this chapter. Understanding the general cases considered here will give
C programmers some understanding of the specific ways in which C is simpler than
other languages. In addition, C programmers who want the effect of passing functions and their environments to other functions may use the ideas described in this
chapter in their programs.
Some commercial implementations of C and C++ actually do support function
parameters and return values, with preservation of static scope by use of closures.
(We will discuss closures in Section 7.4.) In addition, the C++ Standard Template
Library (covered in Subsection 9.4.3) provides a form of function closure as many
programmers find function arguments and return values useful.
7.2 IN-LINE BLOCKS
An in-line block is a block that is not the body of a function or procedure. We
study in-line blocks first, as these are simpler than blocks associated with function
calls.
7.2.1 Activation Records and Local Variables
When a running program enters an in-line block, space must be allocated for variables
that are declared in the block. We do this by allocating a set of memory locations called
an activation record on the run-time stack. An activation record is also sometimes
called a stack frame.
To see how this works, consider the following code example. If this code is part of
a larger program, the stack may contain space for other variables before this block
is executed. When the outer block is entered, an activation record containing space
for x and y is pushed onto the stack. Then the statements that set values of x and y
will be executed, causing values of x and y to be stored in the activation record. On
entry into the inner block, a separate activation record containing space for z will be
added to the stack. After the value of z is set, the activation record containing this
value will be popped off the stack. Finally, on exiting the outer block, the activation
record containing space for x and y will be popped off the stack:
{ int x=0;
int y=x+1;
{ int z=(x+y)*(x-y);
};
};
165
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Scope, Functions, and Storage Management
Space for global
variables
Space for global
variables
Space for global
variables
Space for global
variables
Space for x,y
Space for x,y
Space for x,y
Space for z
Execution time
Figure 7.2. Stack grows and shrinks during program execution.
We can visualize this by using a sequence of figures of the stack. As in Figure 7.1, the
stack is shown growing downward in memory in Figure 7.2.
A simple optimization involves combining small nested blocks into a single block.
For the preceding example program, this would save the run time spent in pushing
and popping the inner block for z, as z could be stored in the same activation record
as that of x and y. However, because we plan to illustrate the general case by using
small examples, we do not use this optimization in further discussion of stack storage allocation. In all of the program examples we consider, we assume that a new
activation record is allocated on the run-time stack each time the program enters a
block.
The number of locations that need to be allocated at run time depends on the
number of variables declared in the block and their types. Because these quantities
are known at compile time, the compiler can determine the format of each activation
record and store this information as part of the compiled code.
Intermediate Results
In general, an activation record may also contain space for intermediate results.
These are values that are not given names in the code, but that may need to be saved
temporarily. For example, the activation record for this block,
{ int z = (x+y)*(x-y);
}
may have the form
Space for z
Space for x+y
Space for x-y
because the values of subexpressions x+y and x-y may have to be evaluated and stored
somewhere before they are multiplied.
On modern computers, there are enough registers that many intermediate results are stored in registers and not placed on the stack. However, because register
7.2 In-Line Blocks
allocation is an implementation technique that does not affect programming language
design, we do not discuss registers or register allocation.
Scope and Lifetime
It is important to distinguish the scope of a declaration from the lifetime of a location:
Scope: a region of text in which a declaration is visible.
Lifetime: the duration, during a run of a program, during which a location is
allocated as the result of a specific declaration.
We may compare lifetime and scope by using the following example, with vertical
lines used to indicate matching block entries and exits.
{ int x = . . . ;
{ int y = . . . ;
{ int x = . . . ;
... .
};
};
};
In this example, the inner declaration of x hides the outer one. The inner block is
called a hole in the scope of the outer declaration of x, as the outer x cannot be
accessed within the inner block. This example shows that lifetime does not coincide
with scope because the lifetime of the outer x includes time when inner block is being
executed, but the scope of the outer x does not include the scope of the inner one.
Blocks and Activation Records for ML
Throughout our discussion of blocks and activation records, we follow the convention
that, whenever the program enters a new block, a new activation record is allocated
on the run-time stack. In ML code that has sequences of declarations, we treat each
declaration as a separate block. For example, in the code
fun f(x) = x+1;
fun g(y) = f(y) +2;
g(3);
we consider the declaration of f one block and the declaration of g another block
inside the outer block. If this code is not inside some other construct, then these
blocks will both end at the end of the program.
When an ML expression contains declarations as part of the let-in-end construct,
we consider the declarations to be part of the same block. For example, consider this
example expression:
167
168
Scope, Functions, and Storage Management
let fun g(y) = y+3
fun h(z) = g(g(z))
in
h(3)
end;
This expression contains a block, beginning with let and ending with end. This block
contains two declarations, functions g and h, and one expression, h(x), calling one
of these functions. The construct let . . . in . . . end is approximately equivalent to
{ . . . ; . . . } in C. The main syntactic difference is that declarations appear between
the keywords let and in, and expressions using these declarations appear between
keywords in and end. Because the declarations of functions g and h appear in the
same block, the names g and h will be given values in the same activation record.
7.2.2 Global Variables and Control Links
Because different activation records have different sizes, operations that push and
pop activation records from the run-time stack store a pointer in each activation
record to the top of the preceding activation record. The pointer to the top of the
previous activation record is called the control link, as it is the link that is followed
when control returns to the instructions in the preceding block. This gives us a structure shown in Figure 7.3. Some authors call the control link the dynamic link because
the control links mark the dynamic sequence of function calls created during program
execution.
Control link
Local variables
Intermediate results
Control link
Local variables
Intermediate results
Environment
Pointer
Figure 7.3. Activation records with
control links.
7.2 In-Line Blocks
When a new activation record is added to the stack, the control link of the new
activation record is set to the previous value of the environment pointer, and the
environment pointer is updated to point to the new activation record. When an
activation record is popped off the stack, the environment pointer is reset by following
the control link from the activation record.
The code for pushing and popping activation records from the stack is generated
by the compiler at compile time and becomes part of the compiled code for the
program. Because the size of an activation record can be determined from the text
of the block, the compiler can generate code for block entry that is tailored to the
size of the activation record.
When a global variable occurs in an expression, the compiler must generate code
that will find the location of that variable at run time. However, the compiler can
compute the number of blocks between the current block and the block where the
variable is declared; this is easily determined from the program text. In addition, the
relative position of each variable within its block is known at compile time. Therefore,
the compiler can generate lookup code that follows a predetermined number of links
Example 7.1
{ int x=0;
int y=x+1;
{ int z=(x+y)*(x-y);
};
};
When the expressions x+y and x-y are evaluated during execution of this code, the
run-time stack will have activation records for the inner and outer blocks as shown
below:
Control link
x
0
y
1
Control link
z
-1
x+y
1
x-y
-1
Environment
Pointer
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Scope, Functions, and Storage Management
On a register-based machine, the machine code generated by the compiler will find
the variables x and y, load each into registers, and then add the two values. The code
for loading x uses the environment pointer to find the top of the current activation,
then computes the location of x by adding 1 to the location stored in the control
link of the current activation record. The compiler generates this code by analyzing
the program text at compile time: The variable x is declared one block out from
the current block, and x is the first variable declared in the block. Therefore, the
control link from the current activation record will lead to the activation record
containing x, and the location of x will be one location down from the top of that
block. Similar steps can be used to find y at the second location down from the top
of its activation record. Although the details may vary from one compiler to the
next, the main point is that the compiler can determine the number of control links
to follow and the relative location of the variable within the correct block from the
source code. In particular, it is not necessary to store variable names in activation
records.
7.3 FUNCTIONS AND PROCEDURES
Most block-structured languages have procedures or functions that include parameters, local variables, and a body consisting of an arbitrary expression or sequence of
statements. For example, here are representative Algol-like and C-like forms:
procedure P(<parameters>)
begin
<local variables>;
<procedure body>;
end;
<type> f(<parameters>)
{
<local variables>;
<function body>;
};
The difference between a procedure and a function is that a function has a return
value but a procedure does not. In most languages, functions and procedures may
have side effects. However, a procedure has only side effects; a procedure call is
a statement and not an expression. Because functions and procedures have many
characteristics in common, we use the terms almost interchangeably in the rest of
this chapter. For example, the text may discuss some properties of functions, and
then a code example may illustrate these properties with a procedure. This should
remind you that the discussion applies to functions and procedures in many programming languages, whether or not the language treats procedures as different from
functions.
7.3.1 Activation Records for Functions
The activation record of a function or procedure block must include space for parameters and return values. Because a procedure may be called from different call
sites, it is also necessary to save the return address, which is the location of the next
instruction to execute after the procedure terminates. For functions, the activation
record must also contain the location that the calling routine expects to have filled
with the return value of the function.
7.3 Functions and Procedures
Control link
Return address
Return-result addr
Parameters
Local variables
Intermediate results
Environment
Pointer
Figure 7.4. Activation record associated with funtion call.
The activation record associated with a function (see Figure 7.4) must contain
space for the following information:
control link, pointing to the previous activation record on the stack,
access link, which we will discuss in Subsection 7.3.3,
return address, giving the address of the first instruction to execute when the
function terminates,
return-result address, the location in which to store the function return value,
actual parameters of the function,
local variables declared within the function,
temporary storage for intermediate results computed with the function executes.
This information may be stored in different orders and in different ways in different
language implementations. Also, as mentioned earlier, many compilers perform optimizations that place some of these values in registers. For concreteness, we assume
that no registers are used and that the six components of an activation record are
stored in the order previously listed.
Although the names of variables are eliminated during compilation, we often
draw activation records with the names of variables in the stack. This is just to make
it possible for us to understand the figures.
Example 7.2
We can see how function activation records are added and removed from the run-time
stack by tracing the execution of the familiar factorial function:
fun fact(n) = if n <= 1 then 1 else n * fact(n-1);
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Scope, Functions, and Storage Management
Suppose that some block contains the expression fact(3)+1, leading to a call of fact(3).
Let us assume that the activation record of the calling block contains a location that
will be used to store the value of fact(3) before computing fact(3)+1.
The next activation record that is placed on the stack will be an activation record
for the call fact(3). In this activation record, shown after the list,
the control link points to the activation record of the block containing the call
fact(3),
the return-result link points to the location in the activation record of the calling
block that has been allocated for the intermediate value fact(3) of the calling
expression fact(3)+1,
the actual parameter value 3 is placed in the location allocated for the formal
parameter n,
a location is allocated for the intermediate value fact(n-1) that will be needed
when n>0.
fact(3)
Control link
Return-result addr
n
3
fact(n-1)
Environment
Pointer
After this activation record is allocated on the stack, the code for factorial is executed.
Because n>0, there is a recursive call fact(2). This leads to a recursive call fact(1), which
results in a series of activation records, as shown in the subsequent figure.
fact(3)
Control link
Return-result addr
n
3
fact(n-1)
fact(2)
Control link
Return-result addr
n
2
fact(n-1)
fact(1)
Control link
Return-result addr
n
fact(n-1)
1
7.3 Functions and Procedures
Note that in each of the lower activation records, the return-result address points
to the space allocated in the activation record above it. This is so that, on return
from fact(1), for example, the return result of this call can be stored in the activation
record for fact(2). At that point, the final instruction from the calculation of fact(2)
will be executed, multiplying local variable n by the intermediate result fact(1).
The final illustration of this example shows the situation during return from fact(2)
when the return result of fact(2) has been placed in the activation record of fact(3),
but the activation record for fact(2) has not yet been popped off the stack.
fact(3)
Control link
Return result addr
n
3
fact(n-1)
2
fact(2)
Control link
Return result addr
n
2
fact(n-1)
1
7.3.2 Parameter Passing
The parameter names used in a function declaration are called formal parameters.
When a function is called, expressions called actual parameters are used to compute the parameter values for that call. The distinction between formal and actual
parameters is illustrated in the following code:
proc p (int x, int y) {
if (x > y) then . . . else . . . ;
...
x := y*2 + 3;
...
}
p (z, 4*z+1);
The identifiers x and y are formal parameters of the procedure p. The actual parameters in the call to p are z and 4*z+1.
The way that actual parameters are evaluated and passed to the function depends
on the programming language and the kind of parameter-passing mechanisms it uses.
The main distinctions between different parameter-passing mechanisms are
the time that the actual parameter is evaluated
the location used to store the parameter value.
Most current programming languages evaluate the actual parameters before executing the function body, but there are some exceptions. (One reason that a language or
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Scope, Functions, and Storage Management
program optimization might wish to delay evaluation of an actual parameter is that
evaluation might be expensive and the actual parameter might not be used in some
calls.) Among mechanisms that evaluate the actual parameter before executing the
function body, the most common are
Pass-by-reference: pass the L-value (address) of the actual parameter
Pass-by-value: pass the R-value (contents of address) of the actual parameter
Recall that we discussed L-values and R-values in Subsection 5.4.5 in connection
with ML reference cells (assignable locations) and assignment. We will discuss how
pass-by-value and pass-by-reference work in more detail below. Other mechanisms
such as pass-by-value-result are covered in the exercises.
The difference between pass-by-value and pass-by-reference is important to the
programmer in several ways:
Side Effects. Assignments inside the function body may have different effects
under pass-by-value and pass-by-reference.
Aliasing. Aliasing occurs when two names refer to the same object or location.
Aliasing may occur when two parameters are passed by reference or one parameter
passed by reference has the same location as the global variable of the procedure.
Efficiency. Pass-by-value may be inefficient for large structures if the value of
the large structure must be copied. Pass-by-reference may be less efficient than
pass-by-value for small structures that would fit directly on stack, because when
parameters are passed by reference we must dereference a pointer to get their
value.
There are two ways of explaining the semantics of call-by-reference and call-byvalue. One is to draw pictures of computer memory and the run-time program stack,
showing whether the stack contains a copy of the actual parameter or a reference to it.
The other explanation proceeds by translating code into a language that distinguishes
between L- and R-values. We use the second approach here because the rest of the
chapter gives you ample opportunity to work with pictures of the run-time stack.
Semantics of Pass-by-Value
In pass-by-value, the actual parameter is evaluated. The value of the actual parameter
is then stored in a new location allocated for the function parameter. For example,
consider this function definition and call:
function f (x) = { x := x+1; return x };
. . . .f(y) . . . ;
If the parameter is passed by value and y is an integer variable, then this code has
the same meaning as the following ML code:
fun f (z : int) = let
. . . f(!y) . . . ;
x = ref z
in
x := !x+1; !x
end;
7.3 Functions and Procedures
As you can see from the type, the value passed to the function f is an integer. The
integer is the R-value of the actual parameter y, as indicated by the expression !y
in the call. In the body of f, a new integer location is allocated and initialized to the
R-value of y.
If the value of y is 0 before the call, then the value of f(!y) is 1 because the function
f increments the parameter and returns its value. However, the value of y is still 0
after the call, because the assignment inside the body of f changes the contents of
only a temporary location.
Semantics of Pass-by-Reference
In pass-by-reference, the actual parameter must have an L-value. The L-value of the
actual parameter is then bound to the formal parameter. Consider the same function
definition and call used in the explanation of pass-by-value:
function f (x) = { x := x+1; return x };
. . . .f(y) . . . ;
If the parameter is passed by reference and y is an integer variable, then this code
has the same meaning as the following ML code:
fun f (x : int ref) = ( x := !x+1; !x );
. . . f(y)
As you can see from the type, the value passed to the function f is an integer reference
(L-value).
If the value of y is 0 before the call, then the value of f(!y) is 1 because the function
f increments the parameter and returns its value. However, unlike the situation for
pass-by-value, the value of y is 1 after the call because the assignment inside the body
of f changes the value of the actual parameter.
Example 7.3
Here is an example, written in an Algol-like notation, that combines pass-byreference and pass-by-value:
fun f(pass-by-ref x : int, pass-by-value y : int)
begin
x := 2;
y := 1;
if x = 1 then return 1 else return 2;
end;
var z : int;
z := 0;
print f(z,z);
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Scope, Functions, and Storage Management
Translating the preceding pseudo-Algol example into ML gives us
fun f(x : int ref, y : int) =
let val yy = ref y in
x := 2;
yy := 1;
if (!x = 1) then 1 else 2
end;
val z = ref 0;
f(z,!z);
This code, which treats L- and R-values explicitly, shows that for pass-by-reference
we pass an L-value, the integer reference z. For pass-by-value, we pass an R-value,
the contents !z of z. The pass-by-value is assigned a new temporary location.
With y passed by value as written, z is assigned the value 2. If y is instead passed
by reference, then x and y are aliases and z is assigned the value 1.
Example 7.4
Here is a function that tests whether its two parameters are aliases:
function (y,z){
y := 0;
z :=0;
y := 1;
if z=1 then y :=0; return 1 else y :=0; return 0
}
If y and z are aliases, then setting y to 1 will set z to 1 and the function will return 1.
Otherwise, the function will return 0. Therefore, a call f(x,x) will behave differently
if the parameters are pass-by-value than if the parameters are pass-by-reference.
7.3.3 Global Variables (First-Order Case)
If an identifier x appears in the body of a function, but x is not declared inside the
function, then the value of x depends on some declaration outside the function.
In this situation, the location of x is outside the activation record for the function.
Because x must be declared in some other block, access to a global x involves finding
an appropriate activation record elsewhere on the stack.
There are two main rules for finding the declaration of a global identifier:
Static Scope: A global identifier refers to the identifier with that name that is
declared in the closest enclosing scope of the program text.
Dynamic Scope: A global identifier refers to the identifier associated with the
most recent activation record.
7.3 Functions and Procedures
These definitions can be tricky to understand, so be sure to read the examples below carefully. One important difference between static and dynamic scope is that
finding a declaration under static scope uses the static (unchanging) relationship
between blocks in the program text. In contrast, dynamic scope uses the actual
sequence of calls that are executed in the dynamic (changing) execution of the
program.
Although most current general-purpose programming languages use static scope
for declarations of variables and functions, dynamic scoping is an important concept that is used in special-purpose languages and for specialized constructs such as
exceptions.
Dynamically Scoped
Statically Scoped
Older Lisps
TeX/LaTeX document languages
Exceptions in many languages
Macros
Newer Lisps, Scheme
Algol and Pascal
C
ML
Other current languages
Example 7.5
The difference between static and dynamic scope is illustrated by the following code,
which contains two declarations of x:
int x=1;
function g(z) = x+z;
function f(y) = {
int x = y+1;
return g(y*x)
};
f(3);
The call f(3) leads to a call g(12) inside the function f. This causes the expression
x+z in the body of g to be evaluated. After the call to g, the run-time stack will
contain activation records for the outer declaration of x, the invocation of f, and the
invocation of g, as shown in the following illustration.
outer block
x
1
f(3)
y
3
x
4
z
12
g(12)
At this point, two integers named x are stored on the stack, one from the outer block
declaring x and one from the local declaration of x inside f. Under dynamic scope,
the identifier x in the expression x+z will be interpreted as the one from the most
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Scope, Functions, and Storage Management
Control link
Access link
Return address
Return-result addr
Parameters
Local variables
Intermediate results
Environment
Pointer
Figure 7.5. Activation record with access link for functions call with static
scope.
recently created activation record, namely x=4. Under static scope, the identifier x in
x+z will refer to the declaration of x from the closest program block, looking upward
from the place that x+z appears in the program text. Under static scope, the relevant
declaration of x is the one in the outer block, namely x=1.
Access Links are Used to Maintain Static Scope
The access link of an activation record points to the activation record of the closest
enclosing block in the program. In-line blocks do not need an access link, as the
closest enclosing block will be the most recently entered block – for in-line blocks,
the control link points to the closest enclosing block. For functions, however, the
closest enclosing block is determined by where the function is declared. Because the
point of declaration is often different from the point at which a function is called,
the access link will generally point to a different activation record than the control
link. Some authors call the access link the static link, as the access links represent the
static nesting structure of blocks in the source program.
The general format for activation records with an access link is shown in Figure 7.5.
Example 7.6
Let us look at the activation records and access links for the example code from
Example 7.5, treating each ML declaration as a separate block.
Figure 7.6 shows the run-time stack after the call to g inside the body of f. As
always, the control links each point to the preceding activation record on the stack.
The control links are drawn on the left here to leave room for the access links
on the right. The access link for each block points to the activation record of the
closest enclosing block in the program text. Here are some important points about
7.3 Functions and Procedures
outer block
x
1
control link
access link
g
…
control link
access link
f
f(3)
…
control link
access link
g(12)
y
3
x
4
control link
access link
z
12
Figure 7.6. Run-time stack after call to g inside f.
this illustration, which follows our convention that we begin a new block for each
ML declaration.
The declaration of g occurs inside the scope of the declaration of x. Therefore,
the access link for the declaration of g points to the activation record for the
declaration of x.
The declaration of f is similarly inside the scope of the declaration of g. Therefore,
the access link for the declaration of f points to the activation record for the
declaration of g.
The call f(3) causes an activation record to be allocated for the scope associated
with the body of f. The body of f occurs inside the scope of the declaration of f.
Therefore, the access link for f(3) points to the activation record for the declaration of f.
The call g(12) similarly causes an activation record to be allocated for the scope
associated with the body of g. The body of g occurs inside the scope of the declaration of g. Therefore, the access link for g(12) points to the activation record
for the declaration of g.
We evaluate the expression x+z by adding the value of the parameter z, stored
locally in the activation record of g, to the value of the global variable x. We find
the location of the global variable x by following the access link of the activation
of g to the activation record associated with the declaration of g. We then follow
the access link in that activation record to find the activation record containing
the variable x.
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As for in-line blocks, the compiler can determine how many access links to follow and
where to find a variable within an activation record at compile time. These properties
are easily determined from the structure of the source code.
To summarize, the control link is a link to the activation record of the previous
(calling) block. The access link is a link to the activation record of the closest enclosing
block in program text. The control link depends on the dynamic behavior of program
whereas the access link depends on only the static form of the program text. Access
links are used to find the location of global variables in statically scoped languages
with nested blocks at run time.
Access links are needed only in programming languages in which functions may
be declared inside functions or other nested blocks. In C, in which all functions are
declared in the outermost global scope, access links are not needed.
7.3.4 Tail Recursion (First-Order Case)
In this subsection we look at a useful compiler optimization called tail recursion elimination. For tail recursive functions, which are subsequently described, it is possible
to reuse an activation record for a recursive call to the function. This reduces the
amount of space used by a recursive function.
The main programming language concept we need is the concept of tail call.
Suppose function f calls function g. Functions f and g might be different functions
or f and g could be the same function. A call to f in the body of g is a tail call if g
returns the result of calling f without any further computation. For example, in the
function
fun g(x) = if x=0 then f(x) else f(x)*2
the first call to f in the body of g is a tail call, as the return value of g is exactly the
return value of the call to f. The second call to f in the body of g is not a tail call
because g performs a computation involving the return value of f before g returns.
A function f is tail recursive if all recursive calls in the body of f are tail calls to f.
Example 7.7
Here is a tail recursive function that computes factorial:
fun tlfact(n,a) = if n <= 1 then a else tlfact(n-1, n * a);
More specifically, for any positive integer n, tlfact(n,a) returns n!. We can see that
tlfact is a tail recursive function because the only recursive call in the body of tlfact
is a tail call.
The advantage of tail recursion is that we can use the same activation record for
all recursive calls. Consider the call tlfact(3,1). Figure 7.7 shows the parts of each
activation record in the computation that are relevant to the discussion.
7.3 Functions and Procedures
control
return result
n
3
a
1
control
return result
n
2
a
3
control
return result
n
1
a
6
Figure 7.7. Three calls to tail recursive tlfact without optimization.
After the third call terminates, it passes its return result back to the second call,
which then passes the return result back to the first call. We can simplify this process
by letting the third call return its result to the activation that made the original call,
tlfact(3,1). Under this alternative execution, when the third call finishes, we can pop
the activation record for the second call as well, as we will no longer need it. In fact,
because the activation record for the first call is no longer needed when the second
call begins, we can pop the first activation record from the stack before allocating
the second. Even better, instead of deallocating the first and then allocating a second
activation record identical to the first, we can use one activation record for all three
calls.
Figure 7.8 shows how the same activation record can be used used three times
for three successive calls to tail recursive tlfact. The figure shows the contents of this
activation record for each call. When the first call, tlfact(3,1), begins, an activation
record with parameters (1,3) is created. When the second call, tlfact(2,3), begins,
the parameter values are changed to (2,3). Tail recursion elimination reuses a single
activation record for all recursive calls, using assignment to change the values of the
function parameters on each call.
tlfact(1,3)
tlfact(2,3)
tlfact(4,3)
control
control
control
return val
return val
return val
n
3
n
2
n
1
a
1
a
3
a
6
Figure 7.8. Three calls to tail recursive tlfact with optimization.
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Tail Recursion as Iteration
When a tail recursive function is compiled, it is possible to generate code that computes the function value by use of an iterative loop. When this optimization is used,
tail recursive functions can be evaluated by only one activation record for a first call
and all resulting recursive calls, not one additional activation record per recursive
call. For example, the preceding tail recursive factorial function may be compiled
into the same code as that of the following iterative function:
fun itfact(n,a) = ( while !n > 0 do (a := !n * !a; n := !n - 1 ); !a );
An activation record for itfact looks just like an activation record for tlfact . If
we look at the values of n and a on each iteration of the loop, we find that they
change in exactly the same way is for tail recursive calls to tlfact. The two functions compute the same result by exactly the same sequence of instructions. Put another way, tail recursion elimination compiles tail recursive functions into iterative
loops.
7.4 HIGHER-ORDER FUNCTIONS
7.4.1 First-Class Functions
A language has first-class functions if functions can be
declared within any scope,
passed as arguments to other functions, and
returned as results of functions.
In a language with first-class functions and static scope, a function value is generally
represented by a closure, which is a pair consisting of a pointer to function code and
a pointer to an activation record.
Here is an example ML function that requires a function argument:
fun map (f, nil) = nil
| map(f, x::xs) = f(x) :: map(f, xs)
The map function take a function f and a list m as arguments, applying f to each
element of m in turn. The result of map(f, m) is the list of results f(x) for elements x of
the list m. This function is useful in many programming situations in which lists are
used. For example, if we have a list of expiration times for a sequence of events and
we want to increment each expiration time, we can do this by passing an increment
function to map.
We will see why closures are necessary by considering interactions between static
scoping and function arguments and return values. C and C++ do not support closures because of the implementation costs involved. However, the implementation
of objects in C++ and other languages is related to the implementation of function
7.4 Higher-Order Functions
values discussed in this chapter. The reason is that a closure and an object both
combine data with code for functions.
Although some C programmers may not have much experience with passing
functions as arguments to other functions, there are many situations in which this
can be a useful programming method. One recognized use for functions as function
arguments comes from an influential software organization concept. In systems programming, the term upcall refers to a function call up the stack. In an important
paper called “The Structuring of Systems Using Upcalls,” (ACM Symp. Operating
Systems Principles, 1985) David Clark describes a method for arranging the functions
of a system into layers. This method makes it easier to code, modularize, and reason
about the system. As in the network protocol stack, higher layers are clients of the
services provided by lower layers. In a layered file system, the file hierarchy layer
is built on the vnode, which is in turn built over the inode and disk block layers. In
Clark’s method, which has been widely adopted and used, higher levels pass handler
functions into lower levels. These handler functions are called when the lower level
needs to notify the higher level of something. These calls to a higher layer are called
upcalls. This influential system design method shows the value of language support
for passing functions as arguments.
7.4.2 Passing Functions to Functions
We will see that when a function f is passed to a function g, we may need to
pass the closure for f, which contains a pointer to its activation record. When f
is called within the body of g, the environment pointer of the closure is used to
set the access link in the activation record for the call to f correctly. The need for
closures in this situation has sometimes been called the downward funarg problem, because it results from passing function as arguments downward into nested
scopes.
Example 7.8
An example program with two declarations of a variable x and a function f passed
to another function g is used to illustrate the main issues:
val x = 4;
fun f(y) = x*y;
fun g(h) = let val x=7 in h(3) + x;
g(f);
In this program, the body of f contains a global variable x that is declared outside the body of f. When f is called inside g, the value of x must be retrieved from
the activation record associated with the outer block. Otherwise the body of f would
be evaluated with the local variable x declared inside g, which would violate static
scope. Here is the same program written with C-like syntax (except for the type
expression int →int) for those who find this easier to read:
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Scope, Functions, and Storage Management
{ int x = 4;
{ int f(int y) {return x*y;}
{ int g(int → int h) {
int x=7;
return h(3) + x;
}
g(f);
}}}
The C-like version of the code reflects a decision, used for simplicity throughout this
book, to treat each top-level ML declaration as the beginning of a separate block.
We can see the variable-lookup problem by looking at the run-time stack after
the call to f from the invocation of g.
x
4
Code
for f
f
g
g(f)
h(3)
Code
for g
h
x
7
y
3
x*y
local var
follow access link
This simplified illustration shows only the data contained in each activation record.
In this illustration, the expression x*y from the body of f is shown at the bottom, the
activation record associated with the invocation of f (through formal parameter h of
g). As the illustration shows, the variable y is local to the function and can therefore
be found in the current activation record. The variable x is global, however, and
located several activation records above the current one. Because we find global
variables by following access links, the access link of the bottom activation record
should allow us to reach the activation record at the top of the illustration.
When functions are passed to functions, we must set the access link for the activation record of each function so that we can find the global variables of that function
correctly. We cannot solve this problem easily for our example program without
extending some run-time data structure in some way.
Use of Closures
The standard solution for maintaining static scope when functions are passed to
functions or returned as results is to use a data structure called a closure. A closure is
a pair consisting of a pointer to function code and a pointer to an activation record.
Because each activation record contains an access link pointing to the record for the
7.4 Higher-Order Functions
x
4
Code
for f
access
f
access
Code
g
for g
g(f )
access
h
x
h(3)
7
access link set
from closure
access
y
3
Figure 7.9. Access link set from closure.
closest enclosing scope, a pointer to the scope in which a function is declared also
provides links to activation records for enclosing blocks.
When a function is passed to another function, the actual value that is passed is
a pointer to a closure. The following steps are used for calling a function, given a
closure:
Allocate an activation record for the function that is called, as usual.
Set the access link in the activation record by using the activation record pointer
from the closure.
We can see how this solves the variable-access problem for functions passed to functions by drawing the activation records on the run-time stack when the program in
Figure 7.8 is executed. These are shown in Figure 7.9.
We can understand Figure 7.9 by walking through the sequence of run-time steps
that lead to the configuration shown in the figure.
1. Declaration of x: An activation record for the block where x is declared is
pushed onto the stack. The activation record contains a value for x and a
control link (not shown).
2. Declaration of f: An activation record for the block where f is declared is
pushed onto the stack. This activation record contains a pointer to the runtime representation of f, which is a closure with two pointers. The first pointer
in the closure points to the activation record for the static scope of f, which
is the activation record for the declaration of f. The second closure pointer
points to the code for f, which was produced during compilation and placed
at some location that is known to the compiler when code for this program is
generated.
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Scope, Functions, and Storage Management
3. Declaration of g: As with the declaration of f, an activation record for the block
where g is declared is pushed onto the stack. This activation record contains a
pointer to the run-time representation of g, which is a closure.
4. Call to g(f): The call causes an activation record for the function g to be allocated on the stack. The size and the layout of this record are determined
by the code for g. The access link is set to the activation record for the scope
where g is declared; the access link points to the same activation record as
the activation record in the closure for g. The activation record contains space
for the parameter h and local variable x. Because the actual parameter is the
closure for f, the parameter value for h is a pointer to the closure for f. The
local variable x has value 7, as given in the source code.
5. Call to h(3): The mechanism for executing this call is the main point of this example. Because h is a formal parameter to g, the code for g is compiled without
knowledge of where the function h is declared. As a result, the compiler cannot insert any instructions telling how to set the access link for the activation
record for the call h(3). However, the use of closures provides a value for the
access link – the access link for this activation record is set by the activation
record pointer from the closure of h. Because the actual parameter is f, the access link points to the activation record for the scope where f is declared. When
the code for f is executed, the access link is used to find x. Specifically, the code
will follow the access link up to the second activation record of the illustration,
follow one additional access link because the compiler knew when generating
code for f that the declaration of x lies one scope above the declaration of f,
and find the value 4 for the global x in the body of f.
As described in step 5, the closure for f allows the code executed at run time to find
the activation record containing the global declaration of x.
When we can pass functions as arguments, the access links within the stack form a
tree. The structure is not linear, as the activation record corresponding to the function
call h(3) has to skip the intervening activation record for g(f) to find the necessary
x. However, all the access links point up. Therefore, it remains possible to allocate
and deallocate activation records by use of a stack (last allocated, first deallocated)
discipline.
7.4.3 Returning Functions from Nested Scope
A related but more complex problem is sometimes called the upward funarg problem,
although it might be more accurate to call it the upward fun-result problem because
it occurs when returning a function value from a nested scope, generally as the return
value of a function.
A simple example of a function that returns a function is this ML code for function
composition:
fun compose(f,g) = (fn x => g(f x));
Given two function arguments f and g, function compose returns the function
7.4 Higher-Order Functions
composition of f and g. The body of compose is code that requires a function
parameter x and then computes g(f(x)). This code is useful only if it is associated
with some mechanism for finding values of f and g. Therefore, a closure is used to
represent compose(f,g). The code pointer of this closure points to compiled code
for “get function parameter x and then computes g(f(x))” and the activation record
pointer of this closure points to the activation record of the call compose(f,g) because
this activation record allows the code to find the actual functions f and g needed to
compute their composition.
We will use a slightly more exciting program example to see how closures solve
the variable-lookup problem when functions are returned from nested scopes. The
following example may give some intuition for the similarity between closures and
objects.
Example 7.9
In this example code, a “counter” is a function that has a stored, private integer value.
When called with a specific integer increment, the counter increments its internal
value and returns the new value. This new value becomes the stored private value
for the next call. The followng ML function, make counter, takes an integer argument
and returns a counter initialized to this integer value:
fun make counter (init : int) =
let val count = ref init (* private variable count *)
fun counter(inc:int) = (count := !count + inc; !count)
in
counter
(* return function counter from make counter *)
end;
val c = make counter(1); (* c is a new counter *)
c(2) + c(2);
(* call counter c twice *)
Function make counter allocates a local variable count, initialized to the value of
the parameter init. Function make counter then returns a function that, when called,
increments count’s value by parameter inc, and then returns the new value of count.
The types and values associated with these declarations, as printed by the compiler, are
val make counter = fn : int → (int → int)
val c = fn : int → int
8 : int
Here is the same program example written in a C-like notation for those who prefer
this syntax:
{int→ int mk counter (int init) {
int count = init;
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Scope, Functions, and Storage Management
make c
Code for
make_counter
access
c
1
mk counter(1)
access
init
1
count
counter
c(2)
access
inc
2
Code for
counter
Figure 7.10. Activation records for function closure returned from function.
int counter(int inc) { return count += inc;}
return counter
}
int→ int c = mk counter(1);
print c(2) + c(2);
}
If we trace the storage allocation associated with this compilation and execution, we
can see that the stack discipline fails. Specifically, it is necessary to save an activation
record that would be popped off the stack if we follow the standard last allocated–first
deallocated policy.
Figure 7.10 shows that the records are allocated and deallocated in execution of
the code from Example 7.9. Here are the sequence of steps involved in producing
the activation records shown in Figure 7.10:
1. Declaration of make counter: An activation record for the block consisting of
the declaration of function make counter is allocated on the run-time stack. The
function name make counter is abbreviated here as make c so that the name
fits easily into the figure. The value of make counter is a closure. The activation
record pointer of this closure points to the activation record for make counter.
The code pointer of this closure points to code for make counter produced at
compile time.
2. Declaration of c: An activation record for the block consisting of the declaration of function c is allocated on the run-time stack. The access pointer of
this activation record points to the first activation record, as the declaration of
7.4 Higher-Order Functions
make counter is the previous block. The value of c is a function, represented
by a closure. However, the expression defining function c is not a function
declaration. It is an expression that requires a call to make counter. Therefore,
after this activation record is set up as the program executes, it is necessary to
call make counter.
3. Call to make counter: An activation record is allocated in order to execute the
function make counter. This activation record contains space for the formal
parameter init of make counter, local variable count and function counter that
will be the return result of the call. The code pointer of the closure for counter
points to code that was generated and stored at compile time. The activation
record pointer points to the third activation record, because the global variables of the function reside here. The program still needs activation record
three after the call to make counter returns because the function counter returned from the call to make counter refers to variables init and count that
are stored in (or reachable through) the activation record for make counter. If
activation record three (used for the call to make counter) is popped off the
stack, the counter function will not work properly.
4. First call to c(2): When the first expression c(2) is evaluated, an activation record
is created for the function call. This activation record has an access pointer that
is set from the closure for c. Because the closure points to activation record
three, so does the access pointer for activation record four. This is important
because the code for counter refers to variable count, and count is located
through the pointer in activation record three.
There are two main points to remember from this example:
Closures are used to set the access pointer when a function returned from a nested
scope is called.
When functions are returned from nested scopes, activation records do not obey
a stack discipline. The activation record associated with a function call cannot
be deallocated when the function returns if the return value of the function is
another function that may require this activation record.
The second point here is illustrated by the preceding example: The activation record
for the call to make counter cannot be deallocated when make counter returns, as this
activation record is needed by the function count returned from make counter.
Solution to Storage Management Problem
You may have noted that, after the code in Example 7.9 is executed, following the
steps just described, we have several activation records left on the stack that might
or might not be used in the rest of the program. Specifically, if the function c is called
again in another expression, then we will need the activation records that maintain
its static scope. However, if the function c is not used again, then we do not need
these activation records. How, you may ask, does the compiler or run-time system
determine when an activation record can be deallocated?
There are a number of methods for handling this problem. However, a full discussion is complicated and not necessary for understanding the basic language design trade-offs that are the subject of this chapter. One solution that is relatively
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straightforward and not as inefficient as it sounds is simply to use a garbage-collection
algorithm to find activation records that are no longer needed. In this approach, a
garbage collector follows pointers to activation records and collects unreachable
activation records when there are no more closure pointers pointing to them.
7.5 CHAPTER SUMMARY
A block is a region of program text, identified by begin and end markers, that may
contain declarations local to this region. Blocks may occur inside other blocks or as
the body of a function or procedure. Block-structured languages are implemented
by activation records that contain space for local variables and other block-related
information. Because the only way for one block to overlap another is for one to
contain the other, activation records are generally managed on a run-time stack,
with the most recently allocated activation record deallocated first (last allocated–
first deallocated).
Parameters passed to functions and procedures are stored in the activation record,
just like local variables. The activation record may contain the actual parameter value
(in pass-by-value) or its address (in pass-by-reference). Tail calls may be optimized to
avoid returning to the calling procedure. In the case of tail recursive functions that do
not pass function arguments, this means that the same activation record may be used
for all recursive calls, eliminating the need to allocate, initialize, and then deallocate
an activation record for each call. The result is that a tail recursive function may be
executed with the same efficiency as an iterative loop.
Correct access to statically scoped global variables involves three implementation
concepts:
Activation records of functions and procedures contain access (or static scoping)
links that point to the activation record associated with the immediately enclosing
block.
Functions passed as parameters or returned as results must be represented as
closures, consisting of function code together with a pointer to the correct lexical
environment.
When a function returns a function that relies on variables declared in a nested
scope, static scoping requires a deviation from stack discipline: An activation
record may need to be retained until the function value (closure) is no longer in
use by the program.
Each of the implementation concepts discussed in this chapter may be tied to a
specific language feature, as summarized in the following table.
Language Feature
Implementation construct
Block
Nested blocks
Functions and procedures
Functions with static scope
Function as arguments and results
Functions returned from nested scope
Activation record with local memory
Control link
Return address, return-result address
Access link
Closures
Failure of stack deallocation order for
activation records
Exercises
EXERCISES
7.1 Activation Records for In-Line Blocks
You are helping a friend debug a C program. The debugger gdb, for example, lets
you set breakpoints in the program, so the program stops at certain points. When
the program stops at a breakpoint, you can examine the values of variables. If you
want, you can compile the programs given in this problem and run them under a
debugger yourself. However, you should be able to figure out the answers to the
questions by thinking about how activation records work.
(a) Your friend comes to you with the following program, which is supposed to
calculate the absolute value of a number given by the user:
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
int main( )
{
int num, absVal;
printf(“Absolute Value\n”);
printf(“Please enter a number:”);
scanf(“%d”,&num);
if (num <= 0)
{
int absVal = num;
}
else
{
int absVal = -num;
}
printf(“The absolute value of %d is %d.\n \n”,num,absVal);
return 0;
}
Your friend complains that this program just doesn’t work and shows you some
sample output:
cardinal: ˜ > gcc -g test.c
cardinal: ˜ > ./a.out
Absolute Value
Please enter a number: 5
The absolute value of 5 is 4.
cardinal: ˜ > ./a.out
Absolute Value
Please enter a number: -10
The absolute value of -10 is 4.
Being a careful student, your friend has also used the debugger to try to track
down the problem. Your friend sets a breakpoints at line 11 and at line 18 and
looks at the address of absval. Here is the debugger output:
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Scope, Functions, and Storage Management
cardinal: ˜ > gdb a.out
(gdb) break test.c : 11
(gdb) break test.c : 18
(gdb) run
Starting program:
Absolute Value
Please enter a number: 5
Breakpoint 1, main (argc=1, argv=0xffbefb04) at test.c : 11
11
int absVal = num;
(gdb) print &absVal
$1 = (int *) 0xffbefa84
(gdb) c
Continuing.
Breakpoint 2, main (argc=1, argv=0xffbefb04) at test.c : 18
18
printf(“The absolute value of %d is %d.\n\n”,num absVal);
(gdb) print &absVal
$2 = (int *) 0xffbefa88
Your friend swears that the computer must be broken, as it is changing the address of a program variable. Using the information provided by the debugger,
explain to your friend what the problem is.
(b) Your explanation must not have been that good, because your friend does not
believe you. Your friend brings you another program:
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
int main( )
{
int num;
printf(“Absolute Value\n”);
printf(“Please enter a number:”);
scanf(“%d”,&num);
if (num >= 0)
{
int absVal = num;
}
else
{
int absVal = -num;
}
{
int absVal;
printf(“The absolute value of %d is %d.\n\n”,num, absVal);
}
}
This program works. Explain why.
Exercises
(c) Imagine that line 17 of the program in part (b) was split into three lines:
17a:
17b:
17c:
{
...
}
Write a single line of code to replace the ... that would guarantee this program
would NEVER be right. You may not declare functions or use the word absVal
in this line of code.
(d) Explain why the change you made in part (c) breaks the program in part (b).
7.2 Tail Recursion and Iteration
The following recursive function multiplies two integers by repeated addition:
fun mult(a,b) =
if a=0 then 0
else if a = 1 then b
else b + mult(a-1,b);
This is not tail recursive as written, but it can be transformed into a tail recursive
function if a third parameter, representing the result of the computation done so
far, is added:
fun mult(a,b) =
let fun mult1(a,b,result) =
if a=0 then 0
else if a = 1 then b + result
else mult1(a-1, b, result+b)
in
mult1(a, b, 0)
end;
Translate this tail recursive definition into an equivalent ML function by using
a while loop instead of recursion. An ML while has the usual form, described in
Subsection 5.4.5.
7.3 Time and Space Requirements
This question asks you to compare two functions for finding the middle element of
a list. (In the case of an even-length list of 2n elements, both functions return the
n + 1st.) The first uses two local recursive functions, len and get. The len function
finds the length of a list and get(n,l) returns the nth element of list l. The second
middle function uses a subsidiary function m that recursively traverses two lists,
taking two elements off the first list and one off the second until the first list is
empty. When this occurs, the first element of the second list is returned:
exception Empty;
fun middle1(l) =
let fun len(nil) = 0
|
len(x : : l) = 1+len(l)
and get(n, nil) = raise Empty
|
get(n, x : : l) = if n=1 then x else get(n-1, l)
in
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Scope, Functions, and Storage Management
get((len(l) div 2)+1, l)
end;
fun middle2(l) =
let fun m(x, nil) = raise Empty
|
m(nil,x : : l) = x
|
m([y],x : : l) = x
|
m(y::(z : : l1),x : : l2) = m(l1, l2)
in
m(l, l)
end;
Assume that both are compiled and executed by a compiler that optimizes use of
activation records or “stack frames.”
(a) Describe the approximate running time and space requirements of middle1
for a list of length n. Just count the number of calls to each function and the
maximum number of activation records that must be placed on the stack at
any time during the computation.
(b) Describe the approximate running time and space requirements of middle2 for
a list of length n. Just count the number of calls to m and the maximum number
of activation records that must be placed on the stack at any time during the
computation.
(c) Would an iterative algorithm with two pointers, one moving down the list twice
as fast as the other, be significantly more or less efficient than middle2? Explain
briefly in one or two sentences.
7.4 Parameter Passing
Consider the following procedure, written in an Algol/Pascal-like notation:
proc power(x, y, z : int)
begin
z := 1
while y > 0 do
z := z*x
y := y-1
end
end
The code that makes up the body of power is intended to calculate xy and place the
result in z. However, depending on the actual parameters, power may not behave
correctly for certain combinations of parameter-passing methods. For simplicity,
we only consider call-by-value and call-by-reference.
(a) Assume that a and c are assignable integer variables with distinct L-values.
Which parameter-passing methods make c = aa after a call power(a, a, c). You
may assume that the R-values of a and c are nonnegative integers.
(b) Suppose that a and c are formal parameters to some procedure P and that the
preceding expression power(a, a, c) is evaluated inside the body of P. If a and c
are passed to P by reference and become aliases, then what parameter-passing
Exercises
method(s) will make c = aa after a call power(a, a, c)? If, after the call, c = aa ,
does that mean that power actually performed the correct calculation?
7.5 Aliasing and Static Analysis
The designers of the systems programming language Euclid wanted programs written in Euclid to be easy to verify formally. They based the language on Pascal, changing those those features that made verification complicated. One set of changes
eliminates aliasing, in which two different names denote the same location. Aliasing can arise in several ways in Pascal. For example, suppose that a program contains
the declaration
procedure P(var x,y: real);
Later in the body of the program the procedure call P(z,z) appears. While executing
the body of P for this call, both x and y refer to the same location. In Pascal this call
is perfectly legal, but in Euclid, the program is not considered syntactically correct.
(a) Explain briefly why aliasing might make it difficult to verify (reason about)
programs.
(b) What problems do you think the designers of Euclid had to face in prohibiting
aliasing? Would it be easy to implement a good compile-time test for aliasing?
7.6 Pass-by-Value-Result
In pass-by-value-result, also called call-by-value-result and copy-in/copy-out, parameters are passed by value, with an added twist. More specifically, suppose a
function f with a pass-by-value-result parameter u is called with actual parameter
v. The activation record for f will contain a location for formal parameter u that is
initialized to the R-value of v. Within the body of f, the identifier u is treated as an
assignable variable. On return from the call to f, the actual parameter v is assigned
the R-value of u.
The following pseudo-Algol code illustrates the main properties of pass-byvalue-result:
var x : integer;
x := 0;
procedure p(value-result y : integer)
begin
y := 1;
x := 0;
end;
p(x);
With pass-by-value-result, the final value of x will be 1: Because y is given a new
location distinct from x, the assignment to x does not change the local value of y.
When the function returns, the value of y is assigned to the actual parameter x.
If the parameter were passed by reference, then x and y would be aliases and the
assignment to x would change the value of y to 0. If the parameter were passed by
value, the assignment to y in the body of p would not change the global variable x
and the final value of x would also be 0.
Translate the preceding program fragment into ML (or pseudo-ML if you prefer) in a way that makes the operations on locations, and the differences between
L-values and R-values, explicit. Your solution should have the form
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Scope, Functions, and Storage Management
val x = ref 0;
fun p(y : int ref) =
...;
p(x);
Note that, in ML, like C and unlike Algol or Pascal, a function may be called as a
procedure.
7.7 Parameter-Passing Comparison
For the following Algol-like program, write the number printed by running the
program under each of the listed parameter passing mechanisms. Pass-by-valueresult, also sometimes called copy-in/copy-out, is explained in problem 6:
begin
integer i;
procedure pass ( x, y );
integer x, y; // types of the formal parameters
begin
x := x + 1;
y := x + 1;
x := y;
i := i + 1
end
i := 1;
pass (i, i);
print i
end
(a) pass-by-value
(b) pass-by-reference
(c) pass-by-value-result
7.8 Static and Dynamic Scope
Consider the following program fragment, written both in ML and in pseudo-C:
1
2
3
4
5
6
7
8
let x = 2 in
let val fun f(y) = x + y in
let val x = 7 in
x+
f(x)
end
end
end;
int x = 2; {
int f (int y) { return x + y; } {
int x = 7; {
x+
f(x);
}
}
}
The C version would be legal in a version of C with nested functions.
(a) Under static scoping, what is the value of x + f(x) in this code? During the
execution of this code, the value of x is needed three different times (on lines
2, 4, and 5). For each line where x is used, state what numeric value is used
when the value of x is requested and explain why these are the appropriate
values under static scoping.
Exercises
(b) Under dynamic scoping, what is the value of x + f(x) in this code? For each line
in which x is used, state which value is used for x and explain why these are the
appropriate values under dynamic scoping.
7.9 Static Scope in ML
ML uses static scope. Consider two functions of this form:
fun f(x) = . . . ;
fun g(x) = . . . f . . . ;
If we treat a sequence of declarations as if each begins a new block, static scoping
implies that any mention of f in g will always refer to the declaration that precedes
it, not to future declarations in the same interactive session.
(a) Explain how ML type inference relies on static scope. Would the current typing
algorithm work properly if the language were changed to use dynamic scope?
(b) Suppose that, instead of using the language interactively, we intend to build
a “batch” compiler. This compiler should accept any legal interactive session
as input. The difference is that the entire program will be read and compiled
before printing any type information or output. Will the problem with type
inference and dynamic scope become simpler in this context?
7.10 Eval and Scope
Many compilers look at programs and eliminate any unused variables. For example,
in the following program, x is unused so it could be eliminated:
let x = 5 in f(0) end
Some languages, including Lisp and Scheme, have a way to construct and evaluate
expressions at run time. Constructing programs at run time is useful in certain kinds
of problems, such as symbolic mathematics and genetic algorithms.
The following program evaluates the string bound to s, inside the scope of two
declarations:
let s = read text from user( ) in
let x = 5 and y = 3 in eval s end
end
If s were bound to 1+x*y then eval would return 16. Assume that eval is a special
language feature and not simply a library function.
(a) The “unused variable” optimization and the “eval” construct are not compatible. The identifiers x and y do not appear in the body of the inner let (the
part between in and end), yet an optimizing compiler cannot eliminate them
because the eval might need them. In addition to the values 5 and 3, what information does the language implementation need to store for eval that would
not be needed in a language without eval?
(b) A clever compiler might look for eval in the scope of the let. If eval does not
appear, then it may be safe to perform the optimization. The compiler could
eliminate any variables that do not appear in the scope of the let declaration.
Does this optimization work in a statically scoped language? Why or why not?
(c) Does the optimization suggested in part (b) work in a dynamically scoped
language? Why or why not?
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Scope, Functions, and Storage Management
7.11 Lambda Calculus and Scope
Consider the following ML expression:
fun foo(x : int) =
let fun bar(f) = fn x => f (f (x))
in
bar(fn y => y + x)
end;
In β reduction on lambda terms, the function argument is substituted for the formal
parameter. In this substitution, it is important to rename bound variables to avoid
capture. This question asks about the connection between names of bound variables and static scope. Using a variant of substitution that does not rename bound
variables, we can investigate dynamic scoping.
(a) The following lambda term is equivalent to the function foo:
λx.((λ f.λx. f ( f x)) (λy.y + x))
Use β reduction to reduce this lambda term to normal form.
(b) Using the example reduction from part (a), explain how renaming bound variables provides static scope. In particular, say which preceding variable (x, f ,
or y) must be renamed and how some specific variable reference is therefore
resolved statically.
(c) Under normal ML static scoping, what is the value of the expression foo(3)(2)?
(d) Give a lambda term in normal form that corresponds to the function that the
expression in part (a) would define under dynamic scope. Show how you can
reduce the expression to get this normal form by not renaming bound variables
when you perform substitution.
(e) Under dynamic scoping, what is the value of the expression foo(3)(2)?
(f) In the usual statically scoped lambda calculus, α conversion (renaming bound
variables) does not change the value of an expression. Use the example expression from part (a) to explain why α conversion may change the value of
an expression if variables are dynamically scoped. (This is a general fact about
dynamically scoped languages, not a peculiarity of lambda calculus.)
7.12 Function Calls and Memory Management
This question asks about memory management in the evaluation of the following
statically scoped ML expression:
val x = 5;
fun f(y) = (x+y) -2;
fun g(h) = let val x = 7 in h(x) end;
let val x = 10 in g(f) end;
(a) Fill in the missing information in the following illustration of the run-time
stack after the call to h inside the body of g. Remember that function values are
represented by closures and that a closure is a pair consisting of an environment
(pointer to an activation record) and compiled code.
In this figure, a bullet (•) indicates that a pointer should be drawn from
this slot to the appropriate closure or compiled code. Because the pointers
to activation records cross and could become difficult to read, each activation
Exercises
record is numbered at the far left. In each activation record, place the number of
the activation record of the statically enclosing scope in the slot labeled “access
link.” The first two are done for you. Also use activation record numbers for
the environment pointer part of each closure pair. Write the values of local
variables and function parameters directly in the activation records.
Activation Records
(1)
(2)
(3)
(4)
(5)
(6)
g(f)
h(x)
access link
x
access link
f
access link
g
access link
x
access link
h
x
access link
y
Closures
Compiled Code
(0)
(1)
•
( )
•
( )
(
(
(
),
•
(
),
•
code for f
)
•
code for g
... ...
)
(b) What is the value of this expression? Why?
7.13 Function Returns and Memory Management
This question asks about memory management in the evaluation of the following
statically scoped ML expression:
val x = 5;
fun f(y) =
let val z = [1, 2, 3] (* declare list *)
fun g(w) = w+x+y (* declare local function *)
in
g
(* return local function *)
end;
val h = let val x=7 in f(3) end;
h(2);
(a) Write the type of each of the declared identifiers (x, f, and h).
(b) Because this code involves a function that returns a function, activation records
cannot be deallocated in a last-in/first-out (LIFO) stacklike manner. Instead,
let us just assume that activation records will be garbage collected at some
point. Under this assumption, the activation record for the call f in the expression for h will still be available when the call h(2) is executed.
Fill in the missing information in the following illustration of the run-time
stack after the call to h at the end of this code fragment. Remember that function values are represented by closures and that a closure is a pair consisting
of an environment (pointer to an activation record) and compiled code.
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Scope, Functions, and Storage Management
In this figure, a bullet (•) indicates that a pointer should be drawn from
this slot to the appropriate closure, compiled code, or list cell. Because the
pointers to activation records cross and could become difficult to read, each
activation record is numbered at the far left. In each activation record, place
the number of the activation record of the statically enclosing scope in the slot
labeled “access link.” The first two are done for you. Also use activation record
numbers for the environment pointer part of each closure pair. Write the values
of local variables and function parameters directly in the activation records.
Activation Records
(1)
access link
x
access link
f
access link
h
access link
x
f(3) access link
y
z
g
h(2) access link
w
(2)
(3)
(4)
(5)
(6)
Heap Data and Closures
Compiled Code
(0)
(1)
•
( )
•
( )
(
),
•
(
),
•
code for f
code for g
)
1
•
(
(
•
2
•
3
)
(c) What is the value of this expression? Explain which numbers are added
together and why.
(d) If there is another call to h in this program, then the activation record for
this closure cannot be garbage collected. Using the definition of garbage given
in the Lisp chapter (see Chapter 3), explain why, as long as h is reachable,
mark-and-sweep will fail to collect some garbage that will never be accessed
by the program.
7.14 Recursive Calls and Memory Management
This question asks about memory management in the evaluation of the following
ML expression (with line numbers):
1>
2>
3>
4>
5>
6>
7>
8>
9>
fun myop(x,y) = x*y;
fun recurse(n) =
if n=0 then 1
else myop(n, recurse(n-1));
let
fun myop(x,y) = x + y
in
recurse(1)
end;
(a) Assume that expressions are evaluated by static scope. Fill in the missing information in the following illustration of the run-time stack after the last call
to myop on line 4 of this code fragment. Remember that function values are
Exercises
represented by closures and that a closure is a pair consisting of an environment
(pointer to an activation record) and compiled code. Remember also that in
ML function arguments are evaluated before the function is called.
In this figure, a bullet (•) indicates that a pointer should be drawn from this
slot to the appropriate closure, compiled code, or list cell. Because the pointers
to activation records cross and could become difficult to read, each activation
record is numbered at the far left. In each activation record, place the number
of the activation record of the statically enclosing scope in the slot labeled
“access link” and the number of the activation record of the dynamically enclosing scope in the slot labeled “control link.” The first two are done for you.
Also use activation record numbers for the environment pointer part of each
closure pair. Write the values of local variables and function parameters in the
activation records. Write the line numbers of code inside the brackets [ ].
Activation Records
(1)
(2)
(3)
(4)
recurse(1)
(5)
myop(1,0)
(6)
recurse(0)
access link
control link
myop
access link
control link
recurse
access link
control link
myop
access link
control link
n
access link
control link
x
y
access link
control link
n
Closures
(0)
(0)
•
(1)
(1)
•
( )
( )
•
( )
( )
(
(
)
)
(
(
)
)
Compiled Code
( ),
• code for myop
defined line [ ]
( ),
• code for recurse
( ),
• code for myop
defined line [ ]
(b) What is the value of this expression under static scope? Briefly explain how
your stack diagram from part (a) allows you to find which value of myop to use.
(c) What would be the value if dynamic scope were used? Explain how the stack
diagram would be different from the one you have drawn in part (a) and briefly
explain why.
7.15 Closures
In ANSI C, we can pass and return pointers to functions. Why does the implementation of this language not require closures?
7.16 Closures and Tail Recursion
The function f in this declaration of factorial is formally tail recursive: The only
calls in the body of f are tail calls:
201
202
Scope, Functions, and Storage Management
fun fact(n) =
let f(n, g) = if n=0 then g(1)
else let h(i) = g(i*n)
in
f(n-1, h)
end
in
f(n, fn x => x)
end
This question asks you to consider the problem of applying the ordinary tail recursion elimination optimization to this function.
(a) Fill in the missing information in the following outline of the activation records
resulting from the unoptimized execution of f(2, fn x => x). You may want to
draw closures or other data.
f(2, fn x => x)
f(1, h)
f(0, h)
access link
n
g
access link
n
g
access link
n
g
(b) What makes this function more difficult to optimize than the other examples
discussed in this chapter? Explain.
7.17 Tail Recursion and Order of Operations
This asks about the order of operations when converting a function to tail recursive form and about passing functions from one scope to another. Here are three
versions of our favorite recursive function, factorial:
factA (n) =
Normal recursive version
(if n=0 then 1 else n*fact A (n-1))
fact’(n,a) =
factB (n) =
Tail-recursive version
(if n=0 then a else fact’(n-1,(n*a)))
fact’(n,1)
Another tail-recursive version
(if n=0 then rest(1)
else fact”(n-1,(fn(r)⇒ rest(n*r))))
factC (n) = fact”(n,fn(answer)⇒answer)
Here is the evaluation of fact A (3) and fact B (3):
fact A (3) = (if 3=0 then 1 else 3*fact A (2))
= (if false then 1 else 3*fact A (2))
= (3*fact(2))
= (3*(if 2=0 then 1 else 2*fact A (1)))
= (3*(2*fact(1)))
fact”(n,rest) =
Exercises
=
=
=
=
factB (3) =
=
=
=
=
=
=
=
=
(3*(2*(if 1=0 then 1 else 1*fact A (0))))
(3*(2*(1*fact(0))))
(3*(2*(1*(if 0=0 then 1 else 0*fact A (-1)))))
(3*(2*(1*(1))))
fact’(3,1)
(if 3=0 then 1 else fact’(2,(3*1)))
(fact’(2,(3*1)))
(if 2=0 then (3*1) else fact’(1,(2*(3*1))))
(fact’(1,(2*(3*1))))
(if 1=0 then (2*(3*1)) else fact’(0,(1*(2*(3*1)))))
(fact’(0,(1*(2*(3*1)))))
(if 0=0 then (1*(2*(3*1))) else fact’(-1,(0*(1*(2*(3*1))))))
(1*(2*(3*1)))
The multiplications are not carried out in these symbolic calculations so that we
can compare the order in which the numbers are multiplied. The evaluations show
that fact A (3) multiplies by 1 first and by 3 last; fact B (3) multiplies by 3 first and 1
last. The order of multiplications has changed.
(a) Show that factC (3) multiplies the numbers in the correct order by a symbolic
calculation similar to those above that does not carry out the multiplications.
(b) We can see that fact” is tail recursive in the sense that there is nothing left
to do after the recursive call. If these functions were entered into a compiler
that supported tail recursion elimination, would factC (n) be closer in efficiency
to fact A (n) or fact B (n)? First, ignore any overhead associated with creating
functions, then discuss the cost of passing functions on the recursive calls.
(c) Do you think it is important to preserve the order of operations when translating an arbitrary function into tail recursive form? Why or why not?
203
8
Control in Sequential Languages
After looking briefly at the history of jumps and structured control, we will study
exceptions and continuations. Exceptions are a form of jump that exits a block or
function call, returning to some previously established point for handling the exception. Continuations are a more general form of “return” based on calling a function
that is passed into a block for this purpose. The chapter concludes with a discussion
of force and delay, complimentary techniques for delaying computation by placing it
inside a function and forcing delayed computation with a function call.
8.1 STRUCTURED CONTROL
8.1.1 Spaghetti Code
In Fortran or assembly code, it is easy to write programs with incomprehensible control structure. Here is a short code fragment that illustrates a few of the possibilities.
The fragment includes a Fortran CONTINUE statement, which is an instruction that
does nothing but is used for the purpose of placing a label between two instructions.
If you scan the code from top to bottom, you might get the idea that the instructions
between labels 10 and 20 act together to perform some meaningful task. However,
then as you scan downward, you can see that it is possible later to jump to instruction
11, which is in the middle of this set of instructions:
10 IF (X .GT. 0.000001) GO TO 20
X = -X
11 Y = X*X - SIN(Y)/(X+1)
IF (X .LT. 0.000001) GO TO 50
20 IF (X*Y .LT. 0.00001) GO TO 30
X = X-Y-Y
30 X = X+Y
...
50 CONTINUE
204
8.1 Structured Control
GUY STEELE
An energetic man with sincere conviction and a sense of humor, Guy
Lewis Steele, Jr., is a Distinguished Engineer at Sun Microsystems, Inc. He
received his A.B. in applied mathematics from Harvard College (1975)
and his S.M. and Ph.D. in computer science and artificial intelligence
from MIT (1977, 1980). He is known for his seminal work on Scheme, including the revolutionary continuation-based Rabbit compiler described
in his S.M. thesis, and subsequent years of work on Common Lisp, parallel computing, and Java. In addition to reference books on Common Lisp,
Fortran, C, and Java, Guy was the original lead author of The Hacker’s
Dictionary, now revised as The New Hacker’s Dictionary.
Guy Steele’s interests also include chess and music. A Life Member
of the United States Chess Federation, he has sung in the bass section
of the MIT Choral Society, the Masterworks Chorale, and in choruses
with the Pittsburgh Symphony Orchestra and the Boston Concert Opera.
Guy composed The Telnet Song, published in the April 1984 ACM, which
pokes lighthearted fun at the exponential explosion of ctrl-Q keystrokes
needed to exit cascaded telnet sessions.
In a masterful 1998 plenary address at the Object-Oriented Programming: Systems, Languages, and Applications (OOPSLA) conference, Guy
illustrated the difficulty of growing a language by starting with words of
one syllable and defining every longer word in his talk from one-syllable
words and previously defined words of two or more syllables. You can
find the text of this speech on the web. Guy Steele was awarded the ACM
Grace Murray Hopper Award in 1988.
X=A
Y = B-A + C*C
GO TO 11
...
205
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Control in Sequential Languages
Although no short sequence can begin to approximate the Byzantine control flow
of many archaic Fortran programs, this example may give you some feel for the
kind of confusing control flow that was used in many programs in the early days of
computing.
8.1.2 Structured Control
In the 1960s, programmers began to understand that unstructured jumps could make
it difficult to understand a program. This was partly a realization about programming
style and partly a realization about programming languages: If incomprehensible
control flow is bad programming style, then programming languages should provide
mechanisms that make it easy to organize the control structure of programs. This led
to the development of some constructs that structure jumps:
if . . . then . . . else . . . end
while . . . do . . . end
for . . . { . . . }
case . . .
These are now adopted in virtually all modern languages.
In modern programming style, we group code in logical blocks, avoid explicit
jumps except for function returns, and cannot jump into the middle of a block or
function body.
The restriction on jumps into blocks illustrates the value of leaving a construct
out of a programming language. If a label is placed in the middle of a function
body and a program executes a jump to this label, what should happen? Should an
activation record be created for the function call? If not, then local variables will
not be meaningful. If so, then how will function parameters stored in the activation
record be set? Without executing a call to the function, there are no parameter values
to use in the call. Because these questions have no good, convincing, clear answers,
it is better to design the compiler to reject programs that might jump into the middle
of a function body.
Although most introductory programming books and courses today emphasize
the importance of clean control structure and discourage the use of go to (if the
language even allows it), it took many years of discussion and debate to reach this
modern point of view. One reason for the change in perspective over the years is the
decreasing importance of instruction-level efficiency. In 1960 and even 1970, there
were many applications in which it was useful to save the cost of a test, even if it
meant complicating the control structure of the program. Therefore, programmers
considered it important to be able to jump out of the middle of a loop, avoiding
another test at the top of the loop.
In the 1980s and 1990s, as computer speed increased, the number of applications
in which a small change in efficiency would truly matter decreased significantly, to
the point at which, in the 1990s, Java was introduced without any go to statement.
Those interested in history may enjoy reading E.W. Dijkstra’s March 1968 letter
to the editor of Communications of the ACM, “Go To Considered Harmful,” later
8.2 Exceptions
posted on the Association for Computing Machinery web site as the October 1995
“Classic of the Month.” The letter begins with these sentences:
For a number of years I have been familiar with the observation that the quality
of programmers is a decreasing function of the density of go to statements in
the programs they produce. More recently I discovered why the use of the go
to statement has such disastrous effects, and I became convinced that the go to
statement should be abolished from all “higher level” programming languages. . . .
Simple control structures such as if-then-else have now been in common use since
the rise of Pascal in the late 1970s. Exceptions and continuations, discussed in the
remainder of this chapter, are more recent innovations in programming languages.
8.2 EXCEPTIONS
8.2.1 Purpose of an Exception Mechanism
Exceptions provide a structured form of jump that may be used to exit a construct such
as a block or function invocation. The name exception suggests that exceptions are
to be used in exceptional circumstances. However, programming languages cannot
enforce any sort of intention like this. Exceptions are a basic mechanism that can be
used to achieve the following effects:
jump out of a block or function invocation
pass data as part of the jump
return to a program point that was set up to continue the computation.
In addition to jumping from one program point to another, there is also some memory
management associated with exceptions. Specifically, unnecessary activation records
may be deallocated as the result of the jump. We subsequently see how this works.
Exception mechanisms may be found in many programming languages, including
Ada, C++, Clu, Java, Mesa, ML, and PL/1. Every exception mechanism includes two
constructs:
a statement or expression form for raising an exception, which aborts part of the
current computation and causes a jump (transfer of control),
a handler mechanism, which allows certain statements, expressions, or function
calls to be equipped with code to respond to exceptions raised during their
execution.
Another term for raising an exception is throwing an exception; another term for
handling an exception is catching an exception.
There are several reasons why exceptions have become an accepted language
construct. Many languages do not otherwise have a clean mechanism for jumping
out of a function call, for example, aborting the call. Rather than using go tos, which
can be used in unstructured ways, many programmers prefer exceptions, which can
be used to jump only out of some part of a program, not into some part of the program
that has not been entered yet. Exceptions also allow a programmer to pass data as
part of the jump, which is useful if the program tries to recover from some kind of
error condition. Exceptions provide a useful dynamic way of determining to where
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a jump goes. If more than one handler is declared, the correct handler is determined
according to dynamic scoping rules. This effect would not be easy to achieve with
other forms of control jumps.
The importance of dynamic scoping is illustrated in the following example.
Example 8.1
This example involves computing the inverse of a matrix. For the purpose of this
example, a matrix is an array of real numbers. We compute matrix multiplication, used
in linear algebra, by multiplying the entries of two matrices together in a certain way.
If A is a matrix and I is the identity matrix (with ones along the diagonal and zeros
elsewhere), then the inverse A−1 of A is a matrix such that the product AA−1 = I.
A square matrix has an inverse if and only if something called the determinant of A
is not zero. It is approximately as much computational work to find the determinant
of a matrix as it is to invert a matrix.
Suppose we write a function for inverting matrices that have real-number entries.
Because only matrices with nonzero determinants can be inverted, we cannot correctly compute the inverse of a matrix if the determinant turns out to be zero. One
way of handling this difficulty might be to check the determinant before calling the
invert function. This is not a good idea, however, as computing the determinant of a
matrix is approximately as much work as inverting it. Because we can easily check
the determinant as we try to convert the inverse, it makes more sense to try to invert
the matrix and then raise an exception in the event that we detect a zero determinant.
This leads to code structured as follows:
exception Determinant; (* declare Determinant exception *)
fun invert(aMatrix) =
...
if . . .
then raise Determinant (* if matrix determinant is zero *)
else . . .
end;
invert(myMatrix) handle Determinant . . .;
In this example, the function invert raises an exception if the determinant of
aMatrix turns out to be zero. If the function raises this exception as a result of the
call invert(myMatrix), then because of the associated handler declaration, the call to
invert is aborted and control is transferred to the code immediately following handle
Determinant.
In this example, the exception mechanism is used to handle a condition that makes
it impossible to continue the computation. Specifically, if the determinant is zero, the
matrix has no inverse, and it is impossible for the invert function to return the inverse
of the matrix.
If you think about it, this example illustrates the need for some kind of dynamic
scoping mechanism. More specifically, suppose that there are several calls to invert in a
program that does matrix calculations. Each of these calls could cause the Determinant
8.2 Exceptions
exception to be raised. When a call raises this exception, where would we like the
exception to be handled?
The main idea behind dynamic scoping of exception handlers is that the code
in which the invert function is called is the best place to decide what to do if the
determinant is zero. If exceptions were statically scoped, then raising an exception
would jump to the exception handler that is declared lexically before the definition
of the invert function. This handler would be in the program text before the invert
function. If invert is in a program library, then the handler would have to be in the
library. However, the person who wrote the library has no idea what to do in the cases
in which the exception occurs. For this reason, exceptions are dynamically scoped:
When an exception is raised, control jumps to the handler that was most recently
established in the dynamic call history of the program. We will see how this works in
more detail in after looking at some specific exception mechanisms.
8.2.2 ML Exceptions
The ML mechanism has three parts:
Exceptions declarations, which declare the name of an exception and specify the
type of data passed when this exception is raised,
A raise expression, which raises an exception and passes data to the handler,
Handler declarations, which establish handlers.
An ML exception declaration has the form
exception name of type
where name is the name of the exception and type is the type of data that are
passed to an exception handler when this exception is raised. The last part of this
declaration, of type, is optional. For example, the exception Ovflw below does not
pass data, but the exception Signal requires an integer value that is passed to the
receiving handler:
exception Ovflw;
exception Signal of int;
An ML raise expression has the form
raise name
arguments
where name is a previously declared exception name and arguments have a type
that matches the exception declaration. For example, the exceptions previously declared can be raised with the following expressions:
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raise Ovflw;
raise Signal (x+4);
An ML handler is part of an expression form that allows a handler to be specified.
An expression with handler has the form
exp1 handle pattern => exp2;
We evaluate this expression by evaluating exp1. If the evaluation of exp1 terminates normally, then this is the value of the larger expression. However, if exp1 raises
an exception that matches pattern, then any values passed in raising the exception
are bound according to pattern and exp2 is evaluated. In this case, the value of
exp2 becomes the value of the entire expression. If exp2 raises an exception or
exp2 raises an exception that does not match pattern, then exp1 handle pattern
=> exp2 has an uncaught exception that can be caught by the handler established
by an enclosing expression or function call. A more general form involving multiple
patterns is described below.
Examples
Here is a simple example that uses the “overflow” exception previously mentioned:
exception Ovflw;
(* Declare exception name *)
fun f(x) = if x <= Min (* Function with exception *)
then raise Ovflw
else 1/x;
(* – Expression that handles Ovflw in two ways – *)
( f(x) handle Ovflw => 0 ) / (f(x) handle Ovflw => 1)
Note that the final expression has two different handlers for the Ovflw exception.
In the numerator, the handler returns value 0, making the fraction zero if overflow
occurs. In the denominator, it would cause division by zero if the handler returns
zero; the handler therefore returns 1 if the Ovflw exception is raised. This example
shows how the choice of handler for an exception raised inside a function depends
on how the function is called.
Here is another example, illustrating the way that data may be passed and used:
exception Signal of int;
fun f(x) = if x=0 then raise Signal(0)
else if x=1 then raise Signal(1)
else if x 10 then raise Signal(x-8)
else (x-2) mod 4;
8.2 Exceptions
f(10) handle Signal(0) => 0
| Signal(1) => 1
| Signal(x) => x+8;
The handler in this expression uses pattern matching, which follows the form
established for ML function declarations. More specifically, the meaning of an expression of the form
<exp> handle <pattern1 > => <exp1 >
| <pattern2 > = <exp2 >
...
| <patternn > => <expn >
is determined as follows:
1. The expression to the left of the handle keyword is evaluated.
2. If this expression terminates normally, its value is the value of the entire expression with handler declaration; the handler is never invoked. (If evaluation
of this expression does not terminate, then evaluation of the enclosing expression cannot terminate either.) If the expression raises an exception that
matches <patterni > (and there is no matching handler declared within <exp>),
then the handler is invoked.
3. If the handler is invoked, pattern matching works just as an ordinary ML
function call. The value passed by exception is matched against <pattern1 >,
<pattern2 >, . . ., <patternn > in order until a match is found. If the value matches
<patterni >, this causes any variables in <patterni > to be bound to values. The
corresponding expression <expi > is evaluated with the bindings created by
pattern matching.
8.2.3 C++ Exceptions
C++ exceptions are similar in spirit to exceptions in ML and other languages. The C++
syntax involves try blocks, a throw statement, and a catch block to handle exceptions
that have been thrown within the associated try block. C++ exceptions are slightly
less elegant than ML exceptions because C++ exceptions are not a separate kind of
entity recognized by the type system.
The try block surrounds statements in which exceptions may be thrown. Here is
a code fragment showing the form of a try block:
try {
// statements that may throw exceptions
}
A throw statement may be executed within a try block, either directly by a statement
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in the try block or from a function called directly or indirectly from the block. A
throw statement contains an expression and passes the value of the expression. Here
is an example in which a character value is thrown:
throw “This generates a char * exception”;
A catch block may immediately follow a try block and receive any thrown exceptions.
Here is an example a catch-block receiving char * exceptions:
catch (char *message) {
// statements that process the thrown char * exception
}
C++ uses types to distinguish between different kinds of exceptions. The throw statement may be used to throw different types of values:
throw ”Hello world”;
// throws a char *
throw 18;
// throws an int
throw new String(”hello”); // throws a String *
In a block that may throw more than one type of exceptions, multiple catch blocks
may be used:
try {
// code may throw char pointers and other pointers
}
catch (char *message) {
// code processing the char pointers thrown as exceptions
}
catch (void *whatever) {
// code processing all other pointers thrown as exceptions
}
Although it is possible to throw objects, some care is required because local objects
are deallocated when an exception is thrown. More discussion of objects and storage
allocation in C++ appears in Chapter 12.
Here is a more complete C++ example, combining try, throw, and catch:
void f(char * c) {
. . . if (c == 0) throw exception(”Empty string argument”);
}
main() {
8.2 Exceptions
try {
. . . f(x); . . .
}
catch (exception) {
exit(1);
}
}
As in ML and other languages, throwing a C++ exception causes a control transfer to
the most recently established handler that is appropriate for the exception. Whereas
ML uses pattern matching to determine whether a handler is appropriate for an
exception, C++ uses type matching.
As is generally the case for C/C++, the type-matching issues are a little more
complicated that we would like. To be specific, a handler of the form
catch(T t)
catch(const T t)
catch(T& t)
catch(const T& t)
can catch exception objects of type E if
T
and E are the same type, or
is an accessible base class of E at the throw point, or
T and E are pointer types and there exists a standard pointer conversion from E
to T at the throw point.
T
The rules for standard pointer conversion are also a little complicated, but we do not
discuss them here because they are not critical for understanding the basic idea of
exceptions.
One significant difference between ML and C++ that is important when programming with exceptions is that ML is garbage collected and C++ is not. In both
languages, raising or throwing an exception will cause all run-time stack activation
records between the point of the throw and the point of the catch to be deallocated. However, storage that is reachable from activation records may no longer be
reachable after the exception, as the activation records with pointers no longer exist.
This problem is discussed at the end of Subsection 8.2.4. There are also some details
regarding the way exceptions work in constructors and destructors of objects that
will be of interest to C++ programmers.
8.2.4 More about Exceptions
This section contains some additional examples of exceptions, with ML used as an
illustrative syntax, and we discuss the interaction between exceptions, storage management, and static type checking. If you have used exceptions in a language different from ML, you might think about how the exception mechanism you have used
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is different and whether the difference is a consequence of some basic difference
between the underlying programming languages.
Exceptions for Error Conditions
Exceptions arose as a mechanism for handling errors that occur when a program is
running. One common form of run-time error occurs when an operation is not defined
on some particular arguments. For example, division by zero raises an exception in
many languages that have exception mechanisms. Here is a simple ML example,
involving the left-subtree function that is not meaningful for trees that have only one
node:
datatype ’a tree = Leaf of ’a | Node of ’a tree * ’a tree;
exception No Subtree;
fun lsub ( Leaf x ) = raise No Subtree
| lsub (Node(x,y)) = x;
In this example, a function lsub(t) returns the left subtree of t if the tree has two
subtrees (left and right). However, if there is no left subtree, then the No Subtree
exception is raised.
Exceptions for Efficiency
Sometimes it is useful to terminate a computation when the answer is evident. Exceptions can be useful for this purpose, as an exception terminates a computation.
Consider the following code for computing the product of the integers stored at the
leaves of a tree. This is written for the tree data type previously defined:
fun prod (Leaf x) = x : int
| prod (Node(x,y)) = prod(x) * prod(y);
This correctly computes the product of all the integers stored at the leaves of a tree,
but is inefficient in the case that some of the leaves are zero. If we are frequently
computing the product for large trees that have zero at one or more leaves, then we
might want to optimize this function as follows:
fun prod(aTree) =
let exception Zero
fun p(Leaf x) = if x = 0 then raise Zero else x
| p(Node(x,y)) = p(x) * p(y)
in
p(aTree) handle Zero => 0
end;
In this function, a test is performed at each leaf to see if the value is zero. If it is, then an
8.2 Exceptions
exception is raised and no other leaves of the tree are examined. This function is less
efficient than the preceding one for trees without a zero, but is more efficient if zero is
found. Even if the last leaf is zero, raising an exception avoids all the multiplications
that would otherwise be performed.
Static and Dynamic Scope
We look at two ML code fragments that use identifier X in analogous ways. In the
first example, X is an expression variable and hence is accessed according to static
scoping rules. In the second, X is the name of an exception, and so its handlers are
used according to dynamic scoping rules. The point of these two code fragments is
to see the differences between the two scoping rules and to clearly illustrate that
exception handlers are determined by dynamic scope.
The following code illustrates static scoping:
val x = 6;
let fun f(y) = x
and g(h) = let val x = 2 in h(1) end
in
let val x = 4 in g(f) end
end;
Under static scoping, the value returned by the call to g(f) is the value of x in the
scope where f is declared, which is 6.
If we rewrite this code making X an exception, we will see what value we get
under dynamic scoping. One thing to remember when looking at the following code
is that ML exceptions use a postfix notation, so although the code is structurally quite
similar, it looks somewhat different at first glance:
exception X;
(let fun f(y) = raise X
and g(h) = h(1) handle X => 2
in
g(f) handle X => 4
end ) handle X => 6;
Here, the value of the g(f) expression is 2. The handler X that is used is the latest one
at the time the function raising the exception is called.
The following illustration shows the run-time stack for each code fragment, following the style explained in Chapter 7, with the exception code on the left and the
corresponding code with values in place of exceptions on the right. For brevity, only
the handler (or identifier) values and the access links are shown in each activation
record, except that the parameter value is also shown in the activation record for
each function call. If you begin at the bottom of the stack and search up the stack
for the most recent handler, which is the rule for dynamic scope, this is the one
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Control in Sequential Languages
with value 2. On the other hand, static scoping according to access links leads to
value 6.
handler
6
val x
access
access
fun f
fun f
access
access
fun g
fun g
access
handler
g(f )
access
4
val x
g(f)
access
formal h
handler
f(1)
4
access
formal h
2
access
formal y
6
val x
f(1)
1
2
access
formal y
1
This illustration also clearly shows which activation records will be popped off the
run-time stack when an exception is raised. Specifically, when the exception X is
raised in the code associated with the stack on the left, control transfers to the
handler identified in the top activation record. At this point, all of the activation
records below the activation record with handler X and value 6 are removed from
the run-time stack because they are not needed to continue the computation.
Typing and Exceptions
In ML, the expression
e1 handle A => e2;
returns either the value of e1 or the value of e2. This suggests that the types of e1
and e2 should be the same. To see why in more detail, look at this expression, which
contains the preceding one:
1 + (e1 handle A => e2);
Here, the value of e1 handle A => e2 is added to 1, so both e1 and e2 must be integer
expressions. More generally, the type system can assign only one type to the expression (e1 handle A => e2). Because (e1 handle A => e2) can return either the value of
e1 or the value of e2, both e1 and e2 must have the same type.
8.2 Exceptions
The situation for raise is very different, as raise <exception> does not have a value.
To see how this works, consider the expression
1 + raise No Value
where No Value is a previously declared exception. In this expression, the addition
will never be performed because raising the exception jumps to the nearest handler,
which is outside the expression shown. In ML, the type of raise <exception> is a type
variable ’a, which allows the type-inference algorithm to give raise <exception> any
type.
Exceptions and Resource Allocation
Raising an exception may cause a program to jump out of any number of in-line blocks
and function invocations. In each of these blocks, the program may have allocated
data on the stack or heap. In ML, C++, and other contemporary languages, all data
allocated on the run-time stack will be reclaimed after an exception is raised. This
occurs when the handler is located and intervening activation records are popped off
the run-time stack.
Data allocated on the heap are treated differently in different languages. In ML,
data on the heap that is no longer reachable after the exception will become garbage.
Because ML is garbage collected, the garbage collector will reclaim this unreachable
data. The situation is illustrated by the following code:
exception X;
(let
val y = ref [1,2,3]
in
. . . raise X
end) handle X => . . . ;
The local declaration let val x= . . . in . . . end causes a list [1,2,3] to be built in the
heap. When raise X raises an exception inside the scope of this declaration, control is
transferred to the handler outside this scope. The reference y, stored in the activation
record associated with the local declaration, is popped off the stack. The list [1,2,3]
remains in the program heap and is later collected whenever the garbage collector
happens to run.
In C++, storage that is reachable from activation records may no longer be reachable after an exception is thrown. Because C++ is not garbage collected, it is up to the
programmer to make sure that any data stored on the heap are explicitly deallocated.
However, it may be impossible to explicitly deallocate memory that was previously
allocated by the program if the only pointers to the data were those on the run-time
stack. In particular, in the C++ version of the preceding program, with a linked list
created in the heap, the only pointer to the list is the pointer y on the run-time stack.
After the exception is thrown, the pointer y is gone, and there is no way to reach
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the data. There are two general solutions: Either make sure there is some way of
reaching all heap data from the handler or accept the phenomenon and live with the
memory leak. The C++ implementation provides some assistance for this problem
by invoking the destructor of each object on the run-time stack as the containing
activation record is popped off the stack. This solves the list problem if we build
the list by placing one list object on the stack and by including code in the list destructor that follows the list pointer and invokes the destructor of any reachable list
nodes.
The general problem with managing unreachable memory also occurs with other
resources. For example, if a file is opened between the point of the handler and the
point where an exception is thrown, there may be no way to close the open file.
The same problem may occur with synchronization locks on concurrently accessible
data areas. There is no systematic language solution that seems effective for handling
these situations cleanly. Resource management is simply a complication that must
be handled with care when one is programming with exceptions.
8.3 CONTINUATIONS
Continuations are a programming technique, based on higher-order functions, that
may be used directly by a programmer or may be used in program transformations
in an optimizing compiler. Programming with continuations is also related to the
systems programming concepts of upcall or callback functions. As mentioned in
Section 7.4, a callback is a function that is passed to another function so that the
second function may call the first at a later time.
The concept of continuation originated in denotational semantics in the treatments of jumps (goto) and various forms of loop exit and in systems programming in
the notion of upcall discussed in Section 7.4. Continuations have found application
in continuation-passing-style (CPS) compilers, beginning with the groundbreaking
Rabbit compiler for Scheme, developed by Guy Steele and Gerald Sussman in the
mid-1970s. Since that time, continuations have been essential to many of the competitive optimizing compilers for functional languages.
8.3.1 A Function Representing “The Rest of the Program”
The basic idea of a continuation can be illustrated with simple arithmetic expressions.
For example, consider the function
fun f(x,y) = 2.0*x + 3.0*y + 1.0/x + 2.0/y;
Assume that this body of the function is evaluated from left to right, so that if we stop
evaluation just before the first division, we will have computed 2.0*x + 3.0*y and,
when the quotient 1.0/x is completed, the evaluation will proceed with an addition,
another division, and a final addition. The continuation of the subexpression 1.0/x
is the rest of the computation to be performed after this quotient is computed. We
8.3 Continuations
can write the computation completed before this division and the continuation to be
invoked afterward explicitly as follows:
fun f(x,y) = let val befor = 2.0*x + 3.0*y
fun continu(quot) = befor + quot + 2.0/y
in
continu(1.0/x)
end;
The evaluation order and result for the second definition of f will be the same as the
first; we have just given names to the part to be computed before and the part to be
computed after the division. The function called continu is the continuation of 1.0/x;
it is exactly what the computer will do after dividing 1.0 by x. The continuation of
the subexpression 1.0/x is a function, as the value of f(x,y) depends on the value of
the subexpression 1.0/x.
Let us suppose that, if x is zero, it makes sense to return before/5.2 as the value
of the function, and that otherwise we know that y should be nonzero and we can
proceed to compute the function normally. Rather than change the function in a
special-purpose way, we can illustrate the general idea by defining a division function
that is passed the continuation, applying the continuation only in the case in which
the divisor is nonzero:
fun divide(numerator,denominator,continuation,alternate value) =
if denominator > 0.0001
then continuation(numerator/denominator)
else alternate value;
fun f(x,y) = let val befor = 2.0*x + 3.0*y
fun continu(quot) = befor + quot + 2.0/y
in
divide(1.0, x, continu, befor/5.2)
end;
This version of f now uses an error-avoiding version of division that can exit in either
of two ways, the normal exit applying the continuation to the result of division and
an error exit returning some alternative value passed as a parameter. This is not
the most general version of division, however, because, in general, we might have
one computation we might wish to perform when division is possible and another
to perform when division is not. We can represent the computation on error as a
function also, leading to the following revision. Here we have assumed, for simplicity,
that computation after error is a function that need not be passed any of the other
arguments:
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Control in Sequential Languages
fun divide(numerator,denominator,normal cont,error cont) =
if denominator > 0.0001
then normal cont(numerator/denominator)
else error cont()
fun f(x,y) = let val befor = 2.0*x + 3.0*y
fun continu(quot) = befor + quot + 2.0/y
fun error continu() = befor/5.2
in
divide(1.0, x, continu, error continu)
end
For this specific computation, it is a relatively minor point that the division befor/5.2
is now done only if error continu is invoked. However, in general, it is far more useful
to have normal continuation and error continuation both presented as functions, as
error handling may require some computation after the error has been identified.
The preceding example can be handled more simply with exceptions. Ignoring
the precomputation of (2.0*x + 3.0*y), which a compiler could identify as a common
subexpression anyway, we can write the function as follows:
exception Div;
fun f(x,y) = (2.0*x + 3.0*y +
1.0/(if x > 0.001 then x else raise Div) + 2.0/y
) handle Div => (2.0*x + 3.0*y)/5.2
In general, continuations are more flexible than exceptions, but also may require
more programming effort to get exactly the control you want.
8.3.2 Continuation-Passing Form and Tail Recursion
There is a program form called continuation-passing form in which each function
or operation is passed a continuation. This allows each function or operation to
terminate by calling a continuation. As a consequence, no function needs to return
to the point from which it was called. This property of continuation-passing form may
remind you of tail calls, discussed in Subsection 7.3.4, as a tail call need not return
to the calling function. We will investigate the correspondence after looking at an
example.
There are systematic rules for transforming an expression or program to CPS. The
main idea is that each function or operation should take a continuation representing
the remaining computation after this function completes. If we begin with a program
that does not contain exceptions or other jumps, then each operation will be expected
to terminate normally. Because each function or operation will therefore terminate
by calling the function passed to it as a continuation, each function or operation will
terminate by executing a tail call to another function.
8.3 Continuations
We can transform the standard factorial function
fun fact(n) = if (n=0) then 1 else n*fact(n-1);
to continuation-passing form by first examining the continuation of each call. Consider the computation of fact(9), for example. This computation begins with an activation record for fact(9), then a recursive call to fact(8). The activation record for
fact(8) points to the activation record for fact(9), as the multiplication associated with
fact(9) must be performed after the call for fact(8) returns. The chain of activation
records is shown in the following illustration:
fact(9)
return
n
Order
of
calls
9
...
fact(8)
Order
of
returns
return
n
8
...
fact(7)
return
n
7
...
Because the invocation fact(9) multiplies the result of fact(8) by 9, the continuation
of fact(8) is λx. 9*x.
Similarly, after fact(7) returns, the invocation of fact(8) will multiply the result by
8 and then return to fact(9), which will in turn multiply by 9. Written as a function in
lambda notation, the continuation of fact(7) is λy.(λx. 9*x) (8*y).
By similar reasoning, the continuation of fact(6) is λz.(λy. (λx. 9*x) (8*y)) (7*z).
Generalizing from these examples, the continuation of fact(n-1) is the composition
of (λx. n*x) with the continuation of fact(n).
Using this insight, we can write a general CPS function to compute factorial. The
function
f(n, k) = if n=0 then k(1) else f(n-1, λx. k (n*x) )
takes a number n and a continuation k as arguments. If n=0, then the function passes
1 to the continuation. If n>0, then the function makes a recursive call. Because the
recursive call is to a CPS function, the continuation passed to the call must be the
continuation of factorial of n-1. Given factorial x of n-1, the continuation multiplies
by n and then passes the result to the continuation k of the factorial of n. Here is
a sample calculation, with initial continuation the identity function, illustrating how
the continuation-passing function works:
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Control in Sequential Languages
f(3, λx.x) = f(2, λy.(( λx.x) (3*y)))
= f(1, λx.(( λy.3*y)(2*x)))
= λx.(( λy.3*y)(2*x)) 1
=6
Intuitively, the continuation-passing form of factorial performs a sequence of recursive calls, accumulating a continuation. When the recursion terminates by reaching
n=0, the continuation is applied to 1 and that produces the return value n!. When the
continuation is applied, the continuation consists of a sequence of multiplications
that compute factorial. This is exactly the same sequence of multiplications as would
be done in the ordinary recursive factorial. However, in the case of ordinary recursive
factorial, each multiplication would be done by a separate invocation of the factorial
function.
Continuations and Tail Recursion. The following ML continuation-passing factorial function appears to use a tail recursive function f to do the calculation: fun
factk(n) = let fun:
f(n, k) = if n=0 then k(1) else f(n-1, fn x => k (n*x) )
in
f(n, fn x => x)
end
The inner function f is tail recursive in the sense that the recursive call to f is the
entire value of the function in the case n>0. Therefore, there is no need for the
recursive call to return to the calling invocation. However, this does not mean that
the continuation-passing function factk runs in constant space. The reason has to do
with closures and scoping of variables. Specifically, the continuation
fn x => k (n*x)
contains the identifier x that is a parameter of the function. The continuation is
therefore represented as a closure, with a pointer to the activation record of the
calling invocation of f. Because the activation record is needed as part of the closure,
it is not possible to use tail recursion optimization to eliminate the need for a new
activation record.
Factorial can, however, be written in a tail recursive form:
fun fact1(n) = let fun
f(n,k) = if n=0 then k else f(n-1, n*k)
in
f(n,1)
end;
8.4 Functions and Evaluation Order
As discussed in Subsection 7.3.4, the tail recursive form is more efficient, as the
compiler can generate code that does not allocate a new activation record for each
call. Instead, the same activation record can be used for all invocations of the function
f, with each call effectively assigning new values to local variables n and k.
We can derive the tail recursive form from the continuation-passing form by
using a little insight. The main idea is that each continuation in the continuationpassing form will be a function that multiplies its argument by some number. We
can therefore achieve the same effect by passing the number instead of the function.
Although there is no standard, systematic way of transforming an arbitrary function
into a tail recursive function that can be executed by use of constant space, there is
a systematic transformation to continuation-passing form and, in some instances, a
clever individual can produce a tail recursive function from the continuation-passing
form.
8.3.3 Continuations in Compilation
Continuations are commonly used in compilers for languages with higher-order functions. The main steps in a continuation-based compiler follow the design pioneered
by Steele and Sussman for Scheme (MIT AI MEMO 474, MIT, 1978):
1.
2.
3.
4.
5.
6.
7.
8.
9.
lexical analysis, parsing, type checking
translation to lambda calculus form
conversion to CPS
optimization of CPS
closure conversion – eliminate free variables
elimination of nested scopes
register spilling – no expression with more than n free vars
generation of target-assembly language program
Assembly to produce target-machine program
The core step that makes continuations useful is step 4, optimizing the continuationpassing form of a program. The merit of continuations is that continuations make
control flow explicit – each part of the computation explicitly calls a function to do
the next part of the computation.
Furthermore, because a continuation-passing operation terminates with a call
instead of a return, it is possible to compile calls directly into jumps. Arranging
the target code in the correct order can eliminate many of these jumps. Although
additional discussion of continuation-based compilation is beyond the scope of this
book, there are several compiler books that address this subject in detail and many
additional articles and web sources.
8.4 FUNCTIONS AND EVALUATION ORDER
Exceptions and continuations are forms of jumps that are used in high-level programming languages. A final technique for manipulating the order of execution in
programs is to use function definitions and calls. More specifically, if a calculation can
be put off until later, it may be placed inside a function and passed to code that will
eventually decide when to do the calculation. This is most useful if the calculation
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Control in Sequential Languages
might not be needed at all. In our brief survey of this simple technique, we look at
controlling the order of evaluation for efficiency.
Delay and force are programming forms that can be used together to optimize
program performance. Delay and Force are explicit program constructs in Scheme, but
the main idea can be used in any language with functions and static scope. To see how
Delay and Force are used, we start with an example. Consider a function declaration
and call of the following form:
fun f(x,y) =
... x ... y ...
...
f(e1, e2)
Suppose that
the value of y is needed only if the value of x has some property, and
the evaluation of e2 is expensive.
In these circumstances, it is a good idea to delay the evaluation of e2 until we determine (inside the body of f) that the value of e2 is actually needed. If and when we
make this determination, we may then force the evaluation of e2. In other words, we
would like to be able to write something like the following, in which Delay(e) causes
the evaluation of e to be delayed until we call Force(Delay(e)):
fun f(x,y) =
. . . x . . . Force(y) . . .
...
f(e1, Delay(e2))
If Force(y) occurs only where the value of y is needed and y is needed at most only
once in the body of f, then it should be clear that this form is more efficient than the
preceding one. Specifically, if many calls do not require the value of y, then each of
these calls will run much more quickly, as we avoid evaluating e2 and we are assuming
that evaluation of e2 is expensive. In cases in which the value of e2 is needed, we do the
same amount of work as before, plus the presumably small overhead we introduced
by adding Delay and Force to the computation.
We discuss two remaining questions: How can we express Delay and Force in a
conventional programming language and what should happen if a delayed value is
needed more than once?
Let us begin with the first problem, expressing Delay and Force in a conventional
language, assuming for simplicity that there will only be one reference to the delayed
value. Delay cannot be an ordinary function. To see this, suppose briefly that Delay(e)
is implemented as a function Delay applied to the expression e. In most languages,
the semantics of function calls requires that we evaluate the arguments to a function
before invoking the function. Then in evaluating Delay(e) we will evaluate e first.
8.4 Functions and Evaluation Order
However, this defeats the purpose of Delay, which was to avoid evaluating e unless
its value was actually needed.
Because Delay cannot be a normal function, we are left with two options:
make Delay a built-in operation of the programming language where we wish to
delay evaluation, or
implement the conceptual construct Delay(e) as some program expression or sequence of commands that is not just a function applied to e.
The first option works fine, but only if we are prepared to modify the implementation
of a programming language. Scheme, for example, inherits the concept of special
form, a function that does not evaluate its arguments, from Lisp. When special forms
are used, a Delay operation that does not evaluate its argument can be defined.
In other languages, we can think of Delay(e) as an abbreviation that gets expanded
in some way before the program is compiled. An implementation of Delay and Force
that works in ML is
Delay(e) == λ().e , which is actually written fn() => e
Force(e) == e()
where == means “macro expand to this form before compiling the program.”
Here Delay(e) makes e into a parameterless function. The notation λ ().e indicates
a function that takes no parameters and that returns e when called. This delays
evaluation because the body of a functions is not evaluated until the function is called.
Force evaluates expressions that have previously been delayed. Because delayed
expressions are zero-argument functions, Force calls a zero-argument function to
cause the function body to be evaluated.
Example 8.2
Here is an example in which the Takeuchi function, tak, is used. The function tak
runs for a very long time, without using so much stack space that the run-time stack
overflows. (Try it!) Because of this characteristic of tak, it is often used as benchmark
for testing the speed of function calls in a compiler or interpreter. In this example,
the function f has two arguments; the second argument is used only if the first is odd.
The purpose of Force and Delay in this example is to make f(fib(9), time consuming(9))
run more quickly; fib is the Fibonacci function and time consuming uses tak:
fun time consuming(n) =
let fun tak(x,y,z) = if x <= y then y
else tak(tak(x-1,y,z), tak(y-1,z,x), tak(z-1,x,y))
in
tak(3*n, 2*n, n)
end;
fun fib(n) = if n=0 orelse n = 1 then 1 else fib(n-1) + fib(n-2);
fun odd(n) = (n mod 2 ) = 1;
fun f(x,y) = if odd(x) then 1 else fib(y);
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Control in Sequential Languages
If we evaluate the following expression directly, it will run for a very long time:
f(fib(9), time consuming(9));
Instead we rewrite the function f to expect a Delay-ed value as the second argument,
and Force evaluation only if needed:
fun lazy f(x,y) = if odd(x) then 1 else fib( y() );
Here is the corresponding call, with calculation of time consuming(9) delayed:
lazy f( fib(9), fn () => (time consuming(9)));
Because fib(9) is odd, this expression terminates much more quickly than the expression without Delay.
The versions of Delay and Force described in Example 8.2 rely on static scoping and
save time only if the delayed function argument is used at most once. Static scoping is
necessary to preserve the semantics of the program. Without static scoping, placing
an expression inside a function and passing it to another function might change the
values of identifiers that appear in the expression. The reason why Delay and Force
do not help if the expression is used twice or more is that each occurrence would
involve evaluating the delayed expression. This will take extra time and may give the
wrong result if the expression has side effects.
It is a relatively simple programming exercise to write versions of Delay and Force
that work when the delayed value is needed more than once. The main idea is to
store a flag that indicates whether the expression has been evaluated once or not. If
not, and the value is needed, then the expression is evaluated and stored so that it
can be retrieved without further evaluation when it is needed again. This trick is a
form of evaluation that is referred to as call-by-need in the literature.
Here is a simple ML version of the code needed to delay a value that may be
needed more than once. A delayed value will be a reference cell containing an “
unevaluated delay”:
Delay ( e ) == ref(UN(fn () => e ))
where the constructor UN and the type of “delays” are defined by
datatype ’a delay = EV of ’a | UN of unit -> ’a;
Intuitively, a delayed value is an assignable cell that will contain an unevaluated value
until it is evaluated. After that, the assignable cell will contain an evaluated value.
The type delay is a union of the two possibilities. A “delay” is either an evaluated
8.5 Chapter Summary
value, tagged with constructor EV, or an unevaluated delay, tagged with constructor
UN. The tagged unevaluated delay is a function of no arguments that can be called to
get an evaluated value.
The corresponding force function,
fun force(d) = let val v = ev(!d) in (d := EV(v); v) end;
uses assignment and a subsidiary function ev that evaluates a delay:
fun ev(EV(x)) = x
| ev(UN(f)) = f();
In words, if a delay is already evaluated, then ev has nothing to do. Otherwise, ev calls
the function of no arguments to get an evaluated delay. To give a concrete example,
here is the code for a delayed evaluation of time consuming(9), followed by a call to
force to evaluate it:
val d = ref(UN(fn () => time consuming(9)));
force(d);
This call to force evaluates the delayed expression and has a side effect so that, on
subsequent calls to force, no further evaluation is needed.
8.5 CHAPTER SUMMARY
We looked at several ways of controlling the order of execution and evaluation in
sequential (nonconcurrent) programs. Here are the main topics of the chapter:
structured programming without go to,
exceptions,
continuations,
Delay and Force.
Control and Go to. Because structured programming is commonly accepted and
taught, we did not look at the entire historical controversy surrounding go to statements. The main conclusion in the “Go to considered harmful” debate is that, in
the end, program clarity is often more important than absolute efficiency. This has
always been true to some degree, but as computer speed has increased, the relation
between programmer time and program execution time has shifted. In modern software development, it is not worth several days of programmer time to reduce the
execution time of a large application by one or two instruction cycles. Programmer
time is expensive, and clever programming can lead to costly mistakes and increased
debugging time. An instruction or two takes so little time that for most applications
noone will notice the difference.
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Control in Sequential Languages
Exceptions. Exceptions are a structured form of jump that may be used to exit
a block or function call and pass a return value in the process. Every exception
mechanism includes a statement or expression form for raising an exception and
a mechanism for defining handlers that respond to exceptions. When an exception
is raised and several handlers have been established, control is transferred to the
handler associated with the most recent activation record on the run-time stack. In
other words, handlers are selected according to dynamic scope, not the static scope
rules used for most other declarations in modern general-purpose programming
languages.
Continuations. Continuation is a programming technique based on higher-order
functions that may be used directly in programming or in program transformations
in an optimizing compiler. Some mostly functional languages, such as Scheme and
ML, provide direct support for capturing and invoking continuations.
Intuitively, the continuation of a statement or expression is a function representing the computation remaining to be performed after this statement or expression
is evaluated. There is a general, systematic method for transforming any program
into continuation-passing form. A function in continuation-passing form does not
return, but calls a continuation (passed as an argument to the function) in order to
continue after the function is complete. Continuation-passing form is related to tail
recursion (see Subsection 7.3.4). Formally, a continuation-passing function appears
to be tail recursive, as all calls are tail calls. However, continuation-passing functions
may pass functions are arguments.
This makes it impossible to perform tail recursion elimination – the function created and passed out of a function call may need the activation record of the function
in order to maintain the value of statically scoped global variables. However, a clever
programmer can sometimes use the continuation-passing version of a function to devise a tail recursive function that will be compiled as efficiently as an iterative loop.
Continuation-passing form is used in a number of contemporary compilers for
languages with higher-order functions.
Delay and Force. Delay and Force may be used to delay a computation until it is
needed. When the delayed computation is needed, Force is used. Delay and Force
may be implemented in conventional programming languages by use of functions:
The delayed computation is placed inside a function and Force is implemented by
calling this function. This technique is simplest to apply if functions can be declared
anywhere in a program. If a delayed value may be needed more than once, this value
may be stored in a location that will be used in every subsequent call to Force.
EXERCISES
8.1 Exceptions
Consider the following functions, written in ML:
exception Excpt of int;
fun twice(f,x) = f(f(x)) handle Excpt(x) => x;
fun pred(x) = if x = 0 then raise Excpt(x) else x-1;
fun dumb(x) = raise Excpt(x);
fun smart(x) = 1 + pred(x) handle Excpt(x) => 1;
Exercises
What is the result of evaluating each of the following expressions?
(a) twice(pred,1);
(b) twice(dumb,1);
(c) twice(smart,0);
In each case, be sure to describe which exception gets raised and where.
8.2 Exceptions
ML has functions hd and t1 to return the head (or first element) and tail (or remaining
elements) of a list. These both raise an exception Empty if the list is empty. Suppose
that we redefine these functions without changing their behavior on nonempty lists,
so that hd raises exception Hd and tl raises exception Tl if applied to the empty list
nil:
– hd(nil);
uncaught exception Hd
– tl(nil);
uncaught exception Tl
Consider the function
fun g(l) = hd(l) : : tl(l) handle Hd => nil;
that behaves like the identity function on lists. The result of evaluating g(nil) is
nil. Explain why. What makes the function g return properly without handling the
exception Tl?
8.3 Exceptions
The following two versions of the closest function take an integer x and an integer
tree t and return the integer leaf value from t that is closest in absolute value to x.
The first is a straightforward recursive function, the second uses an exception:
datatype ’a tree = Leaf of ’a | Nd of (’a tree) * (’a tree);
fun closest(x, Leaf(y)) = y:int
| closest(x, Nd(y, z)) = let val lf = closest(x, y) and rt = closest(x, z) in
if abs(x-lf) < abs(x-rt) then lf else rt end;
fun closest(x, t) =
let
exception Found
fun cls (x, Leaf(y)) = if x=y then raise Found else y:int
| cls (x, Nd(y, z)) = let val lf = cls(x, y) and rt = cls(x, z) in
if abs(x-lf) < abs(x-rt) then lf else rt end
in
cls(x, t) handle Found => x
end;
(a) Explain why both give the same answer.
(b) Explain why the second version may be more efficient.
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8.4 Exceptions and Recursion
Here is an ML function that uses an exception called Odd.
fun f(0) = 1
| f(1) = raise Odd
| f(3) = f(3-2)
| f(n) = (f(n-2) handle Odd => ∼ n)
The expression ∼ n is ML for −n, the negative of the integer n.
When f(11) is executed, the following steps will be performed:
call f(11)
call f(9)
call f(7)
...
Write the remaining steps that will be executed. Include only the following kinds of
steps:
function call (with argument)
function return (with return value)
raise an exception
pop activation record of function off stack without returning control to the
function
handle an exception
Assume that if f calls g and g raises an exception that f does not handle, then
the activation record of f is popped off the stack without returning control to the
function f.
8.5 Tail Recursion and Exception Handling
Can we use tail recursion elimination to optimize the following program?
exception OddNum;
let fun f(0,count) = count
| f(1, count) = raise OddNum
| f(x, count) = f(x-2, count+1) handle OddNum => -1
Why or why not? Explain. This is a tricky situation – try to explain succinctly what
the issues are and how they might be resolved.
8.6 Evaluation Order and Exceptions
Suppose we add an exception mechanism similar to the one used in ML to pure
Lisp. Pure Lisp has the property that if every evaluation order for expression e
terminates, then e has the same value under every evaluation order. Does pure Lisp
with exceptions still have this property? [Hint: See if you can find an expression
containing a function call f(e1 , e2 ) so that evaluating e1 before e2 gives you a different
answer than evaluating the expression with e2 before e1 .]
8.7 Control Flow and Memory Management
An exception aborts part of a computation and transfers control to a handler that was
established at some earlier point in the computation. A memory leak occurs when
memory allocated by a program is no longer reachable, and the memory will not be
Exercises
deallocated. (The term “memory leak” is used only in connection with languages that
are not garbage collected, such as C.) Explain why exceptions can lead to memory
leaks in a language that is not garbage collected.
8.8 Tail Recursion and Continuations
(a) Explain why a tail recursive function, as in
fun fact(n) =
let fun f(n, a) = if n=0 then a
else f(n-1, a*n)
in f(n, 1) end;
can be compiled so that the amount of space required for computing fact(n) is
independent of n.
(b) The function f used in the following definition of factorial is “formally” tail
recursive: The only recursive call to f is a call that need not return:
fun fact(n) =
let fun f(n,g) = if n=0 then g(1)
else f(n-1, fn x=>g(x)*n)
in f(n, fn x => x) end;
How much space is required for computing fact(n), measured as a function of
argument n? Explain how this space is allocated during recursive calls to f and
when the space may be freed.
8.9 Continuations
In addition to continuations that represent the “normal” continued execution of a
program, we can use continuations in place of exceptions. For example, consider the
following function f that raises an exception when the argument x is too small:
exception Too Small;
fun f(x) = if x<0 then raise Too Small else x/2;
(1 + f(y)) handle Too Small => 0;
If we use continuations, then f could be written as a function with two extra arguments,
one for normal exit and the other for “exceptional exit,” to be called if the argument
is too small:
fun f(x, k normal, k exn) = if x<0 then k exn() else k normal(x/2);
f(y, (fn z => 1+z), (fn () => 0));
(a) Explain why the final expressions in each program fragment will have the same
value for any value of y.
(b) Why would tail call optimization be helpful when we use the second style of
programming instead of exceptions?
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PART 3
Modularity, Abstraction, and
Object-Oriented Programming
9
Data Abstraction and Modularity
Computer programmers have long recognized the value of building software systems
that consist of a number of program modules. In an effective design, each module
can be designed and tested independently. Two important goals in modularity are to
allow one module to be written with little knowledge of the code in another module
and to allow a module to be redesigned and reimplemented without modifying other
parts of the system. Modern programming languages and software development
environments support modularity in different ways.
In this chapter, we look at some of the ways that programs can be divided into
meaningful parts and the way that programming languages can be designed to support these divisions. Because in Chapters 10–13 we explore object-oriented languages
in detail, in this chapter we are concerned with modularity mechanisms that do not
involve objects. The main topics are structured programming, support for abstraction, and modules. The two examples used to describe module systems and generic
programming are the standard ML module system and the C++ Standard Template
Library (STL).
9.1 STRUCTURED PROGRAMMING
In an influential 1969 paper called Structured Programming, E.W. Dijkstra argued
that one should develop a program by first outlining the major tasks that it should
perform and then successively refining these tasks into smaller subtasks, until a level
is reached at which each remaining task can be expressed easily by basic operations.
This produces subproblems that are small enough to be understood and separate
enough to be solved independently.
In Example 9.1, the data structures passed between separate parts of the program
are simple and straightforward. This makes it possible to identify the main data
structures early in the process. Because the data structures remain invariant through
most of the design process, Dijkstra’s example centers on refinement of procedures
into smaller procedures. In more complex systems, it is necessary to refine data
structures as well as procedures. This is illustrated in Example 9.2.
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EDSGER W DIJKSTRA
An exacting and fundamentally warm-hearted person, Edsger W.
Dijkstra has made many important contributions to the field of computing science. He is known for semaphores, which are commonly used
for concurrency control, algorithms such as his method for finding shortest paths in graphs, his “guarded command” language, and methods for
reasoning about programs.
Over the years, Dijkstra has written a series of carefully handwritten
articles, known commonly as the EWDs. As of the early 2002, he had
written over 1309 EWDs, scanned and available on the web. As Dijkstra
now says on his web page,
My area of interest focuses on the streamlining of the mathematical argument
so as to increase our powers of reasoning, in particular, by the use of formal
techniques.
His interest in streamlining mathematical argument is evident in the
EWDs, each developing an elegant solution to an intriguing problem in
a few pages.
Like many old-school Europeans, and unlike most Americans,
Dijkstra has impeccable handwriting. In part as a joke and in part as
a tribute to Dijkstra, a programming language researcher named Luca
Cardelli carefully copied the handwriting from a set of EWDs and produced the EWD font. If you can find the font on the web, you can try
writing short notes in Dijkstra’s famous handwriting.
9.1 Structured Programming
Example 9.1
Dijkstra considered the problem of computing and printing the first 1000 prime
numbers. The first version of the program contains a little bit of syntax to get us
thinking about writing a program. Otherwise, it just looks like an English description
of the problem we want to solve.
Program 1:
begin
print first thousand prime numbers
end
This task can now be refined into subtasks. To divide the problem in two, some data
structure must be selected for passing the result of the first subtask onto the second.
In Dijkstra’s example, the data structure is a table, which will be filled with the first
1000 primes.
Program 2:
begin variable table p
fill table p with first thousand primes
print table p
end
In the next refinement, each subtask is further elaborated. One important idea in
structured programming is that each subtask is considered independently. In the
example at hand, the problem of filling the table with primes is independent of the
problem of printing the table. Therefore, each subtask could be assigned to a different
programmer, allowing the problems to be solved at the same time by different people.
Even if the program were going to be written by a single person, there is an important
benefit of separating a complex problem into independent subproblems. Specifically,
a single person can think about only so many details at once. Dividing a task into
subtasks makes it possible to think about one task at a time, reducing the number of
details that must be considered at any one time.
Program 3:
begin integer array p[1:1000]
make for k from 1 through 1000
p[k] equal to the kth prime number
print p[k] for k from 1 through 1000
end
At this point, the basic program structure has been determined and the programmer
can concentrate on the algorithm for computing successive primes. Although this example is extremely simple, it should give some idea of the basic idea of programming
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Data Abstraction and Modularity
by stepwise refinement. Stepwise refinement generally leads to programs with a treelike conceptual structure.
Main
Program
Subprogram
Subprogram
Subprogram
Subprogram
Subprogram
Subprogram
One difficult aspect of top-down program development is that it is important to
make the problem simpler on each refinement step. Otherwise, it might be possible to
refine a task and produce a list of programming problems that are each more difficult
than the original task. This means that a designer who uses stepwise refinement must
have a good idea in advance of how tasks will eventually be accomplished.
9.1.1 Data Refinement
In addition to refining tasks into simpler subtasks, evolution in a system design may
lead to changes in the data structures that are used to combine the actions of independent modules.
Example 9.2
Consider the problem of designing a simple banking program. The goal of this program is to process account deposits, process withdrawals, and print monthly bank
statements. In the first pass, we might formulate a system design that looks something like this:
Main Program
Create Account
Deposit/Withdraw
Print Statement
In this design, the main program receives a list of input transactions and calls
the appropriate subprograms. If we assume that statements only contain the account
number and balance, then we could represent a single bank account by an integer
value and store all bank accounts in a single integer array.
If we later refine the task “Print Statement” to include the subtask “Print list of
transactions,” then we will have to maintain a record of bank transactions. For this
refinement, we will have to replace the integer array with some other data structure
that records the sequence of transactions that have occurred since the last statement.
This may require changes in the behavior of all the subprograms, as all of them
perform operations on bank accounts.
9.1 Structured Programming
9.1.2 Modularity
Divide and conquer is one of the fundamental techniques of computer science. Because software systems can be exceedingly complex, it is important to divide programs
into separate parts that can be treated independently. Top-down program development, when it works, is one method for producing programs that consist of separable
parts. In some cases, it is also useful to work bottom-up, designing basic parts that will
be needed in a large software system and then combining them into larger subsystems. Since the 1970s, a number of other program-development methods have been
proposed.
One useful development method, sometimes called prototyping, involves implementing parts of a program in a simple way to understand if the design will really
work. Then, after the design has been tested in some way, one can improve parts
of the program independently by reimplementing them. This process can be carried
out incrementally by a series of progressively more elaborate prototypes to develop
a satisfactory system. There are also related object-oriented design methods, which
we will discuss in Chapter 10.
One important way for programming languages to support modular programming methods is by helping programmers keep track of the dependencies between
different parts of a system. For the purposes of discussion, we call a meaningful part
of a program that is meant to be partially independent of other parts a program
component.
Two important concepts in modular program development are interfaces and
specifications:
Interface: A description of the parts of a component that are visible to other
program components.
Specification: A description of the behavior of a component, as observable
through its interface.
When a program is designed modularly, it should be possible to change the internal
structure of any component, as long as the behavior visible through the interface
remains the same.
A simple example of a program component is a single function. The interface of
a function consists of the function name, the number and types of parameters, and
the type of the return result. A function interface is also called a function header.
A function specification usually describes the relationship between function arguments and the corresponding return result. If a function will work properly on only
certain arguments, then this restriction should be part of the function specification.
For example, the interface of a square-root function might be
function sqrt (float x) returns float
A specification for this function can be written as
If x>0, then sqrt(x)*sqrt(x) ≈ x.
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where the squiggly approximation sign, ≈, is used to mean “approximately equal,”
as computation with floating-point numbers is carried out to only limited precision.
In some forms of modular programming, system designers write a specification
for each component. When a component is implemented, it should be designed to
work correctly when all of the components it interacts with satisfy their specifications.
In other words, the correctness of one component should not depend on any hidden
implementation details of any other component. One reason for striving to achieve
this degree of independence is that it allows components to be reimplemented independently. Specifically, in a system in which each component relies on only stated
specifications of other components, we can replace any component with another that
satisfies the same specification. This allows us to optimize components independently
or to add functionality that does not violate the original specification.
There are many different languages and methods for writing specifications, ranging from English and graphical notations that have little structure to formal languages
that can be manipulated by specification tools. A basic problem associated with program specification is that there is no algorithmic method for testing that a module
satisfies its specification. This is a consequence of a fundamental mathematical limitation, similar to the undecidability of the halting problem. As a result, programming
with specifications requires substantial effort and discipline.
To illustrate the use of data structures and specifications, we look at a sorting
algorithm that uses a general data structure that also serves other purposes.
Example 9.3 A Modular Sorting Algorithm
An integer priority queue is a data structure with three operations:
empty : pqueue
insert : int * pqueue → pqueue
deletemax: pqueue → int * pqueue
In words, there is a way of representing an empty priority queue, an insert operation
that adds an integer to a priority queue, and a deletemax operation that removes an
element from a priority queue. These three operations form the interface to priority
queues. To give more detail, we have the following specifications:
Each priority queue has a multiset of elements. There must be an ordering –< on
the elements that may be placed in a priority queue. (For integer priority queues,
we may use the ordinary –< ordering on integers.)
An empty priority queue has no elements.
insert(elt, pq) returns a priority queue whose elements are elt plus the elements
of pq.
> all other elements of pq, together
deletemax(pq) returns an element of pq that is –
with a data structure representing the priority queue obtained when this element
is removed.
These specifications do not impose any restrictions on the implementation of priority
queues other than properties that are observable through the interface of priority
queues.
9.1 Structured Programming
Knowing in advance that we would like to use priority queues in our sorting
algorithm, we can begin the top-down design process by stating the problem in a
program form:
Program 1:
function sort
begin
sort an array of integers
end
The next step is to refine the statement sort an array of integers into subtasks. One
way to do this, using priority queues, is to transfer the elements of the array into a
priority queue and then remove them one at a time. In addition, we can make the
decision at this point that the function will take an array and its integer length as
separate arguments.
Program 2:
function sort(n:int, A : array [1..n] of int)
begin
place each element of array A in a priority queue
remove elements in decreasing order and place in array A
end
Finally, we can translate these English descriptive statements into some form of program code. (Here, the program is written in a generic Algol- or Pascal-like notation.)
Program 3:
function sort(n:int, A : array [1..n] of int)
begin priority queue s;
s := empty;
for i := 1 to n do s := insert(A[i], s);
for i := n downto 1 do (A[i],s) := deletemax(s);
end
One advantage of this sorting algorithm is that there is a clear separation between
the control structure of the algorithm and the data structure for priority queues. We
could implement priority queues inefficiently to begin with, by using an algorithm
that is easy to code, and then optimize the implementation later if this turns out to
be needed.
As written, it seems difficult to sort an array in place by this algorithm. However,
it is possible to come close to the conventional heapsort algorithm.
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9.2 LANGUAGE SUPPORT FOR ABSTRACTION
Programmers and software designers often speak about “finding the right abstraction” for a problem. This means that they are looking for general concepts, such as
data structures or processing metaphors, that will make a complex, detailed problem
seem more orderly or systematic. One way that a programming language can help
programmers find the right abstraction is by providing a variety of ways to organize
data and computation. Another way that a programming language can help with
finding the right abstraction is to make it possible to build program components that
capture meaningful patterns in computation.
9.2.1 Abstraction
In programming languages, an abstraction mechanism is one that emphasizes the
general properties of some segment of code and hides details. Abstraction mechanisms generally involve separating a program into parts that contain certain details
and parts where these details are hidden. Common terms associated with abstraction
are
client: the part of a program that uses program component
implementation: the part of a program that defines a program component.
The interaction between the client of an abstraction and the implementation of the
abstraction is usually restricted to a specific interface.
Procedural Abstraction
One of the oldest abstraction mechanisms in programming languages is the procedure or function. The client of a function is a program making a function call. The
implementation of a function is the function body, which consists of the instructions
that will be executed each time the function is called.
If we have a few lines of code that store the square root of a variable x in the
variable y, for example, then we can encapsulate this code into a function. This accomplishes several things:
1. The function has a well-defined interface, made explicit in the code. The interface consists of the function name, which is used to call the function, the input
parameters (and their types, if it is a typed programming language) and the
type of the output.
2. If the code for computing the function value uses other variables, then these
can be made local to the function. If variables are declared inside the function body, then they will not be visible to other parts of the program that use
the function. In other words, no assignment or other use of local variables
has any effect on other parts of the program. This provides a form of information hiding: information about how the function computes a result is contained in the function declaration, but hidden from the program that uses the
function.
3. The function may be called on many different arguments. If code to carry out
a computation is written in-line, then the computation is performed on specific
variables. By enclosing the code in a function declaration, we obtain an abstract
9.2 Language Support for Abstraction
entity that makes sense apart from its specific use on these specific variables.
In grandiose terms, enclosing code inside a function makes the code generic
and reusable.
This is an idealistic description of the advantages of enclosing code inside a function.
In most programming languages, a function may read or assign to global variables.
These global variables are not listed in the function interface. Therefore, the behavior
of a function is not always determined by its interface alone. For this reason, some
purists in program design recommend against using global variables in functions.
Data Abstraction
Data abstraction refers to hiding information about the way that data are represented. Common language mechanisms for data abstractions are abstract data-type
declarations (discussed in Subsection 9.2.2) and modules (discussed in Section 9.3).
We saw in Subsection 9.1.2 how a sorting algorithm can be defined by using a data
structure called a priority queue. If a program uses priority queues, then the writer
of that program must know what the operations are on priority queues and their
interfaces. Therefore, the set of operations and their interfaces is called the interface
of a data abstraction. In principle, a program that uses priority queues should not
depend on whether priority queues are represented as binary search trees or sorted
arrays. These implementation details are best hidden by an encapsulation mechanism.
As for procedural abstraction, there are three main goals of data abstraction:
1. Identifying the interface of the data structure. The interface of a data abstraction consists of the operations on the data structure and their arguments and
return results.
2. Providing information hiding by separating implementation decisions from
parts of the program that use the data structure.
3. Allowing the data structure to be used in many different ways by many different
programs. This goal is best supported by generic abstractions, discussed in
Section 9.4.
9.2.2 Abstract Data Types
Interest in data abstraction came to prominence in the 1970s. This led to the development of a programming language construct call the abstract data-type declaration.
This is a common short definition of an abstract data type:
An abstract data type consists of a type together with a specified set of
operations.
Good languages for programming with abstract data types not only allow a programmer to group types and operations, but also use type checking to limit access
to the representation of a data structure. In other words, not only does an abstract
data type have a specific interface that can be used by other parts of a program, but
access is also restricted so that the only use of an abstract data type is through its
interface.
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If a stack is implemented with an array, then programs that use a stack abstract
data type can use only the stack operations (push and pop, say), not array operations such as indexing into the array at arbitrary points. This hides information
about the implementation of a data structure and allows the implementer of the data
structure to make changes without affecting parts of a program that use the data
structure.
We can appreciate some aspects of abstract data types by understanding a historical idea that was in the air at the time of their development. In the early 1970s, there
was a movement to investigate “extensible” languages. The goal of this movement
was to produce programming languages in which the programmer would be able
to define constructs with the same flexibility as a language designer. For example, if
some person or group of programmers wanted to write programs by using a new form
of iterative loop, they could use a “loop declaration” to define one and use it in their
programs. This idea turned out to be rather unsuccessful, as programs littered with
all kinds of programmer-defined syntactic conventions can be extremely difficult to
read or modify. However, the idea that programmers should be able to define types
that have the same status as the types that are provided by the language did prove
useful and has stood the test of time.
A potential confusion about abstract data types is the sense in which they
are abstract. A simple distinction is that a data type whose representation and
operational details are hidden from clients is abstract. In contrast, a data type
whose representation details are visible to clients may be called a transparent
type. ML abstype, discussed in the next subsection, defines abstract data types,
whereas ML data type, discussed in Subsection 6.5.3, is a transparent type-declaration
form.
9.2.3 ML abstype
We use the historical ML abstract data-type construct, called abstype, to discuss
the main ideas associated with abstract data-type mechanisms in programming languages.
As discussed in the preceding subsection, an abstract data-type mechanism associates a type with a data structure in such a way that a specific set of functions has
direct access to the data structure but general code in other parts of a program does
not. We will see how this works in ML by considering a simple example, complex
numbers.
We can represent a complex number as a pair of real numbers. The first real is
the “real” part of the complex number, and the second is the “imaginary” part of the
complex number. If we are going to compute with complex numbers, then we need to
have a way of forming a complex number from two reals and ways of getting the real
and imaginary parts of a complex number. Computation with complex numbers may
also involve complex addition, multiplication, and other standard operations. Here,
simply providing complex addition is discussed. Other operations could be included
in the abstract data type in similar ways.
An ML declaration of an abstract data type of complex numbers may be written
as follows:
9.2 Language Support for Abstraction
abstype cmplx = C of real * real with
fun cmplx(x,y: real) = C(x,y)
fun x coord(C(x,y)) = x
fun y coord(C(x,y)) = y
fun add(C(x1, y1), C(x2, y2)) = C(x1+x2, y1+y2)
end
This declaration binds five identifiers for use outside the declaration: the type cmplx
and the functions cmplx, x coord, y coord, and add. The declaration also binds the
name C to a constructor that can be used only within the bodies of the functions that
are part of the declaration. Specifically, C may appear in the code for cmplx, x coord,
y coord, and add but not in any other part of the program.
The type name cmplx is the type of complex numbers. When a program uses
complex numbers, each complex number will be represented internally as a pair
of real numbers. However, because the type name cmplx is different from the ML
type real*real for a pair of real numbers, a function that is meant to operate on
a pair of real numbers cannot be applied to a value of type cmplx: The ML type
checker will not allow this. This restriction is one of the fundamental properties of
any good abstract data-type mechanism: Programs should be restricted so that only
the declared operations of the abstract type can be applied.
Within the data-type declaration, however, the functions that are part of the abstract data type must be able to treat complex numbers as pairs of real numbers.
Otherwise, it would not be possible to implement many operations. In ML, a constructor is used to distinguish “abstract” from “concrete” uses of complex numbers.
Specifically, if z is a complex number, then matching z against the pattern C (x,y) will
bind x to the real part of z and y to the imaginary part of z. This form of pattern
matching is used in the implementation of complex addition, for example, in which
add combines the real parts of its two arguments and the imaginary parts of its two
arguments. The pair representing the complex sum is then identified as a complex
number by application of the constructor C.
When ML is presented with this declaration of complex numbers, it returns the
following type information:
type cmplx
val cmplx = fn : real * real → cmplx
val x coord = fn : cmplx → real
val y coord = fn : cmplx → real
val add = fn : cmplx * cmplx → cmplx
The first line indicates that the declaration introduces a new type, named cmplx.
The next four lines list the operations allowed on expressions with type cmplx. The
types of these operations involve the type cmplx, not the type real*real that is used to
represent complex numbers. Be sure you understand the code for add, for example,
and why the type checker gives add the type cmplx * cmplx → cmplx.
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In general, an ML abstype declaration has the form
abstype t = <constructor> of <type>
with
val <pattern> = <body>
...
fun f(<pattern>) = <body>
...
end
The syntax
t = <constructor> of <type>
is the same notation used to define data types. The identifier t is the name of the new
type, <constructor> is the name of the constructor for the new type t, and <type> gives
the type used to represent elements of the abstract type. The difference between
an abstype and a data type lies with the rest of the preceding syntax. The value and
function declarations that occur between the with and the end keywords are the only
operations that may be written with the constructor. Other parts of the program may
refer to the type name t and the functions and values declared between with and
end. However, other parts of the program are outside the scope of the declaration
of the constructor and therefore may not convert between the abstract type and
its representation. The operations declared in an abstype declaration are called the
interface of the abstract type, and the hidden data type and associated function bodies
are called its implementation.
As many readers know, there are two common ways of representing complex
numbers. The preceding abstract type uses rectangular coordinates – each complex
number is represented by a pair consisting of its real and imaginary coordinates. The
other standard representation is called polar coordinates. In the polar representation,
each complex number is represented by its distance from the origin and an angle
indicating the direction (relative to the real axis) used to reach the point from the
origin. Because the implementation of the abstract data type cmplx is hidden, a
program that uses a rectangular implementation can be replaced with one that uses
a polar representation without changing the behavior of any program that uses the
abstract data type.
A polar representation of complex numbers is used in this abstract data-type
declaration:
abstype cmplx = C of real * real with
fun cmplx(x,y: real) = C(sqrt(sq(x)+sq(y)), arctan(y/x))
fun x coord(C(r,theta)) = r * cos(theta)
fun y coord(C(r,theta)) = r * sin(theta)
fun add(C(x1,y1), C(x2,y2)) = C(. . . , . . . )
end
9.2 Language Support for Abstraction
where the implementation of add is filled in as appropriate.
Example 9.4 Set Abstract Type
We can also create polymorphic abstypes, as the following abstype declaration
illustrates:
abstype ’a set = SET of ’a list with
val empty = SET(nil)
fun insert(x, SET(elts)) = . . .
fun union(SET(elts1), SET(elts2)) = . . .
fun isMember(x, SET(elts)) = . . .
end
Assuming the preceding . . . ’s are filled in with the appropriate code to implement
insert, union, and isMember, ML returns as the result of evaluating this declaration:
type ’a set
val empty = - : ’a set
val insert = fn : ’a * (’a set) → (’a set)
val union = fn : (’a set) * (’a set) → (’a set)
val isMember = fn : ’a * (’a set) → bool
Note that the value for empty is written as -, instead of nil. This hiding prevents users
of the ’a set abstype from using the fact that the abstype is currently implemented as
a list.
Clu Clusters
The first language with user-declared abstract types was Clu. In Clu, abstract types
are declared with the cluster construct. Here is an example declaration of complex
numbers:
complex = cluster is make complex, real part, imaginary part, plus, times
rep = struct [ re, im : real ]
make complex = proc (x, y : real) returns (cvt)
return (rep${re:x, im:y})
real part = proc (z : cvt) returns (real)
return (x.re)
imaginary part = proc (z : cvt) returns (real)
return (x.im)
plus = proc (z, w : cvt) returns (cvt)
return (rep${re: z.re + w.re, z.im + w.im})
mult = . . .
end complex
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BARBARA LISKOV
Barabara Liskov’s research and teaching interests include programming
languages, programming methodology, distributed computing, and parallel computing. She was the developer of the programming language
Clu, which was described this way when it was developed in the 1970s:
“The programming language Clu is a practical vehicle for study and development of approaches in structured programming. It provides a new
linguistic mechanism, called a cluster, to support the use of data abstractions in program construction.”
When I was a graduate student at MIT, Barbara had huge piles of
papers, many three or four feet high, covering the top of her desk. Whenever I went in to talk with her, I imagined I might find her unconscious
under a fallen heap of printed matter.
In this code, the line rep = struct [ re, im : real ] specifies that each complex number
is represented by a struct with two real (float) parts, called re and im. Inside the
implementations of operations of the cluster, the keywords cvt and rep are used to
convert between types complex and struct [ re, im : real ], in the same way that pattern
matching and the constructor C are used in the ML abstype declaration for cmplex.
9.2.4 Representation Independence
We can understand the significance of abstract type declarations by considering some
of the properties of a typical built-in type such as int:
We can declare variables x : int of this type.
There is a specific set of built-in operations on the type +, -, *, etc.
Only these built-in operations can be applied to values of type int; it is not type
correct to apply string or other types of operations to integers.
Because ints can be accessed only by means of the built-in operations, they enjoy
a property called representation independence, which means that different computer
representations of integers do not affect program behavior. One computer could
represent ints by using 1’s complement, and another by using 2’s complement, and
the same program run on the two machines will produce the same output (assuming
all else is equal).
9.2 Language Support for Abstraction
A type has the representation-independence property if different (correct) underlying representations or implementations for the values of that type are indistinguishable by clients of the type. This property implies that implementations for
such types may be changed without breaking any client code, a useful property for
software engineering.
In a type-safe programming language with abstract data types,
we can declare variables of an abstract type,
we define operations on any abstract type,
the type-checking rules guarantee that only these specified operations can be
applied to values of the abstract type.
For the same reasons as those for built-in types such as int, these properties of
abstract data types imply representation independence for user-defined type. Representation independence means that we can change the representation for our abstract
type without affecting the clients of our abstraction.
In practice, different programming languages provide different degrees of representation independence. In Clu, ML, and other type-safe programming languages
with an abstract data-type mechanism, it is possible to prove a form of representation independence as a theorem about the ideal implementation of the language.
The proof of this theorem relies on the way the programming language restricts access to implementation of an abstract data type. In languages like C or C++ that
have type loopholes, representation independence is an ideal that can be achieved
through good programming style. More specifically, if a program uses only a specific
set of operations on some data structure, then the data structure and implementations
of these operations can be changed in various ways without changing the behavior of
the program that uses them. However, C does not enforce representation independence. This is true for built-in types as well as for user-declared types. For example,
C code that examines the bits of an integer can distinguish 1’s complement from 2’s
complement implementations of integer operations.
9.2.5 Data-Type Induction
Data-type induction is a useful principle for reasoning about abstract data types. We
are not interested here in the formal aspects of this principle, only the intuition that
it provides for thinking about programming and data-type equivalence. Data-type
equivalence is an important relation between abstract data types: We can replace a
data type with any equivalent one without changing the behavior of any client program. This principle is used informally in program development and maintenance.
In particular, it is common to first build a software system with potentially inefficient prototype implementations of a data type and then to replace these with more
efficient implementations as time permits.
Partition Operations
For many data types, it is possible to partition the operations on the type into three
groups:
1. Constructors: operations that build elements of the type.
2. Operators: operations that map elements of the type that are definable only
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with constructors to other elements of the type that are definable with only
constructors.
3. Observers: operations that return a result of some other type.
The main idea is that all elements of the data type can be defined with constructors;
operators are useful for computing with elements of the type, but do not define any
new values. Observers are the functions that let us distinguish one element of the
data type from another. They give us a notion of observable equality, which is usually
different from equality of representation.
Example 9.5 Equivalence of Integer Sets Implementations
For the data type of integer sets with the signature
empty : set
insert : int * set → set
union : set * set → set
isMember: int * set → bool
the operations can be partitioned as follows:
1. Constructors: empty and insert
2. Operator: union
3. Observer: isMember
We may understand some of the intuition behind this partitioning of the operations
by thinking about how sets might be used in a program. Because there is no print
operation on sets, a program cannot produce a set directly as output. Instead, if any
printable output of a program depends on the value of some set expression, it can be
only because of some membership test on sets. Therefore, if two sets, s1 and s2 , have
the property that
For all integers n, isMember(n,s1 ) = isMember(n,s2 )
then no program will be able to distinguish one from the other in any observable
way. This actually gives us a useful equivalence relation on sets: Two sets s1 and s2
are equivalent if isMember(n,s1 ) = isMember(n,s2 ) for every integer n. For sets, this
equivalence principle is actually the usual extensionality axiom from set theory: Two
sets are equal if they have precisely the same elements.
Given extensionality of sets, it is easy to see that any set can be defined by insertion
of some number of elements into the empty set. More specifically, for every set s,
there is a sequence of elements n1 , n2 , . . . , nk , with
s ≈ insert(n1 , insert(n2 , . . . insert(nk , empty) . . . )).
This demonstrates that insert and empty are in fact constructors for the data type of
sets: Every set can be defined with only these two operations.
To formally show that a given method is an operator, we need to demonstrate
that, for any given use of an operator, there exists a sequence of constructor calls
that produces the same result. As we expect, union is a useful operation on sets, but if
s1 and s2 are definable with the operations of this data type, then union(s1 ,s2 ) can be
9.2 Language Support for Abstraction
defined with only insert and empty. For this reason, union is classified as an operator,
not a constructor.
In practice, it is not always easy to partition the operations of a data type into
these three groups. Some functions might appear to fit into two groups. However,
the principle of data-type induction still provides a useful guide for reasoning about
arbitrary abstract data types.
Induction over Constructors
Because all elements of a given abstract type are given by a sequence of constructor
operations, we may prove properties of all elements of an abstract type by induction
on the number of uses of constructors necessary to produce a given element. Once we
show that some function of the signature is an operator, we can generally eliminate
it from further consideration.
Example 9.6
As an illustration of data-type induction, we will go through the outline of a proof that
two different implementations of integer sets are equivalent. The term equivalent
means that, if we replace one implementation with another, then no client program
can detect the change.
Let us begin with a definition of equivalence, the property we are trying to prove:
Two implementations set and set’ are equivalent if, for all values of all parameters, all corresponding applications of observers to set expressions are
equal.
We refer to the operations of set by empty, insert, union, and isMember and the
operations of set’ by empty’, insert’, union’, and isMember’. Some examples of corresponding applications of observers are
isMember(6, insert(n1, . . . insert(nk, empty) . . . )) and
isMember’(6, insert’(n1, . . . insert’(nk, empty’) . . . ))
These expressions correspond in the sense that all the nonset arguments are the same,
but we have replaced the operations of one implementation with another.
The intuition behind this definition of data-type equivalence is similar to the
equivalence relation on set expressions we previously discussed. Specifically, suppose
we have two different implementations of sets. The only way a client program can
use one of them to produce a printable (or observable) output is to use the set
constructors and operators to build up some potentially complicated sets and then
to “observe” the resulting sets by using the observer operations.
Because we have established that union is an operator, not a constructor, and
the only observer function is isMember, proving the equivalence of two different
implementations of sets boils down to showing that
for all z, isMember(z, aSet) = isMember’(z, aSet’)
where aSet and aSet’ are corresponding expressions in which only the constructors
empty and insert (or empty’ and insert’) are used.
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We have now reduced the problem of establishing data-type equivalence to the
problem of showing that
isMember(n, insert(n1, . . . insert(nk, empty) . . . ))
= isMember’(n, insert’(n1, . . . insert’(nk, empty’) . . . ))
for all sequences of natural numbers n, n1, . . . , nk. We can do this by induction on k,
the number of insert operations required for constructing the sets.
The inductive proof proceeds as follows.
Base Case: Zero Insert Operations. In this case, we must show that
for all n, isMember(n, empty) = isMember’(n, empty’)
We must do this by looking at the actual implementations of the data type. However,
in a correct implementation of sets, the empty set has no elements. Therefore, if both
implementations are correct, then isMember(n, empty) = isMember’(n, empty’) = false.
Induction Step. We assume that equivalence holds when k insert operations are
used and consider the case of k+1 insert operations. This reduces to showing that, for
all n, m, we have
isMember(n, insert(m, s)) = isMember’(n, insert’(m, s’))
under the assumption that for all n we have isMember(n,s) = isMember’(n,s’). Again,
we must do this by looking at the actual implementations. However, if both implementations are correct, then we should have isMember(n,s) = isMember’(n,s’).
An interesting aspect of this argument is that we have proved something about
all possible programs that use a data type by using only ordinary induction over
the constructors. The reason this is possible is the assumption that, in a language
with abstract data types, only the operations of the data type can be applied to
values of the type. It would be impossible to use this form of proof if type-checking
rules did not guarantee that only set operations may be applied to a set. In practice,
however, the ideas illustrated here may be useful for programming in languages such
as C that do not enforce data abstraction, as long as the actual programs that are
built do not operate on data structures except through operations designed for this
purpose.
The “proof” previously described is actually just a proof outline that assumes
some properties of each implementation of sets. To understand how data-type induction really works, you may work through the equivalence proof with two specific
implementations in mind. For example, you may use data-type induction to prove the
equivalence of a linked-list implementation and a doubly linked-list implementation
of sets.
9.3 MODULES
Early abstract data-type mechanisms, like Clu clusters, declared only one type. If
you want only an abstract data type of stacks, queues, trees, or other common data
structures, then this form is sufficient: In each of these examples, there is one kind
of data structure that is being defined, and this can be the abstract type. However,
there are situations in which it is useful to define several related structures. More
9.3 Modules
generally, a set of types, functions, exceptions, and other user-definable entities may
be conceptually related and have implementations that depend on each other.
A module is a programming language construct that allows a number of declarations to be grouped together. Early forms of modules, such as in the language
Modula, provide minimal information hiding. However, a good module mechanism
will allow a programmer to control the visibility of items declared in a module. In
addition, parameterized modules, as discussed in the next subsection and in more detail in Section 9.4, make it possible to generalize a set of declarations and instantiate
them together in different ways for different purposes.
9.3.1 Modula and Ada
As mentioned briefly in Subsection 5.1.4, the Modula programming language was a
descendent of Pascal, developed by Pascal designer Niklaus Wirth in Switzerland in
the late 1970s. The main innovation of Modula over Pascal is a module system. We
will use Modula-2, a successful version of the language, to discuss Modula modules.
The basic form for Modula-2 modules is
module <module name>;
import specifications;
declarations;
begin
statements;
end <module name>.
The declarations may be constant, type and procedure declarations, as in Pascal. The
statements are Pascal-like statements. An import specification lists another module
name and lists the constants, types, and procedures used from that other module; for
example,
from Trig import sin, cos, tan
It is also possible to write the module name only, importing all declarations from that
module.
The basic form of the preceding module may be used as a main program, with the
statements performing some task. However, the basic form does not have any parts
that are visible externally. To make declarations of one module visible to another, a
module interface must be given.
In Modula terminology, a module interface is called a definition module and an
implementation an implementation module. An implementation module has the form
given at the beginning of this section, and a definition module contains only the names
and types of the parts of an implementation module that are to be visible to other
modules.
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Example 9.7 Modula-2 Definition of Fractions
definition module Fractions;
type fraction = ARRAY [1 .. 2] OF INTEGER;
procedure add (x, y : fraction) : fraction;
procedure mul (x, y : fraction) : fraction;
end Fractions.
implementation module Fractions;
procedure Add (x, y : Fraction) : Fraction;
VAR temp : Fraction;
BEGIN
temp [1] := x [1] * y [2] + x [2] * y [1];
temp [2] := x [2] * y [2]; RETURN temp;
END Add;
procedure Mul (x, y : Fraction) : Fraction;
...
END Mul;
end Fractions.
In this example, a complete type declaration is included in the interface. As a result,
the client code can see that a fraction is an array of integers. The following example
hides the implementation of a type. In Modula terminology, the type declaration
in Example 9.7 is transparent whereas the declaration in Example 9.8 is abstract or
opaque.
Example 9.8 Modula-2 Stack Module
definition module Stack module
type stack (* an abstract type *)
procedure create stack ( ) : stack
procedure push( x:integer, var s:stack ) : stack
...
end Stack module
implementation module Stack module
type stack =array [1..100] of integer
...
end Stack module
This example code defines stacks of integers. For stacks of various kinds, we would
either need to repeat this definition with other types of elements or to build a generic
stack module that takes the element type as a parameter. Mechanisms for defining
generic modules are included in Modula-2, Ada, and most modern languages (except
Java!). As representative examples, we discuss C++ templates and ML functors in
Section 9.4.
9.3 Modules
Ada Packages
The Ada programming language was designed in the late 1970s and early 1980s
as the result of an initiative by the U.S. Department of Defense (DoD). The DoD
wanted to standardize its software procurement around a common language that
would provide program structuring capabilities and specific features related to realtime programming. A competitive process was used to design the language. Four
teams, each assigned a color as its code name, were each funded to produce a tentative “strawman” design. One of these designs, selected by a process of elimination,
eventually led to the language Ada.
By some measures, Ada has been a successful language. Many Ada programs
have been written and used. Some Ada design issues led to research studies and
improvements in the state of the art of programming language design. However,
in spite of some practical and scientific success, adoption of the language outside
of suppliers of the U.S. government has been limited. One limitation was the lack
of easily available implementations. Most companies who produced Ada compilers,
especially at the height of the language’s popularity, expected to sell them for high
prices to military contractors. As a result, the language received little acceptance in
universities, research laboratories, or in companies concerned primarily with civilian
rather than military markets.
Ada modules are called packages. Packages can be written with a separate interface, called a package specification, and implementation, called a package body.
Here is a sketch of how the fraction package in Example 9.7 would look if translated
into Ada:
package FractionPkg is
type fraction is array . . . of integer;
procedure Add . . .
end FractionPkg;
package body FractionPkg is
procedure Add . . .
end FractionPkg;
9.3.2 ML Modules
The standard ML module system was designed in the mid-1980s as part of a redesign
and standardization effort for the ML programming language. The principal architect
of the ML module system was David MacQueen, who drew on concepts from type
theory as well as his experience with previous programming languages.
The three main parts of the standard ML module system are structures, signatures,
and functors. An ML structure is a module, which is a collection of type, value, and
structure declarations. Signatures are module interfaces. In standard ML, signatures
behave as a form of “type” for a structure, in the sense that a module may have more
than one signature and a signature may have more than one associated module. If
a structure satisfies the description given in a signature, the structure “matches” the
signature.
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Functors are functions from structures to structures. Functors are used to define generic modules. Because ML does not support higher-order functors (functors
taking functors as arguments or yielding functors as results), there is no need for
functor signatures.
Structures are defined with structure expressions, which consist of a sequence
of declarations between keywords struct and end. Structures are not “first class”
in that they may only be bound to structure identifiers or passed as arguments to
functors. The following declaration defines a structure with one type and one value
component:
structure S =
struct
type t = int
val x : t = 3
end
In this example, the structure expression following the equal sign has type component t equal to int and value component x equal to 3. In standard ML this structure is
“time stamped” when the declaration is elaborated, marking it with a unique name
that distinguishes it from any other structure with the same type and value components. Structure expressions are therefore said to be “generative” because each
elaboration may be thought of as “generating” a new one. The reason for making
structure expressions generative is that the module language provides a form of version control based on specifying that two possibly distinct structures or types must be
equal.
The components of a structure are accessed by qualified names, written in a form
used for record access in many languages. For instance, given our preceding structure
declaration for S, above, the name S.x refers to the x component of S, and hence has
value 3. Similarly, S.t refers to the t component of S and is equivalent to the type int
during type checking. In other words, type declarations in structures are transparent
by default. As in Modula and Ada, the distinction between transparent and opaque
type declarations appears in the interface.
ML signatures are structure interfaces and may declared as follows:
signature SIG =
sig
type t
val x : t
end
This signature describes structures that have a type component t and a value component x, whose type is the type bound to t in the structure. Because the structure S
previously introduced satisfies these conditions, it is said to match the signature SIG.
The structure S also matches the following signature SIG’:
9.3 Modules
signature SIG’ =
sig
type t
val x : int
end
This signature is matched by any structure providing a type t and a value x of type
int such as the structure S. However, there are structures that match SIG, but not SIG’,
namely any structure that provides a type other than int and a value of that type.
In addition to ambiguities of this form, there is another, more practically motivated,
reason why a given structure may match a variety of distinct signatures: Signatures
may be used to provide distinct views of a structure. The main idea is that the signature
may specify fewer components than are actually provided. For example, we may
introduce the signature
signature SIG” =
sig
val x : int
end
and subsequently define a view T of the structure S by declaring
structure T : SIG” = S
It should be clear that S matches the signature SIG” because it provides an x component
of type int. The signature SIG” in the declaration of T causes the t component of S to
be hidden, so that subsequently only the identifier T.x is available.
Example 9.9 ML Geometry Signatures and Structures
This example gives signatures and structures for a simple geometry program. An
associated functor, for which structure parameterization is used, appears in Example
9.11. The three following signatures describe points, circles, and rectangles, with each
signature containing a type name and names of associated operations. Two signatures
use the SML include statement to include a previous signature. The effect of include
is the same as copying the body of the named signature and placing it within the
signature expression containing the include statement:
signature Point =
sig
type point
val mk point : real * real → point
val x coord : point - real
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val y coord : point - real
val move p : point * real * real → point
end;
signature Circle =
sig
include Point
type circle
val mk circle : point * real → circle
val center : circle → point
val radius : circle → real
val move c : circle * real * real → circle
end;
signature Rect =
sig
include Point
type rect
(* make rectangle from lower right, upper left corners *)
val mk rect : point * point → rect
val lleft : rect → point
val uright : rect → point
val move r : rect * real * real → rect
end;
Here is the code for the Point, Circle, and Rect structures:
structure pt : Point =
struct
type point = real*real
fun mk point(x,y) = (x,y)
fun x coord(x,y) = x
fun y coord(x,y) = y
fun move p((x,y):point,dx,dy) = (x+dx, y+dy)
end;
structure cr : Circle =
struct
open pt
type circle = point*real
fun mk circle(x,y) = (x,y)
fun center(x,y) = x
fun radius(x,y) = y
fun move c(((x,y),r):circle,dx,dy) = ((x+dx, y+dy),r)
end;
structure rc : Rect =
struct
open pt
9.4 Generic Abstractions
type rect = point * point
fun mk rect(x,y) = (x,y)
fun lleft(x,y) = x
fun uright (x,y) = y
fun move r(((x1,y1),(x2,y2)):rect,dx,dy) =
((x1+dx,y1+dy),(x2+dx,y2+dy))
end;
9.4 GENERIC ABSTRACTIONS
Abstract data types such as stacks or queues are useful for storing many kinds of data.
In typed programming languages, however, the code for stacks of integers is different
from the code for stacks of strings. The two different versions of stacks are written
with different type declarations and may be compiled to code that allocates different
amounts of space for local variables. However, it is time consuming to write different
versions of stacks for different types of elements and essentially pointless because
the code for the two cases is almost identical. Thus, over time, most typed languages
that emphasize abstraction and encapsulation have incorporated some form of type
parameterization.
9.4.1 C++ Function Templates
For many readers, the most familiar type-parameterization mechanism is the C++
template mechanism. Although some C++ programmers associate templates with
classes and object-oriented programming, function templates are also useful for programs that do not declare any classes. We look at function templates briefly before
considering module-parameterization mechanisms from other languages.
Simple Polymorphic Function
Suppose you write a simple function to swap the values of two integer variables:
void swap(int& x, int& y){
int tmp = x; x = y; y = tmp;
}
Although this code is useful for exchanging values of integer variables, it is not written
in the most general way possible. If you wish to swap values of variables of other
types, then you can define a function template that uses a type variable T in place of
the type name int:
template<typename T>
void swap(T& x, T& y){
T tmp = x; x = y; y = tmp;
}
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The main idea is to think of the type name T as a parameter to a function from types
to functions. When applied, or instantiated, to a specific type, the result is a version
of swap that has int replaced with another type. In other words, swap is a general
function that would work perfectly well for many types of arguments, except for the
fact that the code contains the specific type int. Templates allow us to treat swap as a
function with a type argument.
In C++, function templates are instantiated automatically as needed, using the
types of the function arguments to determine which instantiation is needed. This is
illustrated in the following lines of code:
int i,j; . . .
swap(i,j);
// Use swap with T replaced by int
float a,b; . . . . . . swap(a,b); // Use swap with T replaced by float
String s,t; . . . swap(s,t);
// Use swap with T replaced by String
You may have noticed that the C++ keyword associated with a type variable is class.
In C++, some types are classes and some, like int and float, are not. As illustrated
here, the keyword class is misleading, because a template may be used with nonclass
types such as int and float.
C++ templates are instantiated at program link time. More specifically, suppose
that the swap function template is stored in one file and compiled and a program
calling swap is stored in another file and compiled separately. The so-called relocatable object files produced when the calling program is compiled will include information indicating that the compiled code calls a function swap of a certain type. The
program linker is designed to combine the two program parts by linking the calls to
swap in the calling program to the definition of swap in a separate compilation unit. It
does so by instantiating the compiled code for swap in a form that produces code appropriate for the calls to swap. If the calling program calls swap with several different
types, then several different instantiated copies of swap will be produced. A different
copy is needed for each type of call because swap declares a local variable tmp of
type T. Space for tmp must be allocated in the activation record for swap. Therefore,
the compiled code for swap must be modified according to the size of a variable of
type T. If T is a structure or object, for example, then the size might be fairly large.
On the other hand, if T is int, the size will be small. In either case, the compiled code
for swap must “know” the size of the datum so that addressing into the activation
record can be done properly.
Operations on Type Parameters
The swap example is simpler than most generic functions in several respects. The
most important is that the body of swap does not require any operations on the type
parameter T, other than variable declaration and assignment. A more representative
example of a function template is the following generic sort function:
template <typename T>
void sort( int count, T * A[count] ) {
for (int i=0; i<count-1; i++)
9.4 Generic Abstractions
for (int j=i+1; j<count-1; j++)
if (A[j] < A[i]) swap(A[i],A[j]);
}
If A is an array of type T, then sort(n, A) will work only if operator < is defined on
type T. This property of sort is not declared anywhere in the C++ code. However,
when the function template is instantiated at link time, the actual type T must have
an operator < defined or a link-time error will be reported and no executable object
code will be produced.
The linking process for C++ is relatively complex. We will not study it in detail.
However, it is worth noting that if < is an overloaded operator, then the correct version
of < must be identified when compiled code for sort is linked with a calling program.
As a general programming principle, it is more convenient to have program errors
reported at compile time than at link time. One reason is that separate program
modules are compiled independently, but linked together only when the entire system
is assembled. Therefore, compilation is a “local” process that can be carried out by
the designer or implementer of a single component. In contrast, link-time errors
represent global system properties that may not be known until the entire system
is assembled. For this reason, C++ link-time errors associated with operations in
templates can be irritating and a source of frustration.
Somewhat better error reporting for C++ templates could be achieved if the
template syntax included descriptions of the operations needed on type parameters.
However, this is relatively complicated in C++ because of overloading and other
properties of the language. In contrast, ML has simpler overloading and includes
more information in parameterized constructs, as described in Subsection 9.4.2.
Comparison with ML Polymorphism
Here is the ML code for a polymorphic sort function. The algorithm used in this code
is insertion sort, which uses the subsidiary function insert that appears first:
fun insert(less, x, nil) = [x]
| insert(less, x, y::ys) = if x<y then x::y::ys
else y::insert(less,x,ys)
fun sort(less, nil) = nil
| sort(less, x::xs) = insert(less, x, sort(less,xs))
In the ML polymorphic sort function, the less-than operation is passed as a function
argument to sort. No instantiation is needed because all lists are represented in the
same way (with cons cells, as in Lisp).
Uniform data representation has its advantages and disadvantages. Because there
is no need to repeat code for different argument types, uniform data representation
leads to smaller code size and avoids complications associated with C++-style linking.
On the other hand, the resulting code can be less efficient, as uniform data representation often involves using pointers to data instead of storing data directly in
structures.
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9.4.2 Standard ML Functors
In the ML module system, structures may be parameterized in very flexible ways.
We look at this part of the ML module system as a good example of the ways that
modules can usefully be parameterized in programming languages. Module parameterization is different from ML polymorphism and is essentially independent of it.
More specifically, the ML module design could be adapted to languages that do not
have polymorphism or type inference.
In standard ML, a parameterized structure is called a functor. Functors are written
in a functionlike form, with a parameter list containing structure names and signatures. Here is an example that uses the signature SIG defined in Subsection 9.3.2:
functor F ( S : SIG ) : SIG =
struct
type t = S.t * S.t
val x : t = (S.x,S.x)
end
This code declares a functor F that takes as argument a structure matching the
signature SIG and yields a structure matching the same signature. When applied
to a suitable structure S, the functor F yields as its result the structure whose type
component t, is bound to the product of S.t with itself and whose value component
x, is the pair whose components evaluate to the value of S.x.
When free structure variables in signatures are used, certain forms of dependency
of functor results on functor arguments may be expressed. For example, the following
declaration specifies the type of y in the result signature of G in terms of the type
component t of the argument S:
functor G ( S : SIG ) : sig val y : S.t * S.t end =
struct
val y = (S.x,S.x)
end
The rest of this section consists of examples that show how SML module parameterization can be used to solve some programming problems. Example 9.10 discusses
a form of container structure and Example 9.11 completes the geometry example
started in Example 9.9.
Example 9.10 Parameterized Priority Queue Module
Container structures such as sets, queues, dictionaries, and so on are used in many
programs. This example shows how to define a container structure, priority queues,
when one or more container operations require some operation on the type of objects
stored in the container. Because priority queues require an ordering on the elements
stored in a queue, our definition of priority queues is parameterized by a structure
consisting of a type and order relation.
9.4 Generic Abstractions
We begin by defining the signature of a type with an order relation:
signature Ordered =
sig
type t;
val lesseq : t * t → bool;
end;
Any structure that matches signature Ordered must have a type t and a binary relation
lesseq. We assume in the definition of priority queue that this relation is a total order
on the type t. In this functor definition, we declare a type-implementing priority
queue operations and a set of operations on priority queues. Because the insert or
deletemax operations must test whether an element is less than other elements, at least
one of these operations will use the operation elem.lesseq provided by the parameter
structure elem:
functor PQ(elem:Ordered) =
struct
type pq = . . . elem.t . . . ;
(* representation uses element type
*)
val empty : pq
= ...;
fun insert: elem.t * pq → pq = . . . ; (* operations may use elem.lesseq
*)
fun deletemax: pq → elem.t * pq = . . . ;
end;
Example 9.11 Completion of Geometry Example
This example shows how the Point, Circle, and Rect structures in Example 9.9 can
be combined to form a geometry structure with operation on all three kinds of
geometric objects. To show how additional operations can be added when modules
are combined, the Geom signature contains all of the geometric types and operations,
plus a bounding box function, bbox, intended to map a circle to the smallest rectangle
containing it:
signature Geom =
sig
include Circle
(* add signature of Rect, except Point since Point is already defined in Circle*)
type rect
val mk rect : point * point → rect
val lleft : rect → point
val uright : rect - → point
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val move r : rect * real * real → rect
(* include function computing bounding box of circle *)
val bbox : circle → rect
end;
The geometry structure is constructed with a functor that takes Circle and Rect
structures as arguments. The signatures Circle and Rect both name a type called
point. The two point types must be implemented in the same way. Otherwise, operations that involve both circles and rectangles will not work properly. The constraint
that two structures (Circle and Rect) must share a common substructure (point) is
stated explicitly in the sharing constraint in the parameter list of this functor:
functor geom(
structure c:Circle
structure r:Rect
sharing type r.point = c.point (*constraint requires same type in both
structures *)
) : Geom =
struct
type point = c.point
val mk point = c.mk point
val x coord = c.x coord
val y coord = c.y coord
val move p = c.move p
type circle = c.circle
val mk circle = c.mk circle
val center = c.center
val radius = c.radius
val move c = c.move c
type rect = r.rect
val mk rect = r.mk rect
val lleft = r.lleft
val uright = r.uright
val move r = r.move r
(* Bounding box of circle *)
fun bbox(c) =
let val x = x coord(center(c))
and y = y coord(center(c))
and r = radius(c)
in
mk rect(mk point(x-r,y-r),mk point(x+r,y+r))
end
end;
9.4 Generic Abstractions
9.4.3 C++ Standard Template Library
The C++ Standard Template Library (STL) is a large program library developed
by Alex Stepanov and collaborators and adopted as part of the C++ standard. The
STL provides a collection of container classes, such as lists, vectors, sets, and maps,
and a collection of related algorithms and other components. Because STL is a good
example of a large, practical library with generic structures, we survey some of the
main concepts. The goal in this section is not to teach you enough about STL so that
you can use it easily, but rather to explain the concepts so you can appreciate some
of the design ideas and the power of the system.
One striking feature of STL is the run-time efficiency of the code that is generated. STL not only makes it easy to write certain programs by providing useful
structures, STL makes it possible to write library-based code that is as fast or faster
than the code you might produce if you worked much harder and did not use STL.
One reason for the efficiency of STL is that C++ templates are expanded at compile/link time, with a separate code generated (and optimized) for each instantiation.
Another reason is the use of overloading, which is resolved at compile time, similarly
allowing the compiler to optimize code that is selected for exactly the type of data
manipulated in the program. On the other hand, because of code duplication resulting from template expansion, the compiled code of programs that use STL may be
large.
In this subsection, the term object is used to mean a value stored in memory.
This is consistent with the use of the term in C, C++, and STL documentation. STL
relies on C++ templates, overloading, and the typing algorithms that figure out type
parameters to templates. STL does not use virtual functions and, according to its
developer, is not an example of object-oriented programming.
There are six kinds of entities in STL:
Containers, each a collection of typed objects.
Iterators, which provide access to objects of a container. Intuitively, an iterator is
a generalization of pointer or address to some position in a container.
Algorithms.
Adapters, which provide conversions between one form of entity and another.
An example is a reverse iterator, which reverses the direction associated with an
iterator.
Function objects, which are a form of closure (function code and associated environment). These are used extensively in a form that allows function arguments
to templates to be in-lined. More specifically, a function object is passed as two
separate arguments, a template argument carrying the code and a run-time argument carrying the state of the function. The type system is used to make sure
that the code and state match.
Allocators, which encapsulate the memory pool. Different allocators may provide
garbage-collected memory, reference-counted memory, persistent memory, and
so on.
An example will give you some feel for how STL works.
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Example 9.12 STL Lists
List is one of the simplest container types defined in STL. Here is sample code that
declares a list of strings called wordlist:
#include <string>
#include <list>
int main (void) {
list<string> wordlist;
}
The code list<string> wordlist instantiates a template class list<string> and then creates
an object of that type. STL gives you operations for putting elements on the front or
back of a list. Here is sample code to add words to our word list:
wordlist.push
wordlist.push
wordlist.push
wordlist.push
back(“peppercorn”);
back(“funnybone”);
front(“nosegay”);
front(“supercilious”);
We now have a list with four strings in it. As the names suggest, the list member
function push back() places an object onto the back of the list and push front() puts
one on the front. We can print the words in our list by using an iterator. Here is the
declaration of an iterator for lists of strings:
list<string>::iterator wordlistIterator;
As suggested earlier in this section, an STL iterator is like a pointer – an iterator
points to an element of a container, and dereferencing a nonnull iterator produces
an object from the container. An STL iterator is declared to have a specific container
type; this is important because moving a “pointer” through a container works differently for different types of containers. Here is sample code that prints the words in
the list:
for (wordlistIterator=wordlist.begin();
wordlistIterator!=wordlist.end();
++wordlistIterator) {
// dereference the iterator to get the list element
cout << *wordlistIterator << endl;
}
We determine the bounds of this for loop by calling wordlist.begin(), which returns an
9.4 Generic Abstractions
iterator pointing to the beginning of the container, and wordlist.end(), which returns
an iterator pointing to the position one past the end of the container. The loop
variable is incremented by the operator ++, which is defined on each iterator and
moves the iterator from one list element to the next. In the body of the loop, we
dereference the iterator to obtain the list element at the current position.
As Example 9.12 shows, iterators are essential for programming with STL containers. Most STL algorithms use iterators of some kind, and there are several different kinds of iterators. The simplest form, so-called trivial iterators, may be dereferenced to refer to some object, but operations such as increment and comparison
are not guaranteed to be supported. Input iterators may be dereferenced to refer
to some object and may be incremented to obtain the next iterator in a container.
Output iterators are a fairly restricted form of iterator that may be used for storing
(but not necessarily accessing) a sequence of values. The other forms of iterators are
forward iterator, bidirectional iterator, and random-access iterator. Specific STL algorithms may require specific kinds of iterators, depending on whether the algorithm
reads from a container or stores values into a container and whether the access is
sequential (like reading or writing a tape) or random access.
Another basic concept in STL is a range. A range is the portion of a container
defined by two iterators. The elements of a range are all the objects that may be
reached by incrementing the first interator until it reaches the value of the second
iterator. If you think of an array as an example container, then a range is a pair of
integers, the first less than the second and both less than the length of the array,
representing the portion of the array lying between the first index and the second.
The following example discusses the STL merge function, which uses ranges as
inputs and outputs. In general, a merge operation combines two sorted sequential
structures and produces a third. In merge sort, for example, the merge phase combines
two sorted lists into a longer sorted list that contains all the elements of the two input
lists.
Example 9.13 Generic Merge Function
The generic STL merge function takes two ranges as inputs and returns another
range. Conceptually, the type of merge is
merge : range(s) x range(t) x comparison(u) → range(u)
where range(s) means a range of a container that contains objects of type s. Because
C++ has subtyping (explained in Chapters 10 and 12), the types s, t, and u need not be
identical. However, types s and t must be subtypes of u, meaning that every object of
type s or t can be viewed as or converted to an object of type u. The informal notation
comparison(u) means that merge requires a comparison operation (less-than-or-equal)
on type u, so that merge can combine structures in order.
In fact, each input range in STL is passed as two iterators and an output range is
passed as one iterator. The reason we need two iterators for each input range is that
we need to know how many elements in the range. Because the output of merge is
determined by the input, the output range is represented by a single output iterator
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that marks the place to begin writing the merged container. Therefore, merge has the
following parameters:
merge(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator1 last2,
OutputIterator result, Compare comp)
In words, the first two arguments provide the first input range, the next two arguments
the second input range, the fifth argument the output range, and the sixth argument
must provide a comparison function.
Merge is a polymorphic function, merging ranges that contain any types of objects,
as long as the types are consistent with each other. The polymorphism of merge is
implemented by a template with four parameters. The template requires two iterator
types, one for each of the two sequential structures that will be merged, together with
the type of output structure and the type of the comparison. This brings us to the full
header of merge:
template<typename InputIterator1,
typename InputIterator2,
typename OutputIterator,
typename Compare> inline
OutputIterator merge( InputIterator1 first1,
InputIterator1 last1,
InputIterator2 first2,
InputIterator2 last2,
OutputIterator result,
Compare comp)
It is possible to call merge without supplying a comparison function object. In this
case, operator< is used by default. Here is a short program that uses merge, taken from
the Standard Template Library Programmer’s Guide:
int main()
{
int A1[ ] = { 1, 3, 5, 7 };
int A2[ ] = { 2, 4, 6, 8 };
const int N1 = sizeof(A1) / sizeof(int);
const int N2 = sizeof(A2) / sizeof(int);
merge(A1, A1 + N1, A2, A2 + N2,
ostream iterator<int>(cout, ” ”));
// The output is ”1 2 3 4 5 6 7 8”
}
9.5 Chapter Summary
As you can see from this sample code, even though the concepts and design of STL
are subtle and complex, using STL can be very easy and direct once the concepts are
understood.
9.5 CHAPTER SUMMARY
In this chapter, we studied some of the ways that programs can be divided into
meaningful parts and the way that programming languages support these divisions.
The main topics were structured programming, support for abstraction, and modules.
Two examples were used to investigate module systems and generic programming:
the standard ML module system and the C++ Standard Template Library.
Structured Programming
Historically, interest in program organization grew out of the movement for structured programming. In the late 1960s, Dijkstra and others advocated a top-down
approach. Although most programs of that day were small, currently many of the
early ideas continue to be influential. Most important, we are still concerned with
the decomposition of a complex programming task into smaller subproblems that
can be solved independently.
Top-down programming has its limitations. Top-down design is difficult when
the interfaces and data structures used to communicate between modules must be
changed. In addition, the concept of top-down programming does not provide any
help when programmers discover that the initial design must be modified to account
for phenomena that were not fully appreciated in the initial design. As other design methods have emerged, the principle of decomposing a design problem into
separable parts remains central to managing large software development projects.
Two important concepts in modular program development are interfaces and
specifications:
Interface:
A description of the parts of a component that are visible to other program
components.
Specification:
A description of the behavior of a component, as observable through its
interface.
When a program is designed modularly, it should be possible to change the internal
structure of any component as long as the behavior visible through the interface
remains the same.
Language Support for Abstraction
Procedural abstraction and data abstraction are programming language concepts that
allow sequences of statements, or data structures and code to manipulate them, to be
grouped into meaningful program units. Here are three features of data-abstraction
mechanisms:
Identifying the interface of the data structure. The interface of a data abstraction
consists of the operations on the data structure and their arguments and return
results.
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Providing information hiding by separating implementation decisions from parts
of the program that use the data structure.
Allowing the data structure to be used in different ways by many different programs.
In typed programming languages, the typing rules associated with abstract data types
guarantee representation independence, which means that changing the implementation of a data type does not affect the observable behavior of client programs.
Another principle associated with abstract data types is data-type induction, which
may be used to reason about the behavior of abstract data types or their equivalence.
Modules and Generic Programming
A module is a programming language construct that allows a number of declarations to be grouped together. Early forms of modules provided minimal information
hiding. However, good module mechanisms allow a programmer to control the visibility of items declared in a module. In addition, parameterized modules make it
possible to instantiate a set of declarations in different ways for different purposes.
The three main parts of the standard ML module system are structures, signatures,
and functors. An ML structure is a module, which is a collection of type, value,
and structure declarations. Signatures are module interfaces that behave as a form
of “type” for a structure. Functors are parameterized modules, requiring one or
more types or structures for producing a structure. We looked at a set of structures
that provide points, circles, and rectangles, and a functor Geom that combines circles
and rectangles into a geometry structure. One interesting aspect of this example is
that the circle and the rectangle structures that are combined must have the same
point representation. In standard ML, this is enforced by a sharing constraint in the
parameter list of the Geom functor.
The C++ Standard Template Library is a large program library, adopted as part
of the C++ standard, that provides container classes, such as lists, vectors, sets, and
maps. The main concepts in STL are Containers, Iterators, Algorithms, Adapters,
Function objects, and Allocators. We looked at lists, as an example container, and a
merge function that combines two ranges from sorted collections into a single range.
In STL, a range is represented as a pair of iterators, indicating a starting and ending
“address” inside a collection.
Polymorphic functions such as merge are represented as function templates, instantiated and optimized before program execution. Function objects, which we did
not study, are a form of closure used to pass functions to generic STL algorithms. A
function object is passed as two separate arguments, a template argument carrying
the code and a run-time argument carrying the state of the function. The type system
is used to make sure that the code and the state match.
The concept explored in this chapter can be used in the design of general-purpose
programming languages or special-purpose languages developed to solve specialized
programming problems (such as specifying a chemical simulation or specifying the
security policy of a large organization). Because you are much more likely to design
a special-purpose input language in your career as a practicing computer scientist
than you are to design a general-purpose object-oriented language, the non-objectoriented methods explained here may be useful to you in the future.
Exercises
EXERCISES
9.1 Efficiency vs. Modularity
The text describes the following form of heapsort:
function heapsort1(n:int, A:array[1..n])
begin
s := empty;
for i := 1 to n do
s := insert(A[i], s);
for i := n downto 1 do (A[i],s) := deletemax(s);
end
where heaps (also known as priority queues) have the following signature
empty : heap
insert : elt * heap -> heap
deletemax: heap -> elt * heap
and specification:
Each heap has an associated multiset of elements. There is an ordering < on the
elements that may be placed in a heap.
empty has no elements.
insert(elt, heap) returns a heap whose elements are elt plus the elements of heap.
deletemax(heap) returns an element of heap that is –> all other elements of heap,
together with a heap obtained when this element is removed. If heap is empty,
deletemax(heap) raises an exception.
This problem asks you to reformulate the signature of heaps and the sorting algorithm
so that the algorithm is more efficient, without destroying the separation between the
sorting algorithm and the implementation of heaps. You may assume in this problem
that we use pass-by-reference everywhere.
The standard nonmodular heapsort algorithm uses a clever way of representing a
binary tree by an array. Specifically, we can think of an array A of length n as a tree
by regarding A[1] as the root of the tree and elements A[2*k] and A[2*k+1] as the left
and the right children of A[k]. One efficient aspect of this representation is that the
links between tree nodes do not need do be stored; we only need memory locations
for the data stored at the tree nodes.
We say a binary tree is a heap if the value of the root of any subtree is the maximum
of all the node values in that subtree. For example, the following tree is a heap and
can be represented by the array next to it:
26
✁
✁
18
❆
❆
13
✁ ❆
✁ ❆
✁
❆ ✁
❆
15
11 6
9
The standard heapsort algorithm has the form
function heapsort2(n:int, A:array[1..n])
begin
variable heap size:int := n;
✚✚❧
❧
26 18 13 15 11 6 9
❛❛✟
❍
✟✧✧
❍❍✧
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build heap(n,A);
for i := n downto 2 do
swap(A[1], A[i]);
heap size := heap size - 1;
heapify(heap size,A);
end
where the procedure build heap(n,A) reorders the elements of array A so that they
form a heap (binary search tree) and procedure heapify(k,A) restores array elements
A[1], . . . , A[k] to heap form, assuming that only A[1] was out of place.
(a) Assume that procedures insert, deletemax, and heapify all take time O(log n),
i.e., time proportional to the logarithm of the number of elements in the heap,
and that build heap takes time O(n), i.e., time linear in the number of elements
in the heap.∗ Compare the time and space requirements of the two algorithms,
heapsort1 and heapsort2, and explain what circumstances might make you want
to choose one over the other.
(b) Suppose we change the signature of heaps to
make heap : array * int -> heap
deletemax : heap -> elt
where deletemax may have a side effect on the heap and there is no longer empty
or insert.
In addition to the specification of a return value, which is needed for a normal
function, a function f with side effects should specify the before and after values
of any variables that change, using the following format:
f: If xbefore is . . . then xafter is . . .
The idea here is that the behavior of a function that may change the values of its
arguments can be specified by describing the relationship between values before
the call and values after the call. This “if . . . then . . . ” form also allows you to
state any preconditions you might need, such as that the array A has at least n
elements.
Write a specification for this modified version of heaps.
(c) Explain in words (or some kind of pseudocode if you prefer) how you might
implement the modified form of heap with imperative operations described in
part(b). You may assume that procedures such as build heap, heapify, and swap
are available. Assume that you will use your heaps for some form of heapsort.
Try to make your operations efficient, at least for this application if not for all
uses of heaps in general.
[Hint: You will need to specify a representation for heaps and an implementation
of each function. Try representing a heap by a pair i, A, where A is an array
and 0 < i ≤ length(A).]
(d) Write a heapsort algorithm (heapsort3) using the modified form of heap with
imperative operations described in part (b). How will the time and the space
compare with the other two algorithms?
∗
Function build heap works by a form of iterated insertion. This might require O(n log n), but analysis
of the actual code for heapify allows us to show that it takes O(n) time. If this interests or puzzles you,
see Introduction to Algorithms, by Cormen, Leiserson, and Rivest (MIT Press, 1990), Section 7.3.
Exercises
(e) In all likelihood, your algorithm in part (d) (heapsort3) is not as time and space
efficient as the standard heapsort algorithm (heapsort2). See if you can think of
some way of modifying the definition or implementation of heaps that would let
you preserve modularity and meet the optimum efficiency for this algorithm, or
explain some of the obstacles for achieving this goal. (Alternatively, argue that
your algorithm is as efficient as the standard heapsort.)
9.2 Equivalence of Abstract Data Types
Explain why the following two implementations of a point abstract data type are
equivalent. More explicitly, explain why any program using the first would give exactly the same result (except possibly for differences in the speed of computation
and any consequences of round-off error) as the other.
Appeal to the principle of data-type induction, or related ideas, from this chapter.
You need not give a detailed formal inductive proof; a simple informal argument
referring to the principles discussed in class will be sufficient. You may explain what
needs to be calculated or what conditions need to be checked without doing the
calculations. The point of this problem is not to review trigonometry. An answer
consisting of three to five sentences should suffice.
(* —————– Trig function for arctan(y/x) —————- *)
fun atan(x,y) =
let val pi = 3.14159265358979323844
in
if (x > 0.0) then arctan(y / x)
else (if (y > 0.0) then (arctan(y / x) + pi)
else (arctan(y / x) - pi))
handle Div ⇒ (if (y > 0.0) then pi/2.0 else ∼pi/2.0)
end;
(* ——————– Two point abstract types —————— *)
abstype point = Pt of (real ref)*(real ref) (* rectilinear coordinates *)
with fun mk Point(x,y) = Pt(ref x, ref y)
and x coord (Pt(x, y)) = !x
and y coord (Pt(x, y)) = !y
and direction (Pt(x, y)) = atan(!x, !y)
and distance (Pt(x, y)) = sqrt(!x * !x + !y * !y)
and move (Pt (x, y), dx, dy) = (x := !x + dx; y := !y + dy)
end;
abstype point = Pt of (real ref)* (real ref) (* polar coordinates *)
with fun mk Point(x,y) = Pt(ref (sqrt(x*x + y*y)), ref (atan(x,y)))
and x coord (Pt(r, t)) = !r * cos(!t)
and y coord (Pt(r, t)) = !r * sin(!t)
and direction (Pt(r, t)) = !t
and distance (Pt(r, t)) = !r
and move (Pt(r, t), dx, dy) =
let val x = !r * cos(!t) + dx
and y = !r * sin(!t) + dy
in r := sqrt(x*x + y*y); t := atan(x,y) end
end;
The types of the functions given in either declaration are listed in the following
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output from the ML compiler:
type point
val mk Point = fn : real * real -> point
val x coord = fn : point -> real
val y coord = fn : point -> real
val direction = fn : point -> real
val distance = fn : point -> real
val move = fn : point * real * real -> unit
9.3 Equivalence of Closures
Explain why the following two functions that return a form of point “objects,” represented as ML closures of the same type, are equivalent. More explicitly, explain why
any program using the first would give exactly the same result (except possibly for
differences in the speed of computation and any consequences of round-off error)
as the other.
We did not cover any induction or equivalence principle for closures, but you might
think about whether you can use the same sort of reasoning you used for abstract
data types in Problem 9.2.
(* ————— Two point objects (as closures) ————— *)
fun mk point(xval,yval) =
let val x = ref xval and y = ref yval
in {
x coord = fn () ⇒ !x,
y coord = fn () ⇒ !y,
direction = fn () ⇒ atan(!x, !y),
distance = fn () ⇒ sqrt(!x * !x + !y * !y),
move = fn (dx, dy) ⇒ (x := !x + dx; y := !y + dy)
}
end;
fun mk point(x,y) =
let val r = ref (sqrt(x*x + y*y)) and t = ref (atan(x,y))
in {
x coord = fn () ⇒ !r * cos(!t),
y coord = fn () ⇒ !r * sin(!t),
direction = fn () ⇒ !t,
distance = fn () ⇒ !r,
move = fn (dx, dy) ⇒
let val x = !r * cos(!t) + dx
and y = !r * sin(!t) + dy
in r := sqrt(x*x + y*y); t := atan(x,y) end
}
end;
9.4 Modularity of Concrete Data Types
Given a grammar for a language, we can define ML data types for parse trees of
expressions in that language. Consider a grammar for expressions involving binary
(base 2) numbers and the left shift operator (<<) from C. We will use 0b and 1b to
Exercises
distinguish the syntax of bits from the numerals 0 and 1.
B ::= 0b | 1b
N ::= B | N B
E ::= N | E << E
In ML, we can create one data type for each nonterminal in the grammar:
datatype B = zero | one
datatype N = bit of B | many of N * B
datatype E = num of N | leftshift of E * E
A parse tree for an expression in our language can be written in ML with the ap1b 1b would be written
propriate constructors. For example, the expression 0b 1b
leftshift(many(num(bit(zero), one)), many(num(bit(one), one))).
We may have a denotational semantics for expressions in this language. For example, a function to evaluate binary expressions could be written as
eval [[0b ]] = 0,
eval [[1b ]] = 1,
eval [[nb]] = 2 · eval [[n]] + eval [[b]],
eval [[e1
e2 ]] = eval [[e1 ]] · 2eval[[e2 ]] .
In ML, we would have to write one function per data type:
fun evalB(zero) = 0
| evalB(one) = 1
fun evalN(bit(b)) = evalB(b)
| evalN(many(n,b)) = 2 * evalN(n) + evalB(b)
(a) Write evalE, which has type E → int. Assume you can use the power function in
ML, e.g. power(2,3) will return 8.
If you want to try your program in SML, you may define power by
fun power(b,0) = 1
| power(b,e) = b*power(b,e-1)
(b) Discuss what has to be changed to add an operator (such as >>) to expressions.
Do not worry about how >> actually works.
(c) Discuss what has to be changed to add a new denotational semantics function
(such as odd, a function that takes a binary expression and returns true if there
are an odd number of 1b bits in it). Do not worry about how odd actually works.
(d) If Alice modified the program to add a function bitcount and Bob modified it
to add a function odd, how hard is it to combine the two modifications into the
same program? What has to be changed? Explain.
(e) If instead, Alice modified the program to add an operator >> and at the same time
Bob modified the program to add the odd function, how hard is it to combine the
two modifications into the same program? What has to be changed? Explain.
(f) If instead, Alice modified the program to add an operator >> and Bob modified the program to add an operator xor, how hard is it to combine the two
modifications into the same program? What has to be changed? Explain.
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In your discussion of changes, assume that the parts of the program can be classified
as one of four kinds:
data-type declarations for binary expressions
functions that perform pattern matching on binary expressions
functions that use binary expressions but do not perform pattern matching
functions that do not use binary expressions
For each kind of program component, you should discuss whether it has to be
changed, what has to be changed, and why. If it does not have to be changed, explain
why it can be left unchanged.
9.5 Templates and Polymorphism
ML and C++ both have mechanisms for creating a generic stack implementation that
can be used for stacks with any type of element. In ML, polymorphic stacks could
be written as
datatype ’a stack = Empty | Push of ’a * ’a stack
fun top(Empty) = raise EmptyStack
| top(Push(x,s)) = x
fun pop(Empty) = raise EmptyStack
| pop(Push(x,s)) = x
To achieve a similar effect in C++, we could write a template for stack objects of the
following form:
template <typename A> class node {
public:
node(A v,node<A>* n) {val=v; next=n;}
A val;
node<A>* next;
};
template <typename A> class stack {
node<A>* first;
public:
stack () { first=0; }
void push(A x) { node<A>* n = new node<A>(x,first); first=n; }
void pop() { node<A>* n=first; first=first->next; delete n; }
A top() { return(first->val); }
};
Assume we are writing a program that uses five or six different types of stacks.
(a) For which language will the compiler generate a larger amount of code for stack
operations? Why?
(b) For which language will the compiler generate more efficient run-time representations of stacks? Why?
10
Concepts in Object-Oriented Languages
Over the past 30 years, object-oriented programming has become a prominent software design and implementation strategy. The topics covered in this chapter are
object-oriented design, four key concepts in object-oriented languages, and the way
these language concepts support object-oriented design and implementation.
An object consists of a set of operations on some hidden data. An important
characteristic of objects is that they provide a uniform way of encapsulating almost
any combination of data and functionality. An object can be as small as a single
integer or as large as a file system or database. Regardless of its size, all interactions
with an object occur by means of simple operations that are called messages or
member-function calls.
If you look in magazines or research journals, you will find the adjective objectoriented applied to a variety of languages. As object orientation has become more
popular and gained wider commercial acceptance, advocates of specific languages
have decided that their favorite language is now object oriented. This has created
some amount of confusion about the meaning of object oriented. In this book, we are
interested in making meaningful distinctions between different language features and
understanding how specific features support different kinds of programming. Therefore, the term object-oriented language is used to refer to programming languages
that have objects and the four features highlighted in this chapter: dynamic lookup,
abstraction, subtyping, and inheritance.
10.1 OBJECT-ORIENTED DESIGN
Object-oriented design involves identifying important concepts and using objects
to structure the way that these concepts are embodied in a software system. The
following list of steps is taken from one overview of object-oriented design, written
by object-oriented design proponent Grady Booch (Object-Oriented Design with
Applications, Benjamin/Cummings, 1991):
Identify the objects at a given level of abstraction.
Identify the semantics (intended behavior) of these objects.
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Identify the relationships among the objects.
Implement the objects.
Object-oriented design is an iterative process based on associating objects with components or concepts in a system. The process is iterative because typically we implement an object by using a number of subobjects, just as we typically implement a procedure by calling a number of finer-grained procedures. Therefore, after the important objects in a system are identified and implemented at one level of abstraction, the
next iteration will involve identifying additional objects and implementing them. The
“relationships among objects” mentioned here might be relationships between their
interfaces or relationships between their implementations. Modern object-oriented
languages provide mechanisms for using relationships between interfaces and relationships between implementations in the design and implementation process.
The data structures used in the early examples of top-down programming (see
Section 9.1) were very simple and remained invariant under successive refinements
of the program. When refinement involves replacing a procedure with more detailed
procedures, older forms of structured programming languages such as Algol, Pascal,
and C are adequate. For more complex tasks, however, both the procedures and the
data structures of a program need to be refined together. Because objects are a combination of functions and data, object-oriented languages support the joint refinement
of functions and data more effectively than do procedure-oriented languages.
10.2 FOUR BASIC CONCEPTS IN OBJECT-ORIENTED LANGUAGES
All object-oriented languages have some form of object. As mentioned in the preceding section, an object consists of functions and data, accessible only through a specific interface. In common object-oriented languages, including Smalltalk, Modula-3,
C++, and Java, the implementation of an object is determined by its class. In these
languages, we create objects by creating an instance of their classes.
The function parts of an object are called methods or member functions, and the
data parts of an object are called instance variables, fields, or data members.
Programming languages with objects and classes typically provide dynamic
lookup, abstraction, subtyping, and inheritance. These are the four main language
concepts for object-oriented programming. They may be summarized in the following
manner:
Dynamic lookup means that when a message is sent to an object, the function
code (or method) to be executed is determined by the way that the object is
implemented, not some static property of the pointer or variable used to name
the object. In other words, the object “chooses” how to respond to a message,
and different objects may respond to the same message in different ways.
Abstraction means that implementation details are hidden inside a program unit
with a specific interface. For objects, the interface usually consists of a set of public
functions (or public methods) that manipulate hidden data.
Subtyping means that if some object a has all of the functionality of another object
b, then we may use a in any context expecting b.
Inheritance is the ability to reuse the definition of one kind of object to define
another kind of object.
10.2 Four Basic Concepts in Object-Oriented Languages
These terms are defined and these features are explored in more detail in the following subsections.
There are several forms of object-oriented languages that are not covered directly in this book. One form is the delegation-based language. Two delegation-based
languages are Dylan, originally designed to program Apple Newton personal digital
assistants, and Self, a general-purpose language evolving out of research on implementation of object-oriented languages. In delegation-based languages, objects
are defined directly from other objects when new methods are added by means
of method addition and old methods are replaced by means of method override.
Although delegation-based languages do not have classes, they do have the four
essential characteristics required for object-oriented languages.
10.2.1 Dynamic Lookup
In any object-oriented language, there is some way to invoke the operations associated with an object. A general syntax for invoking an operation on an object, possibly
with additional arguments, is
object → operation (arguments)
In Smalltalk, this is called “sending a message to an object,” whereas in C++ it
is called “calling a member function of an object.” To avoid switching back and
forth between different choices of terminology, we will use the Smalltalk terminology for the remainder of this section. In Smalltalk terminology, a message consists of an operation name and set of additional arguments. When a message is sent
to an object, the object responds to the message by executing a function called a
method.
Dynamic lookup means that a method is selected dynamically, at run time, according to the implementation of the object that receives a message. The important
property of dynamic lookup is that different objects may implement the same operation differently. For example, the statement
x → add(y)
sends the message add(y) to the object x. If x is an integer, then the method (code
implementing this operation) may add integer y to x. If x is a set, then the add
method may insert y into the set x. These operations have different effects and are
implemented differently. However, a single line of code x → add(y) inside a loop
could cause integer addition the first time it is executed and set insertion the second
time if the value of the variable x changes from an integer to a set between one pass
through the loop and another.
Dynamic lookup is sometimes confused with overloading, which is a mechanism
based on static types of operands. However, the two are very different, as we will
see.
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Dynamic lookup is a very useful language feature and an important part of
object-oriented programming. Consider, for example, a simple graphics program that
manipulates pictures containing shapes such as squares, circles, and triangles. Each
square object may contain a draw method with code to draw a square, each circle
a draw method that contains code to draw a circle, and so on. When the program
wants to display a given picture, sending a draw message to each shape in the picture
can do this. The part of the program that sends the draw message does not have to
know which kind of shape will receive the message. Instead, each shape receiving
a draw message will know how to draw that shape. This makes sense because the
implementer of a specific shape is in the best position to figure out how to draw that
kind of shape.
We can understand some aspects of dynamic lookup and scoping by using a brief
comparison with abstract data types. Using an abstract data-type mechanism, we
might define matrices as follows:
abstype matrix = . . .
with
create(. . . ) = . . .
update(m, i, j, x) = . . . set m(i, j) = x. . .
add(m1, m2) = . . .
...
end;
A characteristic of this implementation of matrices is that the add function takes two
matrices as arguments, with call of the form
add(x, y)
The declaration of type matrix and associated operations has a specific scope. Within
this scope, add refers specifically to the function declared for matrices. Therefore, in
an expression add(x,y), both x and y must be matrices. If add were defined for complex
numbers in some outer scope, then either the inner declaration hides the outer one
or the language must provide some static overloading mechanism.
With objects in a class-based language, we might instead declare matrices as
follows:
class matrix
...
(representation)
update(i, j, x) = . . . set (i, j) of *this* matrix . . .
add(m) = . . . add m to *this* matrix . . .
end
10.2 Four Basic Concepts in Object-Oriented Languages
The add method of a matrix requires one matrix as an argument. The method might
be invoked by an expression such as
x → add(y)
In this expression, the operation add appears to have only one argument, the matrix
that is to be added to the matrix x receiving the message add(y).
There are several ways that dynamic lookup may be implemented. In one implementation, each object contains a pointer to a method lookup table that associates
a method body with each message defined for that object. When a message is sent
to an object at run time, the corresponding method is retrieved from that object’s
method table. Because different objects may have different method lookup tables,
sending the same message to different objects may result in the execution of different
code.
It is also possible to think of dynamic lookup as a run-time form of overloading.
More specifically, we can think of each method name as the name of an overloaded
function. When a message m is sent to an object named by variable x, then x is
treated as the first argument of an overloaded function named m. Unlike traditional
overloading, though, the code to execute must be chosen according to the run-time
value of x. In contrast, traditional overloading uses the static type of a variable x to
decide which code to use.
Dynamic lookup is an important part of Smalltalk, C++, and Java. In Smalltalk
and Java, method lookup is done dynamically by default. In C++, only virtual member
functions are selected dynamically.
There is a family of object-oriented languages that is based on the “run-time
overloading” view of dynamic lookup. The most prominent design of this form is the
common Lisp object system, sometimes referred to by the acronym CLOS. In CLOS,
an expression corresponding to
x → f(y,z)
is treated as a call f(x, y, z) to an overloaded function with three arguments. Although ordinary dynamic lookup would select a function body for f based on the
implementation of x alone, CLOS method lookup uses all three arguments. This
feature is sometimes called multiple dispatch to distinguish it from more conventional single-dispatch languages in which only one of the arguments of a function
(the object receiving the message) determines the function body that is called at run
time.
Multiple dispatch is useful for implementing operations such as equality, in which
the appropriate comparisons to use depend on the dynamic type of both the receiver
object and the argument object. Although multiple dispatch is in some ways more
general than the single dispatch found in Smalltalk, C++, and Java, there is also some
loss of encapsulation. Specifically, to define a function on different kinds of arguments, that function must have access to the internal data of each function argument.
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Because single-dispatch languages are the object-oriented mainstream, we focus
on single-dispatch languages in this book.
10.2.2 Abstraction
As discussed in Chapter 9, abstraction involves restricting access to a program component according to its specified interface. In most modern object-oriented languages,
access to an object is restricted to a set of public operations that are chosen by the
designer and implementer of the object. For example, in a program that manipulates
geometric shapes, each shape could be represented by an object. We could implement an object representing a circle by storing the center and radius of the circle. The
designer of circle objects could choose to make a function that changes the center of
the circle part of the interface or choose not to put such a function in the interface.
If there is no public function for changing the center of a circle, then no client code
could change the center of a circle, as client code can manipulate objects only through
their interface.
Abstraction based on objects is similar in many ways to abstraction based on abstract data types: Objects and abstract data types both combine functions and data,
and abstraction in both cases involves distinguishing between a public interface and
private implementation. However, other features of object-oriented languages make
abstraction in object-oriented languages more flexible than abstraction in which abstract data types are used. One way of understanding the flexibility of object-oriented
languages is by looking at the way that relationships between similar abstractions
can be used to advantage.
Consider the following two abstract data types, written in ML syntax. The first
is an abstract data type of queues, the second an abstract data type of priority
queues. For simplicity, both queues and priority queues are defined for only integer
data:
exception Empty;
abstype queue = Q of int list
with
fun mk Queue() = Q(nil)
and is empty(Q(l)) = l=nil
and add(x,Q(l)) = Q(l @ [x])
and first (Q(nil)) = raise Empty | first (Q(x::l)) = x
and rest (Q(nil)) = raise Empty | rest (Q(x::l)) = Q(l)
and length (Q(nil)) = 0
| length (Q(x::l))= 1 + length (Q(l))
end;
In this abstract data type, a queue is represented by a list. The add operation uses the
ML append operator @ to add a new element to the end of a list. The first and the
rest operations read and remove an element from the front of a list. Because client
code cannot manipulate the representation of a queue directly, the implementation
maintains an invariant: List elements appear in first-in/first-out order, regardless of
how queues are used in client programs.
10.2 Four Basic Concepts in Object-Oriented Languages
A priority queue is similar to a queue, except that elements are removed according
to some preference ordering. More specifically, some priority is given to elements,
and the first and the remove operations read and remove the queue elements that
have highest priority:
abstype pqueue = Q of int list
with
fun mk PQueue() = Q(nil)
and is empty(Q(l)) = l=nil
and add(x,Q(l)) =
let fun insert(x,nil) = [x:int]
| insert(x,y::l) = if x<y then x::y::l else y::insert(x,l)
in Q(insert(x,l)) end
and first (Q(nil)) = raise Empty | first (Q(x::l)) = x
and rest (Q(nil)) = raise Empty | rest (Q(x::l)) = Q(l)
and length (Q(nil)) = 0
| length (Q(x::l))= 1 + length (Q(l))
end;
The interface of an abstract data type is the list of public functions and their types.
Queues and priority queues have the same interface: Both have the same number
of operations, the operations have the same names, and each operation has the
same type in both cases, except for the difference between the type names pqueue
and queue. The point of this example is that, although the interfaces of queues and
priority queues are identical, this correspondence is not used in traditional languages
with abstract data types. In contrast, if we implement queues and priority queues in
an object-oriented language, then we can take advantage of the similarity between
the interfaces of these data structures.
A drawback to the kind of abstract data types used in ML and similar languages
becomes apparent when we consider a program that uses both queues and priority
queues. For example, suppose that we build a system with several wait queues, such as
a hospital. In a hospital billing department, customers are served on a first-come, firstserve basis. However, in a hospital emergency room, patients are treated in order of
the severity of their injuries or ailments. In a hospital program, we might like to treat
priority queues and ordinary queues uniformly. For example, we might wish to count
the total number of people waiting in any line in the hospital. To write this code, we
would like to have a list of all the queues (both priority and ordinary) in the hospital
and sum the lengths of all the queues in the list. However, if the length operation is
different for queues and priority queues, we have to decide whether to call q length
or pq length, even though the correct operation is uniquely determined by the data.
This shortcoming of ordinary abstract data types is eliminated in object-oriented
programming languages by a combination of subtyping (see Subsection 10.2.3) and
dynamic lookup.
The implementation of priority queues shows us another drawback of traditional
abstract data types. Although the priority queue add function is different from the
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queue add function, the other five functions have identical implementations. In an
object-oriented language, we may use inheritance (see Subsection 10.2.4 ) to define
priority queue from queue (or vice versa), giving only the new add function and reusing
the other functions.
10.2.3 Subtyping
Subtyping is a relation on types that allows values of one type to be used in place
of values of another. Although it is simplest to describe subtyping in the context of
statically typed programming languages, there is also an implicit subtyping relation
in untyped languages. We will discuss subtyping assuming that we are in a typed
language, in most of this section, turning to untyped languages in the final paragraphs.
In most typed languages, the application of a function f to an argument x requires
some relation between the type of f and the type of x. The most common case is that
f must be a function from type A to type B, for some A and B, and x must be a variable
or expression of type A. We can think of this type comparison as a comparison for
equality: The type checker finds a type A → B for function f and a type C for x, and
checks that A=C.
In languages with subtyping, there is a subtype relation on types. The basic
principle associated with subtyping is substitutivity: If A is a subtype of B, then any
expression of type A may be used without type error in any context that requires an
expression of type B. We write A <: B to indicate that A is a subtype of B.
With subtyping, the subtype relation is used in place of equality in type checking.
Specifically, to type the application of f to argument x, the type checker finds a type
A → B for function f and a type C for x, and checks that C is a subtype of A.
The primary advantage of subtyping is that it permits uniform operations over
various types of data. For example, subtyping makes it possible to have heterogeneous
data structures that contain objects that belong to different subtypes of some common type. Consider as an example a queue containing various bank accounts to be
balanced. These accounts could be savings accounts, checking accounts, investment
accounts, and so on. However, if each type of account is a subtype of bank account,
then a queue of elements of type bank account can contain all types of accounts.
Subtyping in an object-oriented language also allows functionality to be added
without modifying general parts of a system. If objects of a type B lack some desired
behavior, then we may wish to replace objects of type B with objects of another
type A that have the desired behavior. In many cases, the type A will be a subtype of B.
By designing the language so that substitutivity is allowed, one may add functionality
in this way without any other modification to the original program.
This use of subtyping helps in building a series of prototypes of an airport scheduling system. In an early prototype, one would define a class airplane with methods such
as position, orientation, and acceleration that would allow a control-tower object to
affect the approach of an airplane. In a later prototype, it is likely that different types
of airplanes would be modeled. If we add classes for Boeing 757s and Beechcrafts,
these would be subtypes of airplane, containing extra methods and fields reflecting
features specific to these aircraft. By virtue of the subtyping relation, Beechcrafts
and Boeings are subtypes of airplane, and the general control algorithms that apply
to all airplanes can be used for Beechcrafts and Boeings without modification.
10.2 Four Basic Concepts in Object-Oriented Languages
10.2.4 Inheritance
Inheritance is a language feature that allows new objects to be defined from existing
ones. We discuss the form of inheritance that appears in most class-based objectoriented languages by using a neutral notation. The following class A defines objects
with private data v and public methods f and g. We define the class B by inheriting
the declarations of A, redefining g, and adding a private variable w:
class A =
private
val v = . . .
public
fun f(x) = . . . g(. . . ) . . .
fun g(y) = . . . original definition . . .
end;
class B = extend A with
private
val w = . . .
public
fun g(y) = . . . new definition . . .
end;
In principle, inheritance can be implemented by code duplication. For every object or
class of objects defined by inheritance, there is a corresponding definition that does
not use inheritance; it is obtained by expansion of the definition so that inherited code
is duplicated. The importance of inheritance is that it saves the effort of duplicating
(or reading duplicated) code and that, when one class is implemented by inheriting
from another, changes to one affect the other. This has a significant impact on code
maintenance and modification.
A straightforward implementation of inheritance that avoids code duplication is
to build linked data structures of method lookup tables. More specifically, for each
class, a lookup table can contain the list of operations associated with the class. When
one class inherits from another, then the second class can contain a pointer to the
lookup table of the first. If a method is inherited, it can be found in the method
lookup table of the class in which it was originally defined. A scheme of this form is
used in the implementation of Smalltalk, as we will see in Subsection 11.5.3.
A significant optimization may be made in statically typed languages such as C++,
in which the set of possible messages to each object can be determined at compile
time. If lookup tables can be constructed so that all subclasses of a given class store
repeated pointers in the same relative positions, then the offset of a method within
a lookup table can be computed at compile time. This reduces the cost of method
lookup to a simple indirection without a search, followed by an ordinary function call.
We will look at this implementation of inheritance in more detail in Subsection 12.3.3.
Inheritance and Abstraction
In ordinary modules or abstract data types, there are two views of an abstraction:
the client view and the implementation view. With inheritance, there are three views
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of classes: the implementation view, the client view, and the inheritance view. The
inheritance view is the view of classes that inherit from a class. Because object definitions have two external clients, there are two interfaces to the outside: The public
interface lists what the general client may see, whereas the protected interface lists
what inheritors may see. (This terminology comes from C++.) In most languages, the
public interface is a subset of the protected one. In Smalltalk, these interfaces are
generated automatically: The public interface includes all the methods of an object,
whereas the protected interface is all methods and all instance variables (data). In
C++, the programmer explicitly declares which components of an object are public,
which are protected, and which are private and visible only in the class definition
itself. We will discuss this in more detail in Section 12.3.
10.2.5 Closures as Objects
The first characteristic of objects, dynamic lookup,is also provided by records (in
Pascal or ML terminology) or structs (in C terminology). In a language with closures, we can simulate objects by using records that have function components. If an
object has private data (or functions), then we can hide them by using static scoping.
This leads us to look at record closures as a simple first model of objects. It turns
out to be instructive to see how useful this notion of object is and where it falls
short.
To see the similarities, consider the following ML code:
exception Empty;
fun newStack(x) =
let val store = ref [x]
in
{push = fn(y) => store := y ::(!store),
pop = fn() => case !store of
nil = raise Empty
| (y::ys) = (store := ys; y)
}
end;
val myStack = newStack(0);
#push(myStack)(1);
#pop(myStack)( );
The notation #field name(record value) is ML notation for field selection. In Pascallike syntax, this expression would be written as record value.field name.
The function newStack returns a record with two function components, the first
called push, the second called pop. Because the fields of this record contain functions,
they are represented at run time as closures. The environment pointers for these
closures point to the activation record for the newStack function, which stores the local
data store. The initial value of store is a list containing only the initial element passed
as an argument to newStack. If you draw out the activation records and closures, you
will obtain a diagram that is very similar to the ones we will be drawing to represent
10.2 Four Basic Concepts in Object-Oriented Languages
objects. However, most object-oriented languages optimize the representation in one
or more ways.
Because closures and objects have essentially the same functionality, it is reasonable to wonder why we talk about “object-oriented” programming, instead of
“closure-oriented” programming. In other words, what do object-oriented programming languages have that languages like ML lack? The answer is subtyping and inheritance. If you try to translate an object-oriented program into a non-object-oriented
language, you will appreciate the language support for subtyping and inheritance.
10.2.6 Inheritance Is Not Subtyping
Perhaps the most common confusion surrounding object-oriented languages is the
difference between subtyping and inheritance. The simplest distinction between subtyping and inheritance is this: Subtyping is a relation on interfaces, inheritance is a
relation on implementations.
One reason subtyping and inheritance are often confused is that some class mechanisms combine the two. A typical example is C++, in which A will be recognized by
the compiler as a subtype of B only if B is a public base class of A. Combining subtyping and inheritance is an elective design decision, however; C++ could have been
designed differently without linking subtyping and public base classes in this way.
We may see that, in principle, subtyping and inheritance do not always go handin-hand by considering an example suggested by object-oriented researcher, Alan
Snyder. Suppose we are interested in writing a program that requires dequeues, stacks,
and queues. These are three similar kinds of data structures, with the following basic
characteristics:
Queues: Data structures with insert and delete operations, such that the first element inserted is the first one removed (first-in, first-out),
Stacks: Data structures with insert and delete operations, such that the first element
inserted is the last one removed (last-in, first-out),
Dequeues: Data structures with two insert and two delete operations. A dequeue,
or doubly ended queue, is essentially a list that allows insertion and deletion
from each end. If an element is inserted at one end, then it will be the first one
returned by a series of removes from that end and the last one returned by a
series of removes from the opposite end.
An important part of the relationship among stacks, queues, and dequeues is that a
dequeue can serve as both a stack and a queue. Specifically, suppose a dequeue d has
insert operations insert front and insert rear and delete operations delete front and
delete rear. If we use only insert front and delete rear, we have a queue. However, if
we use insert front and delete front, we have a stack.
One way to implement these three classes is first to implement dequeue and then
implement stack and queue by appropriately restricting (and perhaps renaming) the
operations of dequeue. For example, we may obtain stack from dequeue by limiting
access to those operations that add and remove elements from one end of a dequeue.
Similarly, we may obtain queue from dequeue by restricting access to those operations
that add elements at one end and remove them from the other. This method of defining stack and queue by inheriting from dequeue is possible in C++ through the use of
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private inheritance. (This is not a recommended style of implementation; this example is used simply to illustrate the differences between subtyping and inheritance.)
Although stack and queue may be implemented from dequeue, they are not subtypes of dequeue. Consider a function f that takes a dequeue d as an argument and
then adds an element to both ends of d. If stack or queue were a subtype of dequeue,
then function f should work equally well when given a stack s or a queue q. However, adding elements to both ends of either a stack or a queue is not legal; hence,
neither stack nor queue is a subtype of dequeue. In fact, the reverse is true. Dequeue
is a subtype of both stack and queue, as any operation valid for either a stack or a
queue would be a legal operation on a dequeue. Thus, inheritance and subtyping are
different relations in principle: it makes perfect sense to define stack and queue by
inheriting from dequeue, but dequeue is a subtype of stack and queue, not the other
way around.
A more detailed comparison of the two mechanisms appears in Section 11.7, in
which the inheritance and subtyping relationships among Smalltalk collection classes
are analyzed.
10.3 PROGRAM STRUCTURE
There are some systematic differences between the structure of function-oriented
(or procedure-oriented) programs and object-oriented programs. One of the main
differences is in the organization of functions and data. In a function-oriented program, data structures and functions are declared separately. If a function will be
applied to many types of data, then it is common to use some form of case or switch
statement within the function body. In an object-oriented program, functions are
associated with the data they are designed to manipulate. Using dynamic lookup,
the programming language implementation will select the correct function for each
kind of data. This basic difference between function-oriented and object-oriented
programs is illustrated by a comparison of Example 10.1 and Example 10.2. A longer
example illustrating this point, written in C and C++, appears in Appendix B.1.
In both Examples 10.1 and 10.2, we consider a hospital simulation. The data
in these examples represent doctors, nurses, and orderlies. The functions that will
be applied to these data include a function to display information about a hospital
employee and a function to set or determine the pay of an employee.
Example 10.1 Conventional Function-Oriented Organization
In a conventional function-oriented program, operations are grouped into function. If
we want a single function to display information about all types of hospital employees,
then we may use run-time tests to determine how to apply each operation to the given
data. In outline, codes for display and pay functions might look like this:
display(x) =
case type(x) of
Doctor : [“display Doctor” ]
Nurse : [“display Nurse” ]
Orderly : [“display Orderly” ]
end;
10.3 Program Structure
end;
pay(x) =
case type(x) of
Doctor : [“pay Doctor a lot” ]
Nurse : [“pay Nurse less” ]
Orderly : [“pay Orderly less than that” ]
end;
end;
Example 10.2 Object-Oriented Organization
In an object-oriented program, functions are grouped with the data they are designed
to manipulate. For the hospital example, the doctor, nurse, and orderly classes will
contain the code for the two functions. In outline, this produces the following program
organization:
class Doctor =
display = “Display Doctor”;
pay = “pay Doctor a lot ”;
end;
class Nurse =
display = “display Nurse”;
pay = “pay Nurse less”;
end;
class Orderly =
display = “display Orderly”;
pay = “pay Orderly less than that”;
end;
Comparison of Examples 10.1 and 10.2
The data and operations used in Examples 10.1 and 10.2 may be arranged into the
following matrix. In the conventional function-oriented organization, the code is
arranged by row into functions that work for all kinds of data. In the object-oriented
organization, the code is arranged by column, grouping each function case with the
data it is designed for.
Operation
Doctor
Nurse
Orderly
Display
Pay
Display Doctor
Pay Doctor
Display Nurse
Pay Nurse
Display Orderly
Pay Orderly
In the function-oriented organization, it is relatively easy to add a new operation, such
as PayBonus or Promote, but difficult to add a new kind of data, such as Administrator
or Intern. In the object-oriented organization, it is easy to add new data, such as
Administrator or Intern, but more cumbersome to add new operations such as PayBonus
or Promote because this involves changes to every class.
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10.4 DESIGN PATTERNS
The design pattern method is a popular approach to software design that has developed along with the rise in popularity of object-oriented programming. In basic terms, a design pattern is a general solution that has come from the repeated
addressing of similar problems. Design patterns are not solutions developed from
first principles or generic code that can simply be instantiated for a variety of purposes. Instead, a design pattern is a guideline or approach to solving a kind of problem that occurs in a number of specific forms. Solutions based on a design pattern
can be similar; applying a design pattern to a specific situation can require some
thought.
The concept of a design pattern can be used in any design discipline, such as
mechanical design or architecture. The work of architect Christopher Alexander is
often cited as an inspiration for software design patterns. Here is an architectural
example, excerpted from one of Alexander’s books (A Pattern Language: Towns,
Buildings, Construction, Oxford Univ. Press, 1977). This passage includes both a
description of the problem context and a solution developed as a result of experience:
Sitting Circle
. . . A group of chairs, a sofa and a chair, a pile of cushions – these are the most
obvious things in everybody’s life – and yet to make them work, so people become
animated and alive in them, is a very subtle business. Most seating arrangements are
sterile, people avoid them, nothing ever happens there. Others seem somehow to
gather life around them, to concentrate and liberate energy. What is the difference
between the two?
. . . Therefore, place each sitting space in a position which is protected, not cut by
paths or movements, roughly circular, made so that the room itself helps suggest
the circle – not too strongly – with paths and activities around it, so that people
naturally gravitate toward the chairs when they get into the mood to sit. Place the
chairs and cushions loosely in the circle, and have a few too many.
When programmers find that they have solved the same kind of problem over and
over again in slightly different ways but using essentially the same design ideas, they
may try to identify the general design pattern of their solutions. The popularity of this
process has led to the identification of a large number of software design patterns.
To quote pattern advocate Jim Coplien, a good pattern does the following:
It solves a problem: Patterns capture solutions, not just abstract principles or
strategies.
It is a proven concept: Patterns capture solutions with a track record, not theories
or speculation.
The solution isn’t obvious: Many problem-solving techniques (such as software
design paradigms or methods) try to derive solutions from first principles. The
best patterns generate a solution to a problem indirectly – a necessary approach
for the most difficult problems of design.
It describes a relationship: Patterns don’t just describe modules, but describe
deeper system structures and mechanisms.
10.4 Design Patterns
The pattern has a significant human component (minimize human intervention).
All software serves human comfort or quality of life; the best patterns explicitly
appeal to aesthetics and utility.
Beyond reading about general principles of design patterns, the best way to learn
about patterns is to study some examples and use patterns in your programming.
Here are a couple of examples. You can find many more in the books and web
pages devoted to design patterns. A widely used book is Design Patterns: Elements
of Reusable Object-Oriented Software by E. Gamma, R. Helm, R. Johnson, and
J. Vlissides (Addison-Wesley, 1994).
Example 10.3 Singleton Design Pattern
The singleton design pattern is a creational design pattern, meaning that it is a pattern
that is used for creating objects in a certain way. Here is a brief overview of the
singleton pattern, that uses the kind of subject headings that are commonly used in
books and other presentations of design patterns.
Motivation
The singleton pattern is useful in situations in which there should be a single instance
(object) of a class. This pattern gives a class direct control over how many instances
can be created. This is better than making the programmer responsible for creating
only one instance, as the restriction is built into the program.
Implementation
Only one class needs to be written to implement the singleton pattern. The class uses
encapsulation to keep the class constructor (the function that returns new objects
of the class) hidden from client code. The class has a public method that calls the
constructor only if an object of the class has not already been created. If an object
has been created, then the public function returns a pointer to this object and does
not create a new object.
Sample Code
Here is how a generic singleton might be written in C++. Readers who are not familiar
with C++ may wish to scan the explanation and return to this example after reading
Chapter 12. The interface to class Singleton provides a public method that lets client
code ask for an instance of the class:
class Singleton {
public:
static Singleton* instance(); // function that returns an instance
protected:
Singleton();
// constructor is not made public
private:
static Singleton* instance; // private pointer to single object
};
Here is the implementation. Initially, the private pointer instance is set to 0. In the
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implementation of public method instance(), a new object is created and assigned to
instance only if a previous call has not already created an object of this class:
Singleton* Singleton:: singleton = 0
Singleton* Singleton::instance() {
if ( instance = = 0){ instance = new Singleton; }
return instance;
}
Example 10.4 Façade
Façade is a structural object pattern, which means it is a pattern related to composing
objects into larger structures containing many objects.
Motivation
The façade pattern provides a single object for accessing a set of objects that have
been combined to form a structure. In effect, the façade provides a higher-level
interface to a collection of objects, making the collection easier to use.
Implementation
There is a façade class, defined for a set of classes that are used to make up a structure
“behind” the facade. In a typical use, a façade object has relatively little actual code,
passing most calls to objects in the structure behind the façade.
Example of Façade Pattern
Façade is a very common pattern when a task is accomplished by a combination of
the results of a number of subtasks. For example, a compiler might be constructed
by implementation of a lexical scanner, parser, semantic analyzer, and other phases
indicated in the figure in Subsection 4.1.1. If each phase is implemented as an object
with methods that perform its main functions, then the compiler itself will be a
façade object that takes a program as input and uses the separate objects that are
implementing each phase to compile the program. A user of the compiler may see
the interface presented by the compiler object. This is a more useful interface than
the more detailed interfaces to the constituent objects that are hidden behind the
façade.
10.5 CHAPTER SUMMARY
This chapter contains a short overview of object-oriented design and summarizes
the four basic concepts associated with object-oriented languages: dynamic lookup,
abstraction, subtyping, and inheritance.
Dynamic lookup means that when a message is sent to an object, the function
code (or method) that is executed is determined by the way that the object is
implemented. Different objects may respond to the same message in different
ways.
10.6 Looking Forward: Simula, Smalltalk, C++, Java
Abstraction means that implementation details are hidden inside a program unit
with a specific interface. The interface of an object is usually a set of public
functions (or public methods) that manipulate hidden data.
Subtyping means that if some object a has all of the functionality of another
object b, then we may use a in any context expecting b.
Inheritance is the ability to reuse the definition of one kind of object to define
another kind of object.
In conventional languages that implement closures and allow records to contain
functions, records provide a form dynamic lookup and abstraction. Subtyping and
inheritance, in the form needed to support object-oriented programming, are generally not found in conventional languages.
Many people confuse subtyping and inheritance. As the term is used in this book,
subtyping is a relation on types that allows values of one type to be used in place
of values of another. (In Section 11.7, Smalltalk is used to discuss subtyping in a
language that does not have a static type system.) As the term is used in this book,
inheritance allows new objects to be defined from existing ones. In class-based languages, inheritance allows the implementation of one class to be reused as part of
the implementation of another. The simplest way to keep subtyping and inheritance
straight is to remember this: Subtyping is a relation on interfaces and inheritance is a
relation on implementations.
In Section 10.3, the difference between the organizational structure of objectoriented programs and the organizational structure of conventional programs is summarized. In conventional languages, functions are designed to operate on many types
of data. In object-oriented programs, functions can be written to operate on a single
type of data, with dynamic lookup finding the right function at run time.
In Section 10.4, we looked at the basic idea behind design patterns and saw two
examples, singleton and façade. A design pattern is a general solution that has come
from the repeated addressing of similar problems. The design pattern method is a
popular approach to software design that has evolved along with object-oriented
programming.
10.6 LOOKING FORWARD: SIMULA, SMALLTALK, C++, JAVA
In the next three chapters, we will look at four object-oriented languages:
Simula, the first object-oriented language. The object model in Simula was based
on procedure activation records, with objects originally described as procedures
that return a pointer to their own activation record. There was no abstraction in
Simula 67, but a later version incorporated abstraction into the object system.
Simula was an important inspiration for C++.
Smalltalk, a dynamically typed object-oriented language. Many object-oriented
ideas originated or were popularized by the Smalltalk group, which built on Alan
Kay’s then-futuristic idea of the Dynabook. The Dynabook, which was never
built by this group, was intended to be a small portable computer capable of
running a user-friendly programming language. We will look at the Smalltalk
implementation of method lookup and later compare this with C++.
C++, a widely used statically typed object-oriented language. This language is
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designed for efficiency around the principle that programs that do not use a certain
feature should run as efficiently as programs written in a language without that
feature. A significant design constraint was backward compatibility with C.
Java, a modern language design in which security and portability are valued as
much as efficiency. Some interesting features are interfaces, which provide explicit
support for abstract base classes, and run-time class loading, intended for use in
a distributed environment.
Because there is not enough time to study all aspects of each language, we will
concentrate on a few important or distinctive features of each one. One general
theme in our investigation of these languages is the trade-off between language
features and implementation complexity.
Simula is primarily important as a historical language and for the way it illustrates
the relationship between objects and activation records. Of the remaining three languages, Smalltalk represents one extreme and C++ the other. Smalltalk is extremely
flexible and based on the notion that everything is an object. C++, on the other hand,
is defined to favor efficiency over conceptual simplicity. Although C++ provides objects, many features of C++ are inherited from C and are not based on objects. Java
is a compromise between Smalltalk and C++ in the sense that the flexibility of the
implementation and organization around objects are closer to Smalltalk than to C++.
Java also contains features not found in either of the other languages, such as dynamic
class loading and a typed intermediate language.
EXERCISES
10.1 Expression Objects
We can represent expressions given by the grammar
e
::=
num | e + e
by using objects from a class called expression. We begin with an “abstract class”
called expression. Although this class has no instances, it lists the operations common to all kinds of expressions. These are a predicate telling whether there are
subexpressions, the left and right subexpressions (if the expression is not atomic),
and a method computing the value of the expression:
class expression() =
private fields:
(* none appear in the interface *)
public methods:
atomic?() (* returns true if no subexpressions *)
lsub()
(* returns “left” subexpression if not atomic *)
rsub()
(* returns “right” subexpression if not atomic *)
value()
(* compute value of expression *)
end
Because the grammar gives two cases, we have two subclasses of expression, one for
numbers and one for sums:
class number(n) = extend expression() with
private fields:
Exercises
val num = n
public methods:
atomic?() = true
lsub
() = none (* not allowed to call this, *)
rsub () = none (* because atomic?() returns true *)
value () = num
end
class sum(e1, e2) = extend expression() with
private fields:
val left = e1
val right = e2
public methods:
atomic?() = false
lsub
() = left
rsub () = right
value () = ( left.value() ) + ( right.value() )
end
(a) Product Class: Extend this class hierarchy by writing a prod class to represent
product expressions of the form
e ::=
. . . | e ∗ e.
(b) Method Calls: Suppose we construct a compound expression by
val a = number(3);
val b = number(5);
val c = number(7);
val d = sum(a,b);
val e = prod(d,c);
and send the message value to e. Explain the sequence of calls that are used to
compute the value of this expression: e.value(). What value is returned?
(c) Unary Expressions: Extend this class hierarchy by writing a square class to
represent squaring expressions of the form
e
::=
. . . | e2 .
What changes will be required in the expression interface? What changes will be
required in subclasses of expression? What changes will be required in functions
that use expressions?∗ What changes will be required in functions that do not
use expressions? (Try to make as few changes as possible to the program.)
(d) Ternary Expressions: Extend this class hierarchy by writing a cond class to
represent conditionals† of the form
e
::=
. . . | e?e : e
What changes will be required if we wish to add this ternary operator? [As in
part (c), try to make as few changes as possible to the program.]
∗
†
Keep in mind that not all functions simply want to evaluate entire expressions. They may call the other
methods as well.
In C, conditional expressions a?b:c evaluate a and then return the value of b if a is nonzero or return
the value of c if a is zero.
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(e) N-Ary Expressions: Explain what kind of interface to expressions we would
need if we would like to support atomic, unary, binary, ternary and n−ary
operators without making further changes to the interface. In this part of the
problem, we are not concerned with minimizing the changes to the program;
instead, we are interested in minimizing the changes that may be needed in
the future.
10.2 Objects vs. Type Case
With object-oriented programming, classes and objects can be used to avoid “typecase” statements. Here is a program in which a form of case statement is used that
inspects a user-defined type tag to distinguish between different classes of shape
objects. This program would not statically type check in most typed languages
because the correspondence between the tag field of an object and the class of
the object is not statically guaranteed and visible to the type checker. However,
in an untyped language such as Smalltalk, a program like this could behave in a
computationally reasonable way:
enum shape tag {s point, s circle, s rectangle };
class point {
shape tag tag;
int x;
int y;
point (int xval, int yval)
{ x = xval; y = yval; tag = s point; }
int x coord () { return x; }
int y coord () { return y; }
void move (int dx, int dy) { x += dy; y += dy; }
};
class circle {
shape tag tag;
point c;
int r;
circle (point center, int radius)
{ c = center; r = radius; tag = s circle }
point center () { return c; }
int radius () { return radius; }
void move (int dx, int dy) { c.move (dx, dy); }
void stretch (int dr) { r += dr; }
};
class rectangle {
shape tag tag;
point tl;
point br;
rectangle (point topleft, point botright)
{ tl = topleft; br = botright; tag = s rectangle; }
point top left () { return tl; }
point bot right () { return br; }
void move (int dx, int dy) { tl.move (dx, dy); br.move (dx, dy); }
void stretch (int dx, int dy) { br.move (dx, dy); }
Exercises
};
/* Rotate shape 90 degrees. */
void rotate (void *shape) {
switch ((shape tag *) shape) {
case s point:
case s circle:
break;
case s rectangle:
{
rectangle *rect = (rectangle *) shape;
int d = ((rect->bot right ().x coord ()
- rect->top left ().x coord ()) (rect->top left ().y coord ()
- rect->bot right ().y coord ()));
rect->move (d, d);
rect->stretch (-2.0 * d, -2.0 * d);
}
}
}
(a) Rewrite this so that, instead of rotate being a function, each class has a rotate
method and the classes do not have a tag.
(b) Discuss, from the point of view of someone maintaining and modifying code,
the differences between adding a triangle class to the first version (as previously
written) and adding a triangle class to the second [produced in part (a) of this
question].
(c) Discuss the differences between changing the definition of rotate (say, from 90◦
to the left to 90◦ to the right) in the first and the second versions. Assume you
have added a triangle class so that there is more than one class with a nontrivial
rotate method.
10.3 Visitor Design Pattern
The extension and maintenance of an object hierarchy can be greatly simplified (or
greatly complicated) by design decisions made early in the life of the hierarchy. This
question explores various design possibilities for an object hierarchy representing
arithmetic expressions.
The designers of the hierarchy have already decided to structure it as
subsequently shown, with a base class Expression and derived classes IntegerExp,
AddExp, MultExp, and so on. They are now contemplating how to implement various
operations on Expressions, such as printing the expression in parenthesized form or
evaluating the expression. They are asking you, a freshly minted language expert, to
help.
The obvious way of implementing such operations is by adding a method to
each class for each operation. The Expression hierarchy would then look like:
class Expression
{
virtual void parenPrint();
virtual void evaluate();
//. . .
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}
class IntegerExp : public Expression
{
virtual void parenPrint();
virtual void evaluate();
//. . .
}
class AddExp : public Expression
{
virtual void parenPrint();
virtual void evaluate();
//. . .
}
Suppose there are n subclasses of Expression altogether, each similar to IntegerExp
and AddExp shown here. How many classes would have to be added or changed to
add each of the following things?
(a) A new class to represent product expressions.
(b) A new operation to graphically draw the expression parse tree.
Another way of implementing expression classes and operations uses a pattern
called the visitor design pattern. In this pattern, each operation is represented by
a visitor class. Each visitor class has a visitCLS method for each expression class
CLS in the hierarchy. The expression class CLS is set up to call the visitCLS method
to perform the operation for that particular class. Each class in the expression
hierarchy has an accept method that accepts a visitor as an argument and “allows
the visitor to visit the class and perform its operation.” The expression class does
not need to know what operation the visitor is performing.
If you write a visitor class ParenPrintVisitor to print an expression tree, it would
be used as follows:
Expression *expTree = . . .some code that builds the expression tree. . .;
Visitor *printer = new ParenPrintVisitor();
expTree->accept(printer);
The first line defines an expression, the second defines an instance of your
ParenPrintVisitor class, and the third passes your visitor object to the accept method
of the expression object.
The expression class hierarchy that uses the visitor design pattern has this
form, with an accept method in each class and possibly other methods:
class Expression
{
virtual void accept(Visitor *vis) = 0; //Abstract class
//...
}
class IntegerExp : public Expression
{
virtual void accept(Visitor *vis) {vis->visitIntExp(this);};
//...
}
Exercises
class AddExp : public Expression
{
virtual void accept(Visitor *vis)
{ lhs->accept(vis); vis->visitAddExp(this); rhs->accept(vis); }
//...
}
The associated Visitor abstract class, naming the methods that must be included in
each visitor and some example subclasses, have this form:
class Visitor
{
virtual void visitIntExp(IntegerExp *exp) = 0;
virtual void visitAddExp(AddExp *exp) = 0; // Abstract class
}
class ParenPrintVisitor : public Visitor
{
virtual void visitIntExp(IntegerExp *exp) {// IntExp print code};
virtual void visitAddExp(AddExp *exp) {// AddExp print code};
}
class EvaluateVisitor : public Visitor
{
virtual void visitIntExp(IntegerExp *exp) {// IntExp eval code};
virtual void visitAddExp(IntegerExp *exp) {// AddExp eval code};
}
Suppose there are n subclasses of Expression and m subclasses of Visitor. How many
classes would have to be added or changed to add each of the following things by
use of the visitor design pattern?
(c) A new class to represent product expressions.
(d) A new operation to graphically draw the expression parse tree.
The designers want your advice.
(e) Under what circumstances would you recommend using the standard design?
(f) Under what circumstances would you recommend using the visitor design
pattern?
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11
History of Objects: Simula and Smalltalk
Objects were invented in the design of Simula and refined in the evolution of
Smalltalk. In this chapter, we look at the origin of object-oriented programming
in Simula, based on the concept of a procedure that returns a pointer to its activation
record, and the development of a purely object-oriented paradigm in the Smalltalk
project and programming language. Twenty years after its development, Smalltalk
provides an important contrast with C++ and Java both in simplicity of concept and
in the way that its implementation provides maximal programming flexibility.
11.1 ORIGIN OF OBJECTS IN SIMULA
As the name suggests, the Simula programming language was originally designed for
the purpose of simulation. The language was designed by O.-J. Dahl and K. Nygaard at
the Norwegian Computing Center, Oslo, in the 1960s. Although the designers began
with a specific interest in simulation, they eventually produced a general-purpose
programming language with widespread impact on the field of computing.
Simula has been extremely influential as the first language with classes, objects, dynamic lookup, subtyping, and inheritance. It was an inspiration to the Xerox Palo Alto
Research Center (PARC) group that developed Smalltalk and to Bjarne Stroustrop
in his development of C++. Although Simula 67 had important object-oriented concepts, much of the popular mystique surrounding objects and object-oriented design
developed later as a result of other efforts, most notably the Smalltalk work of Alan
Kay and his collaborators at Xerox PARC.
11.1.1 Object and Simulation
You may wonder what objects have to do with simulation. A partial answer may be
seen in the outline of an event-based simulation program. An event-based simulation
is one in which the operation of a system is represented as a sequence of events. Here
is pseudocode representing a generic event-based simulation program:
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11.1 Origin of Objects in Simula
KRISTEN NYGAARD
A driving force behind Simula was Kristen Nygaard, an operations research specialist
and a political activist. He wanted a general modeling language that could be used to
describe complex dynamic social and industrial systems in a simple way. Nygaard’s
interest in modeling motivated the development of innovative computing techniques,
including objects, classes, inheritance, and quasi-parallel program execution allowing
every object to have an optional action thread.
Nygaard was concerned by the social consequences of computing and expressed
some of his reservations as follows:
I could see that SIMULA was being used to organize work for people and I could see that it
would contribute to major changes in this area: More routine work, less demand for knowledge
and a skilled labor force, less flexibility at the work place, more pressure. . . . I gradually came
to face a moral dilemma. . . . I realized that the technology I had helped to develop had serious
consequences for other people. . . The question was what to do about it? I had no desire to uninvent SIMULA .. because I was convinced that in the society I wanted to help build, computers
would come to play an immensely important role. [Translation by J.R. Holmevik]
In addition to the ACM Turing Award and the IEEE John von Neumann Medal,
Kristen Nygaard, received the 1990 Norbert Wiener Award for Professional and Social
Responsibility from Computer Professionals for Social Responsibility (CPSR), “For his
pioneering work in Norway to develop ‘participatory design,’ which seeks the direct
involvement of workers in the development of the computer-based tools they use.”
Historical information about Nygaard and the development of Simula can be found
in an article by J.R. Holmevik, (Annals of the History of Computing Vol. 16, Number 4,
1994; pp. 25–37). More recent information may be found on Nygaard’s home page.
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Q := make queue(first event);
repeat
remove next event e from Q
simulate event e
place all events generated by e on Q
until Q is empty
This form of simulation requires a data structure (some form of queue or priority
queue) that may contain a variety of kinds of events. In a typed language, the most
natural way to obtain such a structure is through subtyping. In addition, the operation simulate e must be written in some generic way or involve case statements that
branch according to the kind of event that e actually represents. Objects help because dynamic lookup for simulate e can determine the correct code automatically.
Inheritance arises when we consider ways of implementing related kinds of events.
As this quick example illustrates, event-based simulations can be programmed in
a general-purpose object-oriented language. The designers of Simula discovered this
when they tried to make a special-purpose language, one tailored to simulation only.
Apparently, when someone once asserted that Simula was not a general-purpose
language, Nygaard’s response was that, because Simula had all of the features of
Fortran and Algol, and more, what would he need to remove before Simula could
be called general purpose?
11.1.2 Main Concepts in Simula
Simula was designed as an extension and modification of Algol 60. Here are short
lists of the main features that were added to and removed from Algol 60:
Added to Algol 60:
class concepts and reference variables (pointers to objects),
pass-by-reference,
char, text, and input–output features,
coroutines, a mechanism for writing concurrent programs.
Removed from Algol 60:
changed default parameter passing mechanism from pass-by-name to a combination of pass-by-value and pass-by-result,
some initialization requirements on variables,
own variables (which are analogous to C static variables),
the Algol 60 string type (in favor of a text type).
In addition to objects, concurrency was an important development in Simula. This
arose from an interest in simulations that had several independent parts, each defining a sequence of events. Before representing events by objects, the designers experimented with representing independent sequences of events by independent
11.2 Objects in Simula
processes, with the main simulation loop alternating between these processes to
allow the simulation to progress. Here is a short quote from Nygaard on the incorporation of processes into the language:
In the spring of 1963, we were almost suffocated by the single-stack structure
of Algol. Then Ole-Johan developed a new storage management scheme [the
multistack scheme] in the summer and autumn of 1963. The preprocessor idea
was dropped, and we got a new freedom of choice. In Feb, 1964 the process
concept was created, which [led to] Simula 67’s class and object concept.
This quote appears in “The Development of the Simula Languages” by K. Nygaard
and O.-J. Dahl (published in History of Programming Languages, R. L. Wexelblat,
ed., Academic, New York, 1981.)
11.2 OBJECTS IN SIMULA
Objects arise from a very simple idea: After a procedure call is executed, it is possible
to leave the procedure activation record on the run-time stack and return a pointer
to it. A procedure of this modified form is called a class in Simula and an activation
record left on the stack is an object:
Class: A procedure returning a pointer to its activation record.
Object: An activation record produced by call to a class, called an instance of
the class.
Because a Simula activation record contains pointers to the functions declared in the
block and their local variables, a Simula object is a closure! Pointers, missing from
Algol 60, were needed in Simula because a class returns a pointer to an activation
record. In Simula terminology, a pointer is called a ref.
Although the concept of object begins with the idea of leaving activation records
on the stack, returning a pointer to an activation record means that the activation
record cannot be deallocated until the activation record (object) is no longer used
by the program. Therefore, Simula implementations place objects on the heap, not
the run-time stack used for procedure calls. Simula objects are deallocated by the
garbage collector, which deallocates objects only when they are no longer reachable
from the program that created them.
11.2.1 Basic Object-Oriented Features in Simula
Simula contains most of the main object-oriented features that we use today. In
addition to classes and objects, mentioned in the preceding subsection, Simula has
the following features:
Dynamic lookup, as operations on an object are selected from the activation
record of that object,
Abstraction in later versions of Simula, although not in Simula 67,
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Subtyping, arising from the way types were associated with classes,
Inheritance, in the form of class prefixing, including the ability to redefine parts
of a class in a subclass.
Although Simula 67 did not distinguish between public and private members of
classes, a later version of the language allowed attributes to be made “protected,”
which means that they are accessible for subclasses (but not other classes), or
“hidden,” in which case they are not accessible to subclasses either.
In addition to the features just listed, Simula contains a few object-related features
that are not found in most object-oriented languages:
Inner, which indicates that the method of a subclass should be called in combination with execution of superclass code that contains the inner keyword,
Inspect and qua, which provide the ability to test the type of an object at run time
and to execute appropriate code accordingly. Inspect is a class (type) test, and qua
is a form of type cast that is checked for correctness at run time.
All of these features are discussed in the following subsections. Some features that
are found in other languages but not in early Simula are multiple inheritance, class
variables (as in Smalltalk), and the self / super mechanism found in Smalltalk.
11.2.2 An Example: Points, Lines, Circles
Here is a short program example, taken from a classic book describing early Simula
(Birtwistle, Dahl, Myhrhaug, and Nygaard, Simula Begin, Auerbach, 1973). This
example illustrates some characteristics of Simula and shows how closely early objectoriented programming in Simula resembles object-oriented programming today.
Problem
Given three distinct points p, q, and r in the plane, find the center and the radius of
the circle passing through p, q, and r. The situation is drawn in Figure 11.1.
q
p
r
Figure 11.1. Points, circles, and their
lines of intersection.
11.2 Objects in Simula
Algorithm
An algorithm for solving this problem is suggested by 11.1. The algorithm has the
following steps:
1. Draw intersecting circles Cp and Cq, centered at points p and q, respectively.
2. Draw intersecting circles Cq and Cr , centered at q and r , respectively. For
simplicity, we assume that Cq and Cq are the same circle.
3. Draw line L1 through the points of intersection of Cp and Cq.
4. Draw line L2 through the points of intersection of Cq and Cr .
5. The intersection of L1 and L2 is the center of the desired circle.
This method will fail if the three points are colinear, as there is no circle passing
through three colinear points.
Methodology
We can code this algorithm by representing points, lines, and circles as objects and
equipping each class of objects with the necessary operations. Here is a sketch of the
classes and operations we need:
Point
Representation
x, y coordinates
Operations
equality(anotherPoint) : boolean
distance(anotherPoint) : real
Line
Representation
All lines have the form ax + by + c = 0. We may store a, b, and c, normalized so
that all three numbers are not too large. When we call the Line class to build a Line
object, we will normalize the values of a, b, and c.
Operations
parallelto(anotherLine) : boolean
meets(anotherLine) : ref(Point)
The parallelto operation is used to see if two lines will intersect. The meets operation
is used to find the intersection of two lines that are not parallel.
Circle
Representation
center : ref(Point)
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Operations
intersects(anotherCircle) : ref(Line)
It should be clear how to solve the problem with these classes of objects. Given
two points, we can pass one to the distance function of the other and obtain the
distance between the two points. This lets us calculate the radius for intersecting
circles centered at the two points. Given two circles, the intersects function finds the
line passing through the two points of intersection, and so on.
11.2.3 Sample Code and Representation of Objects
In Simula, the Point class can be written and used to create a new point, as follows:
class Point(x,y); real x,y;
begin
boolean procedure equals(p); ref(Point) p;
if p =/= none then
equals := abs(x - p.x) + abs(y - p.y) < 0.00001;
real procedure distance(p); ref(Point) p;
if p == none then error else
distance := sqrt(( x - p.x )**2 + (y - p.y) ** 2);
end ***Point***
p :- new Point(1.0, 2.5);
Because all objects are manipulated by refs (the Simula term for pointers), the type
of an object variable has the form ref(Class), where Class is the class of the object.
Because the equals procedure requires another point as an argument, the formal
parameter p is declared to have type ref(Point). Simula references are initialized to
a special value none and :- is used for pointer assignment. The test p =/= none (read
“p not-equal none”) tests whether the pointer p refers to an object.
In the body of the Point class, we access parts of the object, which are locally
declared variables and procedures, simply by naming them. Parts of other objects,
such as the object passed as a parameter to distance, are accessed with a dot notation,
as in p.x and p.y.
When the statement at the bottom of the code example above is executed, it
produces a run-time structure that we may draw as follows:
p
access link
real x
1.0
real y
2.5
code for
equals
equals
distance
code for
distance
11.2 Objects in Simula
This Point object contains pointers to the closures for Point class procedures and
the environment pointer of each closure points to the activation record that is the
object. (This means the codes for equals and distance can be shared among all points,
but the closure pairs must be different for each object.) When one of these procedures is called, as in p.equals(q), an activation record for the call is created and
its access link is set according to the closure for the procedure. This way, when
the code for equals refers to x, the x that will be used is the x stored inside the
object p.
There are several ways that this representation of objects may be optimized;
you should assume only that this representation captures the behavior of Simula objects, not that every Simula compiler actually uses precisely this storage
layout. Some useful optimizations are shown in the chapters on Smalltalk and
C++, in which each object stores only a pointer to a table storing pointers to the
methods.
To give a little more sample code, the line class may be written in Simula as
follows:
class Line(a,b,c); real a,b,c;
begin
boolean procedure parallelto(l); ref(Line) l;
if l =/= none then
parallelto := abs(a*l.b - b* l.a) < 0.00001;
ref(Point) procedure meets(l); ref(Line) l;
begin real t;
if l =/= none and ∼parallelto(l) then
begin
t := 1/(l.a * b - l.b * a);
meets :- new Point(. . . , . . . );
end;
end; ***meets***
real d;
d := sqrt(a**2 + b**2);
if d = 0.0 then error else
begin
d := 1/d;
a := a * d; b := b * d; c := c * d;
end;
end *** Line***
The procedure meets invokes another procedure of the same object, parallelto. The
code following the procedure meets is initialization code; it is executed whenever a
Line object is instantiated. You might want to think about how this initialization code,
which is written as if a class were an ordinary procedure, corresponds to constructor
code in object-oriented languages you are familiar with.
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11.3 SUBCLASSES AND SUBTYPES IN SIMULA
11.3.1 Subclasses and Inheritance
The classes in a Simula program are arranged in a hierarchy. One class is a superclass
of a second if the second is defined by inheritance from the first. We also say that a
class is a subclass of its superclass.
Simula syntax for a class C1 with subclasses C2 and C3 is
class C1
<declarations 1> ;
C1 class C2
<declarations 2>
C1 class C3
<declarations 3>
When we create a C2 object, for example, we do this by first creating a C1 object
(activation record) and then appending a C2 object (activation record). In a picture,
a C2 object looks like this:
C1 parts of object
C2 parts of object
The structure is essentially the same as if procedure called C2 was declared and called
within C1: The access links of the second activation record refer to the first.
Here is an example code that uses Simula class prefixing to define a colored point
subclass of the Point class defined in Subsection 11.2.3:
Point class ColorPt(c); color c;
! List new parameter only
begin
boolean procedure equals(q); ref(ColorPt) q;
...;
end ***ColorPt***
ref(Point) p;
! Class reference variables
ref(ColorPt) cp;
p :- new Point(2.7, 4.2);
cp :- new ColorPt(3.6, 4.9, red);
! Include parent class parameters
The ColorPt class adds a color field c to points. Because Simula 67 did not hide fields,
the c field of a ColorPt object can be accessed and changed directly by use of the dot
notation. For example, cp.c := green changes to color of the point named by cp. The
ColorPt class redefines equals so that cp.equals can compare color as well as x and y.
The statement p :- New Point(2.7, 4.2) causes an activation record to be created
with locations for parameters x and y. These are set to 2.7 and 4.2, respectively and
11.3 Subclasses and Subtypes in Simula
then the body of the Point class is executed. Because the body of the Point} class
is empty, nothing happens at this stage for points. After the body is executed, a
pointer to the activation record is returned. The activation record contains pointers
to function values (closures) equals and distance.
A prefixed class object is created by a similar sequence of steps that involves calls
to the parent class before the child class. More specifically, an activation record is
created for the parent class and an activation record is created for the child class.
Then parameter values are copied to the activation records, parent class first, and the
class bodies are executed, parent class first. Some additional details are considered
in the exercises.
11.3.2 Object Types and Subtypes
All instances of a class are given the same type. The name of this type is the same as
the name of the class. For example, if p and q are variables referring to objects created
by the Point class, then they will have type ref(Point). As mentioned in Section 11.2,
ref arises because all Simula objects are manipulated through pointers.
The class names (types of objects) are arranged in a subtype hierarchy corresponding exactly to the subclass hierarchy. In other words, the only subtype relations
that are recognized in Simula 67 are exactly those that arise from inheritance: If class
A is derived from class B, then the Simula type checker treats type A as a subtype of
type B.
There are some interesting subtleties regarding assignment and subtyping that
also apply to other languages. For example, look carefully at the following legal
Simula code, in which Simula syntax :- is used for reference (pointer) assignment:
class A;
A class B; /* B is a subclass of A */
ref (A) a;
ref (B) b;
a :- b;
/* legal since B is a subclass of A */
...
b :- a;
/* also legal but checked at run-time to make sure a points to a B
object*/
Both assignments are accepted at compile time and satisfy Simula’s notion of type
compatibility. However, a run-time check is needed for the second assignment to
guarantee type safety. The same run-time checking also appears in Beta, a cultural
descendant of Simula, and in Java array array assignment. The reason for the run-time
test is that the object reference a might point to an object of class B or it might point
to an object of class A that is not an object of class B. In the first case, the assignment
is OK; in the second case, it is not. If an assignment causes an object reference b with
static type ref(B) to point to an object that is not from class B or some subclass of B,
this is a type error.
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It is possible to rewrite the assignment to b in the code we just looked at by using
Simula inspect:
inspect a
when B do b :- a
otherwise . . . /* some appropriate action */
If a does not refer to a B object, then the otherwise clause will catch the error and let the
programmer take appropriate action. However, in the case in which the programmer
knows that a refers to a B object, the syntax with an implicit run-time test is obviously
simpler.
There was an error in the original Simula type checker surrounding the relationship between subtyping and inheritance. This is illustrated in the following code,
extracted from a running DEC-20 Simula program written by Alan Borning. (The
DEC-20 version of Simula uses := instead of :- for pointer assignment.)
class A;
A class B; /* B is a subclass of A */
ref (A) a;
ref (B) b;
proc assignA (ref (A) x)
begin
x := a
end;
assignA(b);
In the terminology of Chapter 10, Simula subclassing produces the subtype relation
B <: A. Simula also uses the semantically incorrect principle that, if B <: A, then
ref (B) <: ref (A). Therefore, this code will statically type check, but will cause a type
error at run time. The same type error occurs in the original implementation of Eiffel,
a language designed many years later.
11.4 DEVELOPMENT OF SMALLTALK
The Smalltalk project at Xerox PARC in the 1970s streamlined the object metaphor
and drew attention to object-oriented programming. Although the central objectoriented concepts in Smalltalk resemble their Simula ancestors, Smalltalk was not
an incremental modification of any previously existing language. It was a completely
new language, with new terminology and an original syntax.
This is a list of some of the main advances of Smalltalk over Simula:
The object metaphor was extended and refined
In Smalltalk, everything is an object, even a class.
All operations are therefore messages to objects.
11.4 Development of Smalltalk
Objects and classes were shown to be useful organizing concepts for building
an entire programming environment and system.
Abstraction was added, using the distinction between
private instance variables (data associated with an object) and
public methods (code for performing operations)
These short lists, however, do not do justice to the Smalltalk project. Smalltalk was
conceived as part of an innovative and forward-looking effort that designed an entire
computing system around a new programming language.
Smalltalk is more flexible and more powerful than Simula or other languages
that preceded it, and many Smalltalk fans consider it a better language than most
that followed. Smalltalk is untyped, allowing greater flexibility in the modification
of programs and greater flexibility in the use of general-purpose algorithms on a
variety of data objects. In addition, the importance of treating everything as an object
cannot be underestimated. This is the source of much of the flexibility of Smalltalk.
Smalltalk has many interesting and powerful facilities, including a mechanism that
allows objects to detect a message they do not understand and to respond to the
message.
Dynabook
Part of the inspiration for the Smalltallk language and system came from Alan Kay’s
concept of a Dynabook. The Dynabook was to be a small portable computer that was
capable of storing useful and interesting personal information and running programs
for a variety of applications. It was a laptop computer with business and entertainment software, and sufficient storage space to store telephone numbers, business
data, or a copy of a popular novel. It is almost impossible to explain, in the present
day, how utterly improbable this sounded in the 1970s. At the end of the 1970s, a
minicomputer, typically shared by 15 or 20 people, required the same floor space as
a small laundromat and had roughly the same appearance – rows of machines, some
with visible spinning drums. Very-large-scale integrated (VLSI) circuits were not a
reality until the late 1970s, so there was really no way the Dynabook could conceivably have been manufactured at the time it was imagined. To give another historical
data point, the Xerox Alto of the 1970s was an advanced personal computer that
cost $32,000 at the time, more than the annual salary of a computer professional. It
had removable disks, which each person would use to store personal working data.
Each disk was 2ft in diameter, a couple inches thick, and weighed 5–10 lb! Hardly
the sort of thing you could carry around in your pocket or even imagine inserting
into a portable computer. Not only was Kay’s prediction of hardware miniaturization
revolutionary, but the idea that people would use a computer for a wide range of
personal activities was shocking.
Smalltalk was intended to be the operating system interface as well as the standard
programming language for Dynabook; all of a user’s interaction with the Dynabook
would be through Smalltalk. With this in mind, the syntax of Smalltalk was designed
to be used with a special-purpose Smalltalk editor. Because the Xerox group believed
that hardware would eventually catch up with the need for computing power, the
Smalltalk implementation emphasized flexibility and ease of use over efficiency.
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11.5 SMALLTALK LANGUAGE FEATURES
11.5.1 Terminology
The Smalltalk project developed precise terminology for the main Smalltalk concepts
and constructs. Here is a list of the main object-related terms and brief descriptions:
Object: A combination of private data and functions. Each object is an instance
of some class.
Class: A template defining the implementation of a set of objects.
Subclass: Class defined by inheriting from its superclass.
Selector: The name of a message (analogous to a function name).
Message: A selector together with actual parameter values (analogous to a function call).
Method: The code in a class for responding to a message.
Instance Variable: Data stored in an individual object (instance of a class).
The way these terms were used in the Smalltalk language and documentation is close
to current informal use among many object-oriented professionals, but not entirely
universal. For example, some people use “method” to refer to both the name of a
method (which Smalltalk would call a selector) and the code used to respond to a
message.
11.5.2 Classes and Objects
The Smalltalk environment included a special editor designed for the Smalltalk language. With this editor, classes were defined by filling in a table of a certain form,
illustrated with the definition of a Point class in Figure 11.2.
In Smalltalk terminology, a class variable is a variable that is defined in the class
and shared among all objects of the class. In this example, the value of pi may be used
in geometric calculations involving points. Therefore, the class includes a variable
called pi. An instance variable is a variable that is repeated for each object. Each
point object, for example, will have two instance variables, x and y, as each point may
class name
Point
super class
Object
class var
pi
instance var
x y
class messages and methods
〈…names and code for methods...〉
instance messages and methods
〈…names and code for methods...〉
Figure 11.2. Definition of Point class.
11.5 Smalltalk Language Features
be located at a different position in the x–y plane. Methods are functions that may
be used to manipulate objects, and messages contain the names and parameters of
methods.
Here are some example class messages and methods for point objects (subsequently explained):
newX:xvalue Y:yvalue ||
∧
self new x: xvalue y: yvalue
newOrigin ||
∧
self new x: 0 y: 0
initialize ||
pi <- 3.14159
and some example instance messages and methods:
x: xcoord y: ycoord ||
x <- xcoord
y <- ycoord
moveDx: dx Dy: dy ||
x <- dx + x
y <- dy + y
x || ∧ x
y || ∧ y
draw ||< < . . . code to draw point . . . > >
These are written with Smalltalk syntax, which includes the following:
|| −place to list local declarations for method body
∧
- unary prefix operator giving return value of method
<- - assignment
Smalltalk message names include positions for message arguments, indicated by
colons.
The first class method, newX:Y:, is a two-argument method that creates a new
point. The two arguments are the x coordinate and the y coordinate of the new
point. A new point at coordinates (3, 2) is created when the message
newX: 3 Y:2
is sent to the Point class, with x coordinate 3 following the first colon and y coordinate
2 following the second coordinate. When we look at inheritance and other situations
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in which the name of a method is important, it will be useful to remember that
the official Smalltalk name of a message contains all the words and colons, as in
“newX:Y:.”
The newX:Y: method is a class method of the Point class, which means that if you
want to create a new point with specific x and y coordinates, you may send a newX:Y:
message to the Point class (which is itself an object). The newX:Y: method has no local
variables, as any local declarations would appear between the vertical bars following
the name. The effect of the method is indicated by the expression
∧
self new x: xvalue y: yvalue
As previously mentioned, the first symbol, ∧ , means “return.” The value that is returned is the object produced when the message new x: xvalue y: yvalue is sent to
the Point class. (Self refers to the object that contains the method; this is discussed in
Subsection 11.6.3.) This message is handled by a predefined class method, provided
by the system, that allows any class to create instances of it.
To continue the example, suppose we execute the following code:
p <- Point newOrigin
p moveDx: 3 Dy: 4
This will create a point at the origin, (0, 0), and then move the point to position (3, 4).
Thereafter, the value of expression
px
that sends the message x to the object p will be the object 3 and the value of
py
that sends the message y to the object p will be the object 4.
Run-Time Representation of Objects
The run-time representation of a Smalltalk object has a number of parts. Each point
object, for example, may have a different x coordinate and y coordinate. Therefore,
each object must contain space for an x coordinate and a y coordinate. The methods,
on the other hand, are shared among all point objects. To save space and take advantage of the similarity between objects of the same class, each object contains a pointer
to a data structure containing information associated with its class. The basic run-time
structure for points is illustrated in Figure 11.3. This illustration is intended to show
the basic functionality of Smalltalk classes and objects; actual implementations might
store things differently or perform various optimizations to improve performance.
11.5 Smalltalk Language Features
to superclass Object
Point class
Template
Point object
x
class
x
3
y
2
y
Method dictionary
newX:Y:
...
move
code
...
code
Figure 11.3. Run-time representation of Point object and class.
When a message, such as move, is sent to a point object, the code for move is located
by following a pointer to the class of the object, following another pointer to the
method dictionary, and then looking up the code in this data structure. If the code for
a method is not found in the method dictionary for the class, the method dictionaries
of superclasses are searched, as described in the next subsection on inheritance.
11.5.3 Inheritance
We look at Smalltalk inheritance by considering a subclass of Point that implements
colored points. The main difference between ColoredPoint and Point is that each colored point has an associated color. If we want to provide methods for the same
geometric calculations, then we can make colored points a subclass of points. In
Smalltalk, the ColoredPoint class can be defined as shown in Figure 11.4.
class name
ColoredPoint
super class
Point
class var
instance var
color
class messages and methods
newX:xv Y:yv C:cv
〈 … code … 〉
instance messages and methods
color
| | ^color
draw
〈 … code … 〉
Figure 11.4. Definition of ColoredPoint class.
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The second line of Figure 11.4 shows that Point is the superclass of ColoredPoint.
As a result, all instance variables and methods of Point become instance variables
and methods of ColoredPoint by default. In Smalltalk terminology, ColoredPoint inherits instance variables x and y, methods x:y:, moveDx:Dy:, and so on. In addition,
ColoredPoint adds an instance variable color and a method color to return the color of
a ColoredPoint.
Although ColoredPoint inherits the draw method from Point, the draw method for
ColoredPoint must be implemented differently. When we draw a ColoredPoint, the point
should be drawn in color. Therefore, Figure 7.1 shows a draw method, overriding the
one inherited from Point. (Redefinition of a method is sometimes called overriding
the method.) An option that is available in Smalltalk, but not in most other objectoriented languages, is to specify that a superclass method should be undefined on a
subclass.
Because colored points should have a color specified when they are created, the
ColoredPoint class will have a class method that sets the color of a colored point:
newX: xvalue Y:yvalue C:cvalue ||
∧
self new x: xvalue y:yvalue color:cvalue
As an example use of ColoredPoint, suppose we execute the following code:
cp <- ColoredPoint newX:1 Y:2 C:red
cp moveDx: 3 Dy: 4
Then the value of the expression
cp x
that sends the message x to object cp will be the object 4, and the value of the
expression
cp color
that sends the message color to the object cp will be the object red. Note that even
though move is an inherited method, defined originally for points without color, the
result of moving a ColoredPoint is a ColoredPoint.
Run-Time Structure to Support Inheritance
The run-time structures for a ColoredPoint object and ColoredPoint class are illustrated in Figure 11.5. The illustration shows a ColoredPoint object, with the object’s
class pointer pointing to the ColoredPoint class object. The ColoredPoint class contains
a superclass pointer that points to the Point class object. Like the Point class, the
11.5 Smalltalk Language Features
Point object
Point class
Template
x
y
2
Method dictionary
newX:Y:
...
3
move
ColoredPoint object
ColoredPoint class
Template
x
Method dictionary
4
y
newX:Y:C:
5
color
color
red
draw
Figure 11.5. Run-time representation of ColoredPoint object and classs.
ColoredPoint class also has a pointer to a template, which contains the names of the
instance variables, and a method dictionary, which contains pointers to code that
implements each of the methods. Like Figure 11.3, this figure is schematic. An actual
compiler could store things slightly differently or perform optimizations to improve
performance.
If a draw message is sent to a ColoredPoint object, then we find the code for
executing this method by following the pointer in the object to the ColoredPoint
class structure and by following the pointer there to the method dictionary for the
ColoredPoint class. The dictionary is then searched for a method with the right name.
If a message other than new, color, and draw is sent, then the method name will not
be found in the ColoredPoint method dictionary. In this case, the superclass pointer in
the ColoredPoint class is used to locate the Point class and the Point method dictionary.
Then the search continues as if the object were an instance of its superclass.
It should be clear from these data structures that looking up a method at run time
can involve a large number of steps. If class A inherits from class B and class B inherits
from class C, for example, then searching for an inherited method could involve
searching all three method dictionaries at run time. If the operation performed by
this method is simple, then it might take more time to find a method than to execute
it.
This run-time cost of Smalltalk method lookup illustrates one of the basic principles of programming language design: You can’t get something for nothing. Dynamic
lookup is a powerful programming feature, but it has its costs. Fortunately, there
are optimizations that reduce the cost of method lookup in practice. One common
optimization (also used in Java) is to cache recently found methods at the method
invocation site in a program. If this is done, then a second invocation of the same
method can be executed by a jump directly to the code location stored in the cache.
As discussed in the next chapter, static type information can also be used to significantly reduce the cost of method lookup. This makes lookup more efficient in C++.
However, optimizations based on static type information do not work in Smalltalk
because Smalltalk has no static type system.
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There are some interesting details surrounding the way that instance variables are
accessed by methods. Because the code for the draw method in the ColoredPoint class
is compiled at the same time that the layout of ColoredPoint objects is determined, the
ColoredPoint method can be compiled so that if it needs to access instance variables
of a ColoredPoint object, it can locate them easily (given a pointer to the object).
Specifically, the offset of the x coordinate, y coordinate, and color, relative to the
starting address of the ColoredPoint object, is known at compile time and can be used
in any ColoredPoint method. The same is true for Point methods. However, because a
Point method can be inherited by ColoredPoint and invoked on a ColoredPoint object,
it is necessary for Point instance variables to appear at the same relative offset in both
Point and ColoredPoint objects. More generally, accessing instance variables by use of
offsets known at compile time requires the layout of a subclass object to resemble
the layout of a superclass object. You might want to think about the advantages and
disadvantages of compiling methods in this way compared with run-time search for
instance variables by using the class templates. One consequence of using compiletime offsets is that, when a superclass is changed and a new instance variable is added,
all subclasses must be recompiled. (This was the case for the Xerox PARC Smalltalk
compiler.)
11.5.4 Abstraction in Smalltalk
In most object-oriented languages, a programmer can decide which parts of an object
are visible to clients of the class and which parts are not. Smalltalk makes this decision
for the programmer by using particularly simple rules:
Methods are Public: Any code with a pointer to an object may send any message
to that object. If the corresponding method is defined in the class of the object, or
any superclass, the method will be invoked. This makes all methods of an object
visible to any code that can access the object.
Instance Variables are Protected: The instance variables of an object are accessible
only to methods of the class of the object and to methods of its subclasses.
This is not a fully general approach to abstraction. In particular, there are practical
situations in which it would be useful to have private methods, accessible only to
other methods of the same class. However, the Smalltalk visibility rules are simple,
easy to remember, and sufficient in many situations.
11.6 SMALLTALK FLEXIBILITY
11.6.1 Dynamic Lookup and Polymorphism
Advocates of object-oriented languages often speak about the “polymorphism” inherent in object-oriented code. When they say this, they mean that the same message name invokes different code, depending on the object that receives the message. In this book, we refer to this phenomenon as dynamic lookup, as explained in
Chapter 10.
Here is an example that illustrates the value of dynamic lookup. Consider a list
of objects, some of which are points and others of which are colored points. If we
11.6 Smalltalk Flexibility
want to display every point on the list, we may traverse the list, sending each object
the draw method. Because of dynamic lookup, the Point objects will execute the Point
draw method and the ColoredPoint objects will execute the ColoredPoint draw method.
Dynamic lookup supports code reuse, as one routine will display lists of Points, lists
of ColoredPoint, and lists that have both Points and ColoredPoints.
The run-time structures used for Smalltalk classes and objects support dynamic
lookup in two ways. First, methods are selected through the receiver object. Second,
method lookup starts with the method dictionary of the class of the receiver and then
proceeds upward. The first method found with the appropriate name is selected.
The following code sends a draw message to a Point and a ColoredPoint:
p <- Point newOrigin
p draw
p <- ColoredPoint newX:1 Y:2 C:red
p draw
Be sure you understand how and why different code is used in lines two and four to
draw the two different kinds of objects.
11.6.2 Booleans and Blocks
An assumption that many people make when learning about Smalltalk is that it has
the same kind of syntax, with specific reserved words and special forms, as other
languages.
However, many constructs that would be handled with a special-purpose statement form in other languages can be handled uniformly in Smalltalk with the principle
that everything is an object. A good example is the Smalltalk treatment of Booleans,
which uses the built-in construct of blocks.
A Smalltalk Boolean object (true or false) has a selector named ifTrue:ifFalse:. The
method associated with this selector requires two parameters, one to be executed if
the object is true and the other if the object is false. The arguments to this method are
generally Smalltalk blocks, which are objects that have a message, “execute yourself.”
Although blocks may be defined by special in-line syntax, for convenience, they are
not essentially different from other Smalltalk objects.
The reason for using blocks in Smalltalk Booleans is easy to understand if we
think about conditionals in other languages. In general, the construct
if . . . then . . . else . . .
cannot be defined as an ordinary function, as an ordinary function call would involve
evaluating all arguments before calling the function. If we evaluated A, B, and C first in
if A then B else C
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this is inefficient and incorrect if B and C have side effects or fail to terminate. We want
to evaluate B only in the case that A is true and evaluate C only in the case that A is false.
For example, consider the following expression:
i < j ifTrue: [i add: 1] ifFalse: [j subtract: 1]
This is the Smalltalk way of writing
if i < j then i+1 else j-1
The first part of the expression, i < j, is a < j message to object i, asking it to compare
itself with j. The result will be either the object true or the object false, in either case an
object that understands an ifTrue:ifFalse: message. Therefore, after the comparison,
i < j, one of the two subsequent blocks will be executed, depending on the result of
the comparison.
The way Smalltalk actually works is that the class Boolean has two subclasses,
True and False, each with one instance, called true and false, respectively. In these
classes, the ifTrue:ifFalse: method is implemented as follows:
True class . . .
ifTrue: trueBlock ifFalse: falseBlock
trueBlock value
False class . . .
ifTrue: trueBlock ifFalse: falseBlock
falseBlock value
where value is the message you send to a block if you want it to evaluate itself and
return its value.
There is a little bit of “special treatment” here in that the system will not allow
more instances of classes True and False to be created. This is to guarantee correctness
of an optimization. For efficiency, the standard idiom ifTrue: [. . .] ifFalse: [. . .] is
optimized by the compiler in most Smalltalk implementation. However, this is again
simply an optimization for efficiency. In principle, Booleans could be defined within
the language. In addition, the unoptimized methods previously given are called if
any arguments are variables that might hold blocks.
11.6.3 Self and Super
The special symbol self may be used in the body of a Smalltalk method. The special
property of self is that it always refers to the object that contains this method, whether
directly or by inheritance. The way this works is most easily explained by example.
There are several classes of numbers in Smalltalk. Three are arranged in the
following hierarchy:
11.6 Smalltalk Flexibility
Integer
SmallInt
LargeInt
Even if the Integer class does not create any objects, we can define the following
method in the Integer class:
factorial ||
self <= 1
ifTrue: [∧ 1]
ifFalse: [∧ (self - 1) factorial * self]
This method computes the factorial of the integer object to which it belongs. The
method first asks the object to compare itself with the predefined number object 1.
If it is less-than-or-equal, the result will be the object true and the block [∧ 1] will be
evaluated, giving the result 1. If the object is greater than 1, then the second block
[∧ (self - 1) factorial * self] will be evaluated, causing a recursive call to the factorial
method on a smaller integer.
There are two operations in the ifFalse case of the factorial method that could be
implemented in different ways in the two subclasses of Integer. The first is - (minus).
Because small integers may have a different representation from large integers, subtraction may be implemented differently in the two classes. Multiplication, *, may
also be implemented differently in the two classes. These methods do not have to
return objects of the same class. For example, n*m could be a SmallInt if n and m are
both small, or could be a LargeInt if both are SmallInts and their product is larger than
the maximum size for SmallInt.
In Smalltalk, we begin a message to self by beginning the search for each method
at the object where the first message is received. For example, suppose we want to
compute the factorial of a small integer n. The computation will begin by sending the
factorial message to n. Because the first step of factorial is self <= 1, the message <= 1
will be sent to the object n. It is important that dynamic lookup is used here, as the
correct method for comparing n with 1 is the <= method defined in the SmallInt class.
In fact, the Integer class does not need to have a <= method defined in order for the
calculation of factorial to work properly. To appreciate the value of self, you might
think about whether you could write a factorial method in class Integer that would
work properly on SmallInt and LargeInt without using self.
The special symbol super is similar to self, except that, when a message is sent to
super, the search for the appropriate method body starts with the superclass of the
object instead of the class of the object. This mechanism provides a way of accessing
a superclass version of a method that has been overidden in a subclass.
11.6.4 System Extensibility: The Ingalls Test
Dan Ingalls, a leader of the Smalltalk group, proposed the following test for determining whether a programming language is truly object oriented:
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Can you define a new kind of integer, put your new integers into rectangles, ask
the system to blacken a rectangle, and have everything work?
This test is probably the most extreme test of an object-oriented language. Essentially
he is asking “Is everything an object?” Or, more specifically, “Is every operation
determined by dynamic lookup through objects?”
Why is This Important?
The main issue addressed by this test is the extensibility of programs and systems. If
we build a graphic-oriented system that manipulates and displays various kinds of
objects, then the design and implementation of basic objects like points and coordinates might be determined very early in the design process. If a complex system is
built around one basic construct, like standard language-supplied integers, then how
easy will it be to modify the system to use another form of integers?
The answer, of course, is that it depends on how you build the system. In Smalltalk,
however, the only way of building a system is by representing basic concepts as
objects. Because all operations in Smalltalk are done by sending messages and every
message results in run-time lookup of the appropriate method body, a lot of flexibility
is inherently built into any system. In C++ and Java, by contrast, it is possible to build
a system with integers that are not objects. As a result, it is possible to build systems
that will not immediately accept other forms of integer without some change to the
code. In particular, most Java or C++-based systems will not be able to handle a
rectangle that has some coordinates given by built-in integers that are not objects
and some coordinates given by objects.
What Are the Implementation Costs?
If a system is built so that every operation on integers is accompanied by method
lookup, then an expression x add y sending a message to an integer cannot be safely
optimized to a static function call such as integer add(x,y). If the compiler or programmer replaces x add y with integer add(x,y), then we cannot replace a built-in
integer x with a user-defined integer object x.
If a program uses integers repeatedly and each integer operation involves a runtime search for an appropriate method, this is a substantial run-time cost, even if the
best-known optimizations are used to reduce running time as much as possible. In
fact, for essentially this reason, the efficiency of many Smalltalk programs is 5 to 10
times less than corresponding programs in C. Of course, program running time is not
everything. If a system will be used in many ways, and flexibility and adaptability
are important, then flexibility and adaptability may be more important than running
speed.
11.7 RELATIONSHIP BETWEEN SUBTYPING AND INHERITANCE
11.7.1 Interfaces as Object Types
Although the Smalltalk language does not use any static type checking, there is an
implicit form of type that every Smalltalk programmer uses in some way. The type
of an object in Smalltalk is its interface, the set of messages that can be sent to the
11.7 Relationship between Subtyping and Inheritance
object without receiving the error “message not understood.” The interface of an
object is determined by its class, as a class lists the messages that each object will
answer. However, different classes may implement the same messages, as there are
no Smalltalk rules to keep different classes from using the same selector names.
The Smalltalk library contains a number of implemented classes for a variety of
purposes. One part of the library that provides interesting examples of interfaces and
inheritance is the collection-class library. The collection classes implement general
collections like Set, Bag, Dictionary, LinkedList, and a number of more specialized kinds
of collections. Some example interfaces from the collection classes are the following
sets of selector names:
Set Interface = {isEmpty, size, includes:, removeEvery:}
Bag Interface = {isEmpty, size, includes:, occurrencesOf:, add:, remove:}
Dictionary Interface = {at:put:, add:, values, remove:}
Array Interface = {at:put:, atAllPut:, replaceFrom:to:with:}
Remember that a colon indicates a position for a parameter. For example, if s is a set
object, then s includes:3 sends the message includes:3 to s. The result will be true or
false, depending on whether the set s contains the element 3.
Here is a short explanation of the intended meaning of each of the selectors:
isEmpty: return true if the collection is empty, else false
size: return number of elements in the collection
includes:x: return true if x is in the collection, false otherwise
removeEvery:x: remove all occurrences of x from the collection
occurrencesOf:x: return a number telling how many occurrences of x are in the
collection
add:x: add x to the collection
remove:x: remove x from the collection
at:x put:y: insert value y into an indexed collection at position indexed by x
values: return the set of indexed values in the indexed collection (e.g., list all the
words in a dictionary)
atAllPut:x: in finite updateable collection such as an array, set every value to x
replaceFrom:x t:y with:v: in range from x to y, set every indexable value to v (e.g.,
for array A, set A[x]=A[x+1]= . . . =A[y]=v)
11.7.2 Subtyping
Because Smalltalk is an untyped language, it may not seem relevant to compare
subtyping and inheritance. However, if we recall the definition of a subtype,
Type A is a subtype of a type B if any context expecting an expression of type
B may take any expression of type A without introducing a type error,
we can see that subtyping is about substitutivity. If we build a Smalltalk program
by using one class of objects and then try to use another class in its place, this can
work if the interface of the second class contains all of the methods of the first,
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but not necessarily otherwise. Therefore, it makes sense to associate subtyping with
the superset relation on Smalltalk class interfaces. Subtyping is sometimes called
conformance when discussing interfaces instead of types of some typed programming
language.
An example of subtyping arises if we consider extensible collections. To be specific, let us define ExtensibleCollection Interface by
ExtensibleCollection Interface = {isEmpty, size, includes:, add:, removeEvery:}
This is the same as Set Interface, with the method add: added. Using <: for subtyping,
we have
ExtensibleCollection Interface <: Set Interface
In words, this means that every object that implements the ExtensibleCollection interface also implements the Set interface. Semantically, this makes sense because a
program that uses Set objects can send any of the messages listed in the Set interface
to Set objects. Because each one of these messages is also implemented in every
ExtensibleCollection object, we can substitute ExtensibleCollection objects for Set
objects without obtaining “message not understood” errors.
11.7.3 Subtyping and Inheritance
In Smalltalk, the interface of a subclass is often a subtype of the interface of its
superclass. The reason is that a subclass will ordinarily inherit all of the methods
of its superclass, possibly adding more methods. This is certainly the case for Point
and ColoredPoint, as described earlier in this chapter. Points have the five methods
x:y:, moveDx:Dy:, x,y, and draw. Colored points have these five methods and a color
method, so every colored point has all of the point methods.
In general, however, subclassing does not always lead to subtyping in Smalltalk.
The simplest way that subtyping may fail is when a subclass does not implement all
of the methods of the superclass. In Smalltalk, there is simply a way of saying, in
a subclass, that one or more superclass methods should not be defined on subclass
objects. Because it is possible to delete a method from a superclass, a subclass may
not produce a subtype.
On the other hand, it is easy to have subtyping without inheritance. Because
subtyping, as we have defined it, is based only on the names of methods, two classes
could be defined independently (neither a subclass of the other) but have exactly the
same methods. Then, according to our definitions, each is a subtype of the other. But
there is no inheritance.
William Cook, a researcher at Hewlett-Packard Labs and later a developer at
Apple Computer, did an empirical study of the collection classes in the Smalltalk-80
class library to determine which relationships between subtyping and inheritance
11.7 Relationship between Subtyping and Inheritance
arise in Smalltalk practice (Interfaces and Specifications for the Smalltalk-80
Collection Classes, ACM Conf. Object.Oriented Programming: Systems, Languages,
and Applications, ACM SIGPLAN Notices 27(10), 1992). Among a relatively small
number of classes, he found all possible relationships between subtyping and inheritance. A diagram from his study is shown in Figure 11.6. In this diagram, solid lines
indicate subtyping between interfaces, and dotted curves show actual places where
inheritance is used in the construction of the collection classes of the Smalltalk-80
library. In studying this diagram, it is important to keep in mind that there are more
interfaces listed here than were actually implemented in the library. To show the
structural relationship between classes more clearly, Cook included some interfaces
that are intersections of the interfaces of classes in the library.
Collection
Extensible
Indexed
Updatable
Ordered
Set
Mapped
Collection
CopyReplace
Dictionary
Bag
Interval
Poppable
Arrayed
Collection
Sequenceab
Pushable
String
Adjustable
Sorted
Collection
Ordered
Collection
Linked List
Figure 11.6. Smalltalk collection class hierarchy.
RunArray
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Generally speaking, the subtyping relation between a set of classes is independent
of how the classes are implemented, but the inheritance relation is not. Subtyping is
a relation between types, which is a property that is important to users of an object,
whereas inheritance is a property of implementations. Imagine a group of people
sitting down to design the Smalltalk collection classes. Once they have agreed on
the names of the classes and the operations on each class, they have agreed on the
subtyping relation between the classes. However, the inheritance relation is not determined at this point. If each person implemented one of the classes independently,
then all of the classes could be built without using inheritance at all. At the other
extreme, it is often possible to factor the functionality of a set of related classes so
that inheritance is used extensively. To give a specific example, arrays and dictionaries are similar kinds of collections. Abstractly, both are a form of function from
a finite set to an arbitrary set. Therefore, it is conceivable that a programmer could
build a general finite-function class and use this as the dictionary class. This programmer can inherit from dictionary to produce the special case of arrays in which the
index set is always a range of integers. On the other hand, a different programmer
could first build arrays and then use arrays to implement dictionaries by hashing each
index word to an integer value and using this to store the indexed value in a specific array location. This illustrates the independence of subtyping and inheritance in
Smalltalk.
11.8 CHAPTER SUMMARY
In this chapter, we studied Simula, the first object-oriented language, and Smalltalk,
an influential language that generalized and popularized object-oriented programming.
Simula was designed in Norway, beginning as part of a project to develop a simulation language. Smalltalk was designed at Xerox PARC (in Palo Alto, California) and
was intended to accompany a futuristic concept called the Dynabook. An important
unifying concept in Smalltalk is that everything should be an object.
Simula
Objects: A Simula object is an activation record produced by call to a class.
Classes: A Simula class is a procedure that returns a pointer to its activation
record. The body of a class may initialize the object it creates.
Abstraction: Hiding was not provided in 1967, but was added later and used as
the basis for C++.
Subtyping: Objects are typed according to the classes that create them. Subtyping
is determined by class hierarchy.
Inheritance: A Simula class may be defined as an extension of a class that has
already been defined.
Smalltalk
Objects: A Smalltalk object is created by a class. At run time, an object stores its
instance variables and a pointer to the instantiating class.
Classes: A Smalltalk class defines class variables, class methods, and the instance
methods that are shared by all objects of the class. At run time, the class data
Exercises
structure contains pointers to an instance variable template, a method dictionary,
and the superclass.
Abstraction: Abstraction is provided through protected instance variables. All
methods are public but instance variables may be accessed only by the methods
of the class and methods of subclasses.
Subtyping: Smalltalk does not have a compile-time type system. Subtyping arises
implicitly through relations between the interfaces of objects. Subtyping depends
on the set of messages that are understood by an object, not the representation
of objects or whether inheritance is used.
Inheritance: Smalltalk subclasses inherit all instance variables and methods of
their superclasses. Methods defined in a superclass may be redefined in a subclass
or deleted.
Because Smalltalk has no compile-time type system, a variable may point to any object of any type at run time. This makes it more difficult to implement efficient method
lookup algorithms for Smalltalk than for some other object-oriented languages. You
may wish to review the data structure and algorithms for locating a method of an
object as illustrated in Subsection 11.5.3. Be sure you understand how search for
a method proceeds from subclass to superclass in the event that the method is not
defined in the subclass.
EXERCISES
11.1 Simula Inheritance and Access Links
In Simula, a class is a procedure that returns a pointer to its activation record.
Simula prefixed classes were a precursor to C++ -derived classes, providing a form
of inheritance. This question asks about how inheritance might work in an early
version of Simula, assuming that the standard static scoping mechanism associated
with activation records is used to link the derived class part of an object with the
base class part of the object.
Sample Point and ColorPt classes are given in the text. For the purpose of this
problem, assume that if cp is a ColorPt object, consisting of a Point activation record
followed by a ColorPt activation record, the access link of the parent-class (Point)
activation record points to the activation record of the scope in which the class
declaration occurs, and the access link of the child-class (ColorPt) activation record
points to activation record of the parent class.
(a) Fill in the missing information in the following activation records, created by
execution of the following code:
ref(Point) p;
ref(ColorPt) cp;
r :- new Point(2.7, 4.2);
cp :- new ColorPt(3.6, 4.9, red);
cp.distance(r);
Remember that function values are represented by closures and that a closure
is a pair consisting of an environment (pointer to an activation record) and a
compiled code.
In the following illustration, a bullet (•) indicates that a pointer should be
drawn from this slot to the appropriate closure or compiled code. Because the
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pointers to activation records cross and could become difficult to read, each
activation record is numbered at the far left. In each activation record, place
the number of the activation record of the statically enclosing scope in the
slot labeled “access link.” The first two are done for you. Also use activation
record numbers for the environment pointer part of each closure pair. Write
the values of local variables and function parameters directly in the activation
records.
Activation Records
(0)
(1)
(2)
(3)
(4)
r→
Point part of cp
cp →
cp.distance(r)
x
y
access link
x
y
equals
distance
access link
x
y
equals
distance
access link
c
equals
access link
q
Closures
Compiled Code
(0)
•
•
(0)
•
•
( )
•
( )
(r )
(
),
•
(
),
•
(
),
•
(
),
•
(
),
•
code for
equals
code for
distance
code for
cpt equals
(b) The body of distance contains the expression
sqrt(( x - q.x )**2 + (y - q.y) ** 2)
that compares the coordinates of the point containing this distance procedure
to the coordinate of the point q passed as an argument. Explain how the value
of x is found when cp.distance(r) is executed. Mention specific pointers in your
diagram. What value of x is used?
(c) This illustration shows that a reference cp to a colored point object points to
the ColorPt part of the object. Assuming this implementation, explain how the
expression cp.x can be evaluated. Explain the steps used to find the right x
value on the stack, starting by following the pointer cp to activation record (3).
(d) Explain why the call cp.distance(r) needs access to only the Point part of cp and
not the ColorPt part of cp.
(e) If you were implementing Simula, would you place the activation records representing objects r and cp on the stack, as shown here? Explain briefly why you
might consider allocating memory for them elsewhere.
11.2 Loophole in Encapsulation
A priority queue (also known as a heap) is a form of queue that allows the minimum
element to be retrieved. Assuming that Simula has a list type, a Simula pq class might
Exercises
look something like this:
class pq();
begin
(int list) contents;
bool procedure isempty();
< . . . return true if pq empty, else false . . . >
procedure insert(x); int x;
< . . . put x at appropriate place in list . . . >
int procedure deletemin();
< . . . delete first list elt and return it . . . >
contents := nil
end
where the sections <. . . > would be filled in with appropriate executable Simula
code. We would create a priority queue object by writing
h :- new pq();
and access the components by writing something like
h.contents;
h.isempty;
h.insert(x);
h.deletemin();
In ML we can write a function that produces a closure with essentially the same
behavior as that of a Simula priority queue object. This is subsequently written,
together with the associated exception declaration:
exception Empty;
fun new pq() =
let val contents = (ref nil) : int list ref
fun insert(x,nil) = [x]
| insert(x, y : : l) = if x<y then x : : (y : : l)
else y : : insert(x, l)
in
{
emp meth = fn () => !contents = nil,
ins meth = fn x => (contents := insert(x, !contents)),
del meth = fn () => case !contents of
nil => raise Empty
| y : : l => (contents := l; y)
}
end;
This question asks you to think about a “loophole” in the Simula object system that
allows a client program to interfere with the correct behavior of a priority queue
object.
(a) The main property of a priority queue is that if n elements are inserted and k
are removed (for any k < n), then these k elements are returned in increasing
order. Explain in two or three sentences why both the Simula and ML forms of
priority queue objects should exhibit this behavior in a “reasonable” program.
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(b) Explain in a few sentences how a “devious” program that uses a Simula priority
queue object h could cause the following behavior: After 0 and 1 are inserted
into the priority queue by calls h.insert(0) and h.insert(1), the first deletemin operation on the priority queue returns some number other than 0. (This will not
be possible for an ML priority queue object.) Do not use any pointer arithmetic
or other operations that would not be allowed in a language like Pascal.
(c) Write a short sequence of priority queue operations of the form
h :- new pq();
h.insert(0);
h.insert(1);
...
h.deletemin();
where the ellipsis (. . .) may contain h operations, but not insert or deletemin,
demonstrating the behavior you described in part (b).
11.3 Subtyping of Refs in Simula
In Simula, the procedure call assignA(b) in the following context is considered
statically type correct:
class A . . . ;
/* A is a class */
A class B . . . ; /* B is a subclass of A */
ref (A) a;
ref (B) b;
/* a is a variable pointing to an A object */
/* b is a variable pointing to a B object */
proc assignA (ref (A) x)
begin
x := a
end;
assignA(b);
(a) Assume that if B <: A then ref (B) <: ref (A). Using this principle, explain why
both the procedure assignA and the call assignA(b) can be considered statically
type correct.
(b) Explain why actually executing the call assignA(b) and performing the assignment given in the procedure may lead to a type error at run time.
(c) The problem is that the “principle,” if B <: A then ref (B) <: ref (A), is not
semantically sound. However, we can make type checking by using this principle sound by inserting run-time tests. Explain the run-time test you think
a Simula compiler should insert in the compiled code for procedure assignA.
Can you think of a reason why the designers of Simula might have decided to
use run-time tests instead of disallowing ref subtyping in this situation? (You
do not have to agree with them; just try to imagine what rationale might have
been used at the time.)
11.4 Smalltalk Run-Time Structures
Here is a Smalltalk Point class whose instances represents points in the twodimensional Cartesian plane. In addition to accessing instance variables, an instance
method allows point objects to be added together.
Exercises
Class name
Superclass
Class variables
Instance variables
Class messages and methods
Point
Object
Comment: none
xy
Comment: instance creation
newX: xValue Y: yValue | |
↑ self new x: xValue y: yValue
Instance messages and methods
Comment: accessing instance vars
x: xCoordinate y: yCoordinate | |
x ← xCoordinate
y ← yCoordinate
x||↑x
y||↑y
comment: arithmetic
+ aPoint | |
↑ Point newX: (x + aPoint x) Y:
(y + aPoint y)
(a) Complete the top half of the illustration of the Smalltalk run-time structure
shown in Figure P.11.4.1 for a point object with coordinates (3, 4) and its class.
Label each of the parts of the top half of the figure, adding to the drawing as
needed.
(b) A Smalltalk programmer has access to a library containing the Point class, but
she cannot modify the Point class code. In her program, she wants to be able
to create points by using either Cartesian or polar coordinates, and she wants
to calculate both the polar coordinates (radius and angle) and the Cartesian
coordinates of points. Given a point (x, y) in Cartesian coordinates, the radius
is ((x * x) + (y * y)) squareRoot and the angle is (x/y) arctan. Given a point (r, θ )
in polar coordinates, the x coordinate is r * (θ cos) and the y coordinate is r *
(θ sin).
i. Write out a subclass PolarPoint, of Point and explain how this solves the
programming problem.
ii. Which parts of Point could you reuse and which would you have to define
differently for PolarPoint?
(c) Complete the drawing of the Smalltalk run-time structure by adding a PolarPoint
object and its class to the bottom half of the figure you already filled in with
Point structures. Label each of the parts and add to the drawing as needed.
11.5 Smalltalk Implementation Decisions
In Smalltalk, each class contains a pointer to the class template. This template stores
the names of all the instance variables that belong to objects created by the class.
(a) The names of the methods are stored next to the method pointers. Why are the
names of instance variables stored in the class, instead of in the objects (next
to the values for the instance variables)?
(b) Each class’s method dictionary stores the names of only the methods explicitly
written for that class; inherited methods are found by searching up the superclass pointers at run time. What optimization could be done if each subclass
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Figure P.11.4.1. Smalltalk run-time structures for Point and PolarPoint.
contained all of the methods of its superclasses in its method dictionary? What
are some of the advantages and disadvantages of this optimization?
(c) The class template stores the names of all the instance variables, even those
inherited from parent classes. These are used to compile the methods of the
class. Specifically, when a subclass is added to a Smalltalk system, the methods
of the new class are compiled so that, when an instance variable is accessed, the
method can access this directly without searching through the method dictionary and without searching through superclasses. Can you see some advantages
and disadvantages of this implementation decision compared with looking up
the relative position of an instance variable in the appropriate class template
each time the variable is accessed? Keep in mind that in Smalltalk, a set of
classes could remain running for days, while new classes are added incrementally and, conceivably, existing classes could be rewritten and recompiled.
11.6 Protocol Conformance
We can compare Smalltalk interfaces to classes that use protocols, which are lists
of operation names (selectors). When a selector allows parameters, as in at: put: ,
Exercises
the selector name includes the colons but not the spaces. More specifically, if dict
is an updatable collection object, such as a dictionary, then we could send dict a
message by writing dict at:‘cross’ put:‘angry’. (This makes our dictionary definition of
“cross” the single word “angry.”) The protocol for updatable collections will therefore contain the seven-character selector name at:put: . Here are some example
protocols:
stack : {isEmpty, push:, pop }
queue : {isEmpty, insert:, remove }
priority queue : {isEmpty, insert:, remove }
dequeue : {isEmpty, insert:,insertFront:, remove, removeLast }
simple collection : {isEmpty }
Briefly, a stack can be sent the message isEmpty, returning true if empty, false otherwise; push: requires an argument (the object to be pushed onto the stack), pop
removes the top element from the stack and returns it. Queues work similarly, except they are first-in, first-out instead of first-in, last-out. Priority queues are first-in,
minimum-out and dequeues are doubly ended queues with the possibility of adding
and removing from either end. The simple collection class just collects methods that
are common to all the other classes. We say that the protocol for A conforms to
the protocol for B if the set of A selector names contains the set of B selector
names.
(a) Draw a diagram of these classes, ordered by protocol conformance. You should
end up with a graph that looks like William Cook’s drawing shown in Figure
11.6.
(b) Describe briefly, in words, a way of implementing each class so that you make
B a subclass of A only if the protocol for B conforms to the protocol set for A.
(c) For some classes A and B that are unrelated in the graph, describe a strategy
for implementing A as a subclass of B in a way that keeps them unrelated.
(d) Describe implementation strategies for two classes A and B (from the preceding
set of classes) so that B is a subclass of A, but A conforms to B, not the other
way around.
11.7 Removing a Method
Smalltalk has a mechanism for “undefining” a method. Specifically, if a class A has
method m, then a programmer may cancel m in subclass B by writing
m:
self shouldNotImplement
With this declaration of m in subclass B, any invocation of m on a B object will result
in a special error indicating that the method should not be used.
(a) What effect does this feature of Smalltalk have on the relationship between
inheritance and subtyping?
(b) Suppose class A has methods m and n, and method m is canceled in subclass
B. Method n is inherited and not changed, but method n sends the message m
to self. What do you think happens if a B object b is sent a message n? There
are two possible outcomes. See if you can identify both, and explain which one
you think the designers of Smalltalk would have chosen and why.
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11.8 Subtyping and Binary Methods
This question is about the relationship between subtyping and inheritance. Recall
that the main principle associated with subtyping is substitutivity: If A is a subtype
of B, then wherever a B object is required in a program, an A object may be used
instead without producing a type error. For the purpose of this question, we will use
Message not understood as our Smalltalk type error. This is the most common error
message resulting from a dynamic type failure in Smalltalk; it is the object-oriented
analog of the error Cannot take car of an atom that every Lisp programmer has
generated at some time. Remember that Smalltalk is a dynamically typed language.
This question asks you to show how substitutivity can fail, using the fact that a
method given in a superclass can be redefined in a subclass.
If there are no restrictions on how a method (member function) may be redefined
in a subclass, then it is easy to redefine a method so that it requires a different
number of arguments. This will make it impossible to meaningfully substitute a
subclass object for a superclass object. A more subtle fact is that subtyping may
fail when a method is redefined in a way that appears natural and (unless you’ve
seen this before) unproblematic. This is illustrated by the following Point class and
ColoredPoint subclass:
Class name
Class variables
Instance variables
Instance messages and methods
Point
Class name
Class variables
Instance variables
Instance messages and methods
ColoredPoint
xval yval
xcoord ↑ xval
ycoord ↑ yval
origin
xval ← 0
yval ← 0
movex: dx movey: dy
xval ← xval + dx.
yval ← yval + dy
equal: pt
↑ (xval = pt xcoord & yval = pt ycoord)
color
color ↑ color
changecolor: newc color ← newc
equal: cpt
↑ (xval = cpt xcoord & yval = cpt ycoord &
color = cpt color)
The important part here is the way that equal is redefined in the colored point class.
This change would not be allowed in Simula or C++, but is allowed in Smalltalk.
(The C++ compiler does not consider this an error, but it would not treat it as
redefinition of a member function either.) The intuitive reason for redefining equal
is that two colored points are equal only if they have the same coordinates and are
the same color.
Exercises
Problem: Consider the expression p1 equal:p2, where p1 and p2 are either Point
objects or ColoredPoint objects. It is guaranteed not to produce Message not understood if both p1 and p2 are either Points or ColoredPoints, but may produce an error if
one is a Point and the other a ColoredPoint. Consider all four combinations of p1 and
p2 as Points and ColoredPoints, and explain briefly how each message is interpreted.
11.9 Delegation-Based Object-Oriented Languages
In this problem, we explore a delegation-based object-oriented language, Self, in
which objects can be defined directly from other objects. Classes are not needed
and not supported. Self has run-time type checking and garbage collection.
Here is the Self language description:
(extracted from D. Ungar and R. B. Smith, Self: The power of simplicity, ACM
SIGPLAN Notices 22(12), 1987, pp. 227–242.)
In Smalltalk, . . . everything is an object and every object contains a
pointer to its class, an object that describes its format and holds its behavior. In Self too, everything is an object. But, instead of a class pointer,
a Self object contains named slots with may store either state or behavior. If an object receives a message and it has no matching slot, the search
continues via a parent pointer. This is how Self implements inheritance.
Inheritance in Self allows objects to share behavior, which in turn allows
the programmer to alter the behavior of many objects with a single change.
For instance, a [Cartesian] point object would have slots for its non-shared
characteristics: x and y. Its parent would be an object that held the behavior
shared among all points: +, etc.
In Self, there is no direct way to access a variable: instead, objects send
messages to access data residing in name slots. So to access its “x” value,
a point sends itself the “x” message. The message finds the “x” slot, and
evaluates the object found therein. . . . In order to change contents of the
“x” slot to, say, 17, instead of performing an assignment like “x←17,” the
point must send itself the “x:” message with 17 as the argument. The point
object must contain a slot named “x:” containing the assignment [function].
(a) Using this description, draw the run-time data structure of the Self version of
a (3,4) Point object, as described in the last two sentences of the first paragraph,
and it parent object. Assume that assignments to the x and y characteristics
are permitted.
(b) Self’s lack of classes and instance variables make inheritance more powerful.
For example, to create two points sharing the same x coordinate, the x and x:
slots can be put in a separate object that is a parent of each of the two points.
Draw a picture of the run-time data structures for two points sharing the same
x coordinate.
(c) To create a new Point object, the clone message is sent to an existing point. The
language description continues:
Creating new objects . . . is accomplished by a simple operation,
copying. . . . [In Smalltalk,] creating new objects from classes is accomplished by instantiation, which includes the interpretation of format
information in a class. Instantiation is similar to building a house from
a plan.
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Cloning an object does not clone its parent, however.
If a point object contains fields x and y and methods x:, y:, move, then cloning
the object will create another object with two fields and three methods. Each
point will have a parent pointer, two fields, and three methods – six entries in
all. The Self point will be twice the size of the corresponding Smalltalk point.
Explain how you would structure your Self program so that each point can
be cloned without cloning its methods.
(d) Self also allows a change-parent operation on an object. The parent pointer
of an object can be set to point to any other object. (An exception is that the
first object must not be an ancestor of the second – we do not want a cycle.)
The change-parent operation is useful for objects that change behavior
in different states. For example, a window can be in the visible or iconified
(minimized) state. When iconified, mouse clicks and window display work
differently than when the window is visible. A window’s parent pointer can
be set to VisibleWindow initially, then changed to IconifiedWindow when the
“minimized” button is pressed. Another example is a file object that can be
in the open or closed state. The open method changes the parent pointer from
ClosedFile to OpenFile.
The change-parent operation adds a lot of flexibility to Self. Can you think
of disadvantages of this feature?
12
Objects and Run-Time Efficiency: C++
C++ is an object-oriented extension of the C language. It was originally called C with
classes, with the name C++ originating around 1984. A Bell Laboratories researcher
interested in simulation, Bjarne Stroustrup began the C++ project in the early 1980s.
His goal was to add objects and classes to C, using his experience with Simula as
the basis for the design. The design and implementation of C++ was originally a
one-person effort, with no apparent intent to produce a commercial product. However, as interest in objects and program structure grew over the course of the 1980s,
C++ became popular and widely used. In the 1990s, C++ became the most widely
used object-oriented language, with good compilers and development environments
available for the Macintosh, PC, and Unix-based workstations.
C, discussed in Section 5.2, was originally designed by Dennis Ritchie. The C
programming language was used for writing the Unix operating system at Bell
Laboratories. The original implementation of C++ was a preprocessor that converted
C++ to C.
12.1 DESIGN GOALS AND CONSTRAINTS
The main goal of C++ is to provide object-oriented features in a C-based language
without compromising the efficiency of C. In the process of adding objects to C, some
other improvements were also made. In more detail, the main design goals of C++
may be summarized as follows:
data abstraction and object-oriented features,
better static type checking,
backwards compatibility with C. In other words, most C code should compile as
legal C++, without requiring significant changes to the code,
efficiency of compiled code, according to the principle “If you do not use a feature,
you should not pay for it.”
The principle stated in the final goal is significant and may require some thought
to appreciate. This principle suggests that C programs should compile as efficiently
under the C++ compiler as under the C compiler. It would violate this principle to
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BJARNE STROUSTRUP
Bjarne Stroustrup is the principal designer of C++. Bjarne came to Bell
Laboratories in 1979 after finishing his M.S. degree in Aarhus, Denmark,
and his Ph.D. on design of distributed systems in Cambridge, U.K.
Working on distributed computing, in the same Computing Science
Center as C and Unix designers Ritchie and Thompson, Stroustrup decided to add object-oriented features from Simula to C. His goal was to
further his personal research objectives, which involved simulation.
An occasionally reserved but essentially gregarious and friendly person, Stroustrup was catapulted into the technological limelight by the
success of C++, first within AT&T and later in the wider community of
practicing software developers. He has written several books on C++, including The Design and Evolution of C++ (Addison-Wesley, 1994), which
contains an interesting history, explanation, and commentary on the
language. Bjarne’s nonresearch interests include reading books that are
not about computer science, photography, hiking and running, travel,
and music.
implement C integers as objects, for example, and use dynamic method lookup to
find integer functions at run time as in Smalltalk, as this would significantly reduce
the performance of C integer calculations. The principle does not mean that C++
statements that also appear in C must be implemented in exactly the same way in
both languages, but whatever changes are adopted in C++, they must not slow down
execution of compiled code unless some slower features of C++ are also used in the
program.
12.1.1 Compatibility with C
The decision to maintain compatibility with C has a pervasive effect on the design of
C++. Those familiar with C know that C has a specific machine model, revealing much
of the structure of the underlying computer architecture. In particular, C operations
that return the address of a variable and place any bit pattern in any location make it
possible for C programs to rely on the exact representation of data. Therefore C++
must adhere to the same data representations as C.
12.1 Design Goals and Constraints
Most other object-oriented languages, including those designed before C++ and
those designed after, use garbage collection to relieve programmers from the task
of identifying inaccessible objects and deallocating the associated memory locations.
However, there is no inherent reason why objects must be garbage collected. The
strongest connection is that with the increased emphasis on abstraction and type
correctness, it would have been consistent with other goals of C++ to offer some
form of garbage collection where feasible in a form that does not affect the running
time of programs that do not use garbage-collected objects. However, because there
are features of C that make garbage collection extremely difficult, it would have been
difficult to base C++ objects on garbage collection. Not only is garbage collection
counter to the C philosophy of providing programmer control over memory, but the
similarity between pointers and integers makes it effectively impossible to build a
garbage collector that works on C++ programs that use pointer arithmetic or cast
integers to pointers.
An important specific decision is to treat C++ objects as a generalization of C
structs. This makes it necessary to allow objects to be declared and manipulated in the
same way as structs. In particular, objects can be allocated in the activation records of
functions or local blocks, as well as on the heap, and can be manipulated directly (i.e.,
not through pointers). This is one place where C++ deviates from Simula, as Simula
objects can be manipulated only through pointers. More specifically, C++ allows a
form of object assignment that copies one object into the space previously occupied by another, whereas most other object-oriented languages allow only pointer
assignment for objects. Some aspects of C++ objects on the stack are explored in the
homework exercises.
12.1.2 Success of C++
C++ is a very carefully designed language that has succeeded admirably, in spite
of difficult design constraints. Measured by the number of users, C++ is without
question the most successful language of the decade from its development in the
mid-1980s until the release of Java in the mid-1990s. However, the design goals and
backward compatibility with C do not allow much room for additional aesthetic
consideration. Some aspects of C++ have become complex and difficult for many
programmers to understand. On the other hand, many C programmers use the C++
compiler and appreciate the benefits of better type checking. Perhaps a fair summary
of the success of C++ is that it is widely used, with most users choosing to program in
some subset of the language that they feel they understand and that they find suitable
for their programming tasks. In other words, C++ is a useful programming tool that
allows designers to craft good object-oriented programs, but it does not enforce good
programming style in the way that other language designs have attempted to do. This
is intended to be a statement of fact and not a count against C++. In fact, much of
its success seems attributable to the way that C++ is designed to give programmers
choices and not to restrict programmers to a particular style.
There are many published style guides that advocate use of certain features of C++
and caution against the use of others. Those interested in serious C++ programming
or interested in understanding how many programmers view the language may wish
to visit their library or bookstore and take a look at some of the current guides.
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12.2 OVERVIEW OF C++
Before looking at the object-oriented features of C++ in Subsection 12.2.2, we will
take a quick tour of some of the additions to C that are not related to objects.
Evaluation and commentary on C++ appear in Subsection 12.2.3.
12.2.1 Additions to C Not Related to Objects
There are a number of differences between C++ and C that are not related to objects.
Although we are primarily interested in the C++ object system, it is worth considering
a few of the more significant changes. The most interesting additions are
type bool
reference types and pass-by-reference
user-defined overloading
function templates
exceptions
There are also changes in memory management calls (new and delete instead of
malloc and free), changes in stream and file input and output, addition of default
parameter values in function declarations, and more minor changes such as the addition of end-of-line comments and elimination of the need for typedef keyword with
struct/union/enum declarations.
The first three additions, bool, pass-by-reference, and overloading, are discussed
in the remainder of this subsection. Function templates are discussed in Subsection
9.4.1 and a general discussion of exceptions appears in Subsection 8.2.
Type bool
In C, the value of a logical test is an integer. For example, the C Reference Manual
defines the comparison operator < and logical operator && as follows:
The operator < (less than) returns 1 if the first operand is less than the second
and 0 otherwise.
The && operator returns 1 if both its operands are nonzero and 0 otherwise.
This makes it possible to write C statements with expressions such as
if ((5 < 3) + 1 && 2 ==3) { . . . }
which combine comparison operations and arithmetic.
To make a syntactic distinction between Booleans and integers, C++ has a separate type: bool with values true and false. Because of conversions, this does not
completely separate integers and Booleans. In particular, integer values may be assigned to variables of type bool, with implicit conversion of nonzero numbers to true
and 0 to false. However, the type bool does improve the readability of many programs
by indicating that a variable or return value of a function will be used as a Boolean
value rather than an integer.
12.2 Overview of C++
The C++ changes can be mimicked in C by the declarations
typedef int bool;
#define false 0
#define true 1
which define the type bool and values true and false. One difference between built-in
Booleans of C++ and Booleans in C is that when C++ Booleans are printed, they
print as true and false, not 1 and 0.
Because of all the implicit conversions, a separate Boolean type does not help with
one of the most common simple programmer mistakes in C. At least for beginners,
conditional statements such as
if (a=b) c;
are often the result of typographic error; the programmer meant to write
if (a==b) c;
The reason why the first statement type checks and compiles in C is that an integer
assignment a=b has type int and has the value of b. Because a C conditional statement
requires an integer (not a Boolean), there is no warning associated with this statement. In most languages with bool different from integer, it would be a type error
to write if (a=b) c. However, because C++ Booleans are automatically converted to
integers, the statement if (a=b) c remains legal in C++.
Because the introduction of bool into C++ has very little effect in the end, you
might wonder why this was done. Before a Boolean type was added, C/C++ often
contained definitions of bool, true, and false by macro, as previously illustrated. However, bool could be defined in different ways, with the resulting types having slightly
different semantics. For example, bool could be defined as int, unsigned int, or short
int. This caused problems when libraries that use slightly different definitions were
combined. Therefore it was useful to standardize code by adding a built-in Boolean
type.
Reference Type and Pass-By Reference
In C, all parameters are passed by value. If you wish to modify a value that is passed
as a parameter, you must pass a pointer to the value. For example, here is C code for
a function that increments an integer:
void increment(int *p) { /* parameter is pointer to int */
(*p)++; }
/* modify value pointed to by p */
main() {
int k = 0;
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increment(&k);
k;
...
}
/* pass address of variable k */
/* value of k is now 1 */
In C++, it is possible to pass-by-reference, as shown here:
void increment(int & n) { /* parameter is reference to int */
n++; }
/* modify reference param */
main() {
int k = 0;
increment(k);
/* no need to explicitly compute address */
k;
/* value of k is now 1 */
...
}
The effect is similar to C’s pointer argument, but the calling function does not have to
provide the address of an argument and the called function does not have to dereference pointers. A mild inconvenience with C pointer arguments is that the programmer must remember whether a given function uses pass-by-value or the pointer in
its parameter list. If pointer arguments are changed to C++ pass-by-reference, then
no address calculation needs to be written as part of the call and a programmer using
functions from a library can avoid this issue entirely.
A benefit of explicit references is sometimes called pass-by-constant-reference. If
a function argument is not to be changed by the function, then it is possible to specify
that the argument is to be a constant, as in void f (const int &x). In the body of a
function with parameter x passed this way, it is illegal to assign to x.
User-Defined Overloading
As discussed in Chapter 6, overloading allows one name to be used for more than
one value. In C++, it is possible to declare several functions with the same name, as
long as the functions have a different number and type of parameters. For example,
here is a C++ program with three types of print function, each called show:
#include <stdio.h>
void show(int val) {
printf(“Integer: %d\n”, val);
}
void show(double val) {
printf(“Double: %lfn”, val);
}
void show(char *val) {
printf(“String: %s\n”, val);
}
int main( ) {
12.2 Overview of C++
show(12);
show(3.1415);
show(“Hello World\n!”);
}
Because the three functions have different types of arguments, the compiler can
determine which function to call by the type of the actual parameter used in the
function call.
C++ does not allow overloaded functions that have the same number and types
of arguments but differ only in their return value because C and C++ functions can
be called as statements. When a function is called as a statement, ignoring the return
value of the function, the compiler would not have any way of determining which
function to call.
One source of potential confusion in C++ arises from combinations of overloading
and automatic conversion. In particular, if the actual parameters in a function call do
not match any version of a function exactly, the compiler will try to produce a match
by promoting and/or converting types. In this example code,
void f(int)
void f(int *)
...
f(’a’)
the call f(’a’) will result in f(int) instead of f(int *) because a char can be promoted
to be an int. When a function is overloaded and there are many possible implicit
conversions, it can be difficult for a programmer to understand which conversions
and call will be used.
12.2.2 Object-Oriented Features
The most significant part of C++ is the set of object-oriented concepts added to C;
these are the main concepts:
Classes, which declare the type associated with all objects constructed from the
class, the data members of each object, and the member functions of the class.
Objects, which consist of private data and public functions for accessing the hidden
data, as in other object-oriented languages.
Dynamic lookup, for member functions that are declared virtual. A virtual function in a derived class (subclass) may be implemented differently from virtual
functions of the same name in a base class (superclass).
Encapsulation, based on programmer designations public, private, and protected
that determine whether data and functions declared in a class are visible outside
the class definition.
Inheritance, using subclassing: One class may be defined by inheriting the data
and functions declared in another. C++ allows single inheritance, in which a class
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has a single base class (superclass) or multiple inheritance in which a class has
more than one base class.
Subtyping, based on subclassing. For one class to define a subtype of the type
defined by another class, inheritance must be used. However, the programmer
may decide whether inheritance results in a subtype or not.
This is only a brief summary; there are many more features, and it would take an entire
book to explain all of the interactions between features of C++. Further description
of classes, inheritance, and objects appear in Section 12.3.
C++ Terminology. Although C++ terminology differs from Smalltalk (and Java)
terminology, there is a fairly close correspondence. The terms class and object are
used similarly. Smalltalk instance variables are called member data and methods are
called member functions. The term subclass is not usually used in connection with
C++. Instead, a superclass is called a base class and a subclass a derived class.The
term inheritance has the same meaning in both languages.
12.2.3 Good Decisions and Problem Areas
C++ is the result of an extensive effort involving criticism and suggestions from
many experienced programmers. In many respects, the language is as good a design
as possible, given the goals of adding objects and better compile-time type checking
to C, without sacrificing efficiency or backwards compatibility. Some particularly
successful parts of the design are
encapsulation: careful attention to visibility and hiding, including public, protected, and private levels of visibility and friend functions and classes,
separation of subtyping and inheritance: classes may have public or private base
classes, giving the programmer some explicit control over the resulting subtype
hierarchy,
templates (described in Section 9.4),
exceptions (described in Section 8.2)
better static type checking than C.
There are also a number of smaller successful design decisions in C++. One example
is the way the scope resolution operator (written as ::) is used in connection with
inheritance and multiple inheritance to resolve ambiguities that are problematic in
other languages.
Problem Areas
There are a number of aspects of C++ that programmers occasionally find difficult.
Some of the main problem areas are
casts and conversions, which can be complex and unpredictable in certain situations,
objects allocated on the stack and other aspects of object memory management,
overloading, a complex code selection mechanism in C++ that can interact unpredicatably with dynamic lookup (virtual function lookup),
multiple inheritance, which is more complex in C++ than in other languages because of the way objects and virtual function tables are configured and accessed.
12.2 Overview of C++
These problem areas exist not because of oversight but because the goals of C++,
followed to their logical conclusion, led to a design with these properties. In other
words, these problems were not the result of carelessness or inattention, but a consequence of decisions that were made with other objectives in mind. To be fair,
most of them have their roots in C, not the C++ extensions to C. Nonetheless, these
features might reasonably lead programmers to prefer other languages when compatibility with C and absolute efficiency of compiled code are not essential to an
application.
Some programmers might also say that lack of garbage collection, or lack of a
standard interface for writing concurrent programs, is a problem in C++. However,
these are facilities that simply lie outside the scope of the language design.
Casts and Conversions. A cast instructs the compiler to treat an expression of one
type as if it is an expression of another. For example, (float) i instructs the compiler
to treat the variable i as a float, regardless of its type otherwise, and (int *)x causes
x to be treated as pointer to an integer. In certain situations, C and C++ compilers
perform implicit type conversions. For example, when a variable is assigned the value
of an expression, the expression may be converted to the type associated with the
variable, if needed. In most object-oriented languages, the automatic conversion
of an object from one type to another does not change the representation of the
object. For example, we have seen from the description of subtyping in Smalltalk in
Chapter 11 that Point objects can be treated as ColoredPoint objects without changing
the representation of objects because ColoredPoint objects are represented in a way
that is compatible with the representation of Point objects. If multiple inheritance is
used in C++, however, converting an object from a subtype to a supertype may require
some change in the value of a pointer to the object. This happens in a way that may
introduce hard-to-find bugs and forces programmers to understand the underlying
representation of objects. More generally, Stroustrup himself says, “Syntactically and
semantically, casts are one of the ugliest features of C and C++” in The Design and
Evolution of C++ (Addison-Wesley, 1994).
Objects Allocated on the Stack. Simula, Smalltalk, and Java allow objects to be
created only on the heap, not on the run-time stack. In these other languages, objects can be accessed through pointers, but not through ordinary stack variables
that contain space for an object of a certain size. Although C++ on-the-stack objects are efficiently allocated and deallocated as part of entry and exit from local
blocks, there are also some disadvantages. The most glaring rough spot is the way
that assignment works in combination with subtyping, truncating an object to the
size that fits when an assignment is performed. This can change the behavior of an
object, as eliminating some of its member data forces the compiler to change the
way that virtual functions are selected. Although the C++ language definition explains what happens and why, the behavior of object assignment can be confusing to
programmers.
Overloading. Overloading in itself is not a bad programming language feature,
but C++ user-defined overloading can be complex. In addition, the interaction with
dynamic lookup (virtual functions) can be unpredictable. Because overloading is
a compile-time code selection mechanism and dynamic lookup is a run-time code
selection mechanism, the two behave very differently. This causes confusion for many
programmers.
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Multiple Inheritance. There are some inherent complications associated with multiple inheritance that are difficult to avoid. These are discussed in Section 12.5. The
C++ language design compounds these problems with a tricky object and method
lookup table (vtable) format. Unfortunately, the details of the implementation of
multiple inheritance seem to intrude on ordinary programming, sometimes forcing
a programmer who is interested in using multiple inheritance to learn some of the
implementation details. Even a programmer not using multiple inheritance may be
affected if a derived class is defined using multiple inheritance.
12.3 CLASSES, INHERITANCE, AND VIRTUAL FUNCTIONS
The main object-oriented features of C++ are illustrated by a repeat of the point and
colored-point classes from Chapter 11.
12.3.1 C++ Classes and Objects
We can define point objects in C++ by declaring the point class, here called Pt,
and separately declaring the member functions of class Pt. The class declaration
without implementation defines both the interface of point objects and the data
used to implement points, but need not include the code for member functions. The
declarations that appear in the Pt class are divided into three parts, public members,
protected members, and the private members of the class.
For simplicity, Pt is a class of one-dimensional points, each point having only an
x coordinate:
/* ---------- Point Class Interface -------------------*/
class Pt {
public:
Pt(int xv);
Pt(Pt* pv);
int getX();
virtual void move(int dx);
protected:
void setX(int xv);
private:
int x;
};
/* ---------- Declarations of Constructors --------*/
Pt::Pt(int xv) { x = xv; }
Pt::Pt(Pt* pv) { x = pv->x; }
/* ---------- Declarations of Member Functions --------*/
int Pt::getX() { return x; }
void Pt::setX(int xv) { x = xv; }
Constructors. A constructor is used to initialize the member data of an object
when a program contains a statement or expression indicating that a new object is to
12.3 Classes, Inheritance, and Virtual Functions
be created. When a new point is created, space is allocated, either on the heap or in
an activation record on the stack, depending on the statement creating the object. A
constructor is then called to initialize the locations allocated for the object. Constructors are declared with the same syntax as that of member functions, except that the
“function” name is the same as the class name. In the point class, two constructors
are declared. The result is an overloaded function Pt, with one constructor called if
the argument is an integer and the other called if the argument is a Pt.
In Smalltalk terminology, constructors are “class methods,” not “instance methods,” as a constructor is called without referring to any object that is an instance of
the class. In particular, constructors are not member functions.
Visibility. As previously mentioned, the declarations that appear in the Pt class are
divided into three parts, public members, protected members, and private members
of the class. These designations, which affect the visibility of a declaration, may be
summarized as follows:
public: members that are visible in any scope in which an object of the class can
be created or accessed,
protected: members that are visible within the class and in any derived classes,
private: members visible only within the class in which these members are declared.
The Pt class is written following one standard visibility convention that is used by
many C++ programmers. Member data is made private, so that if a programmer
wishes to change the way that objects of the class are represented, this may be done
without affecting the way that other classes (including derived classes) depend on
this class. Members that modify private data are made protected, so that derived
classes may change the value of member data, but external code is not allowed to do
so. Finally, member functions that read the value of member data and provide useful
operations on objects are declared public, so that any code with access to an object
of this class can manipulate these objects in useful ways.
A feature of C++ that is not illustrated in this example is the friend designation,
which is used to allow visibility to the private part of a class. A class may declare
friend functions and friend classes. If the Pt class contained the declaration friend
class A, then code written as part of class A (such as member functions of class A)
would have access to the private part of class Pt. The friend mechanism is used when
a pair of classes is closely related, such as matrices and vectors.
Virtual Functions. Member functions may be designated virtual or left nonvirtual, which is the default for member functions that are not preceded by the keyword virtual. If a method is virtual, it may be redefined in derived classes. Because
different objects will then have different member functions of the same name, selection of virtual member functions is by dynamic lookup: There is a run-time code
selection mechanism that is used to find and invoke the correct function. These
extra steps make calls to virtual functions less efficient than calls to nonvirtual
functions.
Nonvirtual methods may not be redefined in derived classes. As a result, calls to
nonvirtual methods are compiled and executed in the same way as ordinary function
calls not associated with objects or classes.
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A common point of confusion is that, syntactically, a redeclaration of a nonvirtual function may appear in a derived class. However, this produces an overloaded
function, with code selection done at compile time. We will discuss this phenomenon
in Subsection 12.3.3.
12.3.2 C++ Derived Classes (Inheritance)
The following ColorPt class defines an extension of the one-dimensional points defined in the Pt class. As the name suggests, ColorPt objects have a color added. For
simplicity, we assume that colors are represented by integers. In addition, to illustrate virtual function definition, moving a colored point causes the color to darken a
little:
/* ---------- Color Point Class Interface -------------------*/
class ColorPt: public Pt {
public:
ColorPt(int xv,int cv);
ColorPt(Pt* pv,int cv);
ColorPt(ColorPt* cp);
int getColor();
virtual void darken(int tint);
virtual void move(int dx);
protected:
void setColor(int cv);
private:
int color;
};
/* ---------- Declarations of Constructors --------*/
ColorPt::ColorPt(int xv,int cv)
: Pt(xv)
/* call parent’s constructor */
{ color = cv; } /* then initialize color */
ColorPt::ColorPt(Pt* pv,int cv): Pt(pv) { color = cv; }
/* ---------- Declarations of Member Functions --------*/
int ColorPt::getColor() { return color; }
void ColorPt::darken(int tint) { color += tint; }
void ColorPt::move(int dx) {Pt::Move(dx); this->darken(1); }
void ColorPt::setColor(int cv) { color=cv; }
Inheritance. The first line of code declares that the ColorPt class has Pt as a public
base class; this is the meaning of the clause : public Pt following the name of the
ColorPt class. If the keyword public is omitted, then the base class is called a private
base class.
When a class has a base class, the class inherits all of the members of the base class.
This means that ColorPt objects have all of the public, protected, and private members
of the Pt class. In particular, although the ColorPt class declares only member data
color, ColorPt objects also have member data x, inherited from Pt.
12.3 Classes, Inheritance, and Virtual Functions
The difference between public base classes and private base classes is that, when
a base class is public, then the declared (derived) class is declared to be a subtype
of the base class. Otherwise, the C++ compiler does not treat the class as a subtype,
even though it has all the members of the base class. This is discussed in more detail
in Section 12.4.
Constructors. The ColorPt class has three constructors. As in the Pt class, the result is an overloaded function ColorPt, with selection between the three functions
completed at compile time according to the type of arguments to the constructor.
The two constructor bodies previously discussed illustrate how a derived-class
constructor may call the constructor of a base class. As with points, when a new
colored point is created, space is allocated, either on the heap or in an activation
record on the stack, depending on the statement creating the object, and a constructor
is called to initialize the locations allocated for the object. Because the derived class
has all of the data members of the base class, most derived-class constructors will call
the base class constructor to initialize the inherited data members. If the base class
has private data members, which is the case for Pt, then the only way for the derived
class ColorPt to initialize the private members is to call the base-class constructor.
Visibility. When one class inherits from another, the members have essentially
the same visibility in the derived class as in the base class. More specifically, public
members of the base class become public members of the derived class, protected
members of the base class are accessible in the derived class and its derived classes,
which is the same visibility as if the member were declared protected in the derived
class. Inherited private members exist in the derived class, but cannot be named
directly in code written as part of the derived class. In particular, every ColorPt object
has an integer member x, but the only way to assign to or read the value of this
member is by calling the protected and public functions from the Pt class.
Virtual Functions. As previously mentioned, virtual functions in a base class may
be redefined in a derived class. The member function move is declared virtual in
the Pt class and redefined above for ColorPt. In this example, the move function for
points just changes the x coordinate of the point, whereas the move function on
colored points changes the x coordinate and the color. If the implementation of
ColorPt::move were omitted, then the move function from Pt would be inherited on
ColorPt. The implementation of virtual functions is discussed in the next subsection.
12.3.3 Virtual Functions
Dynamic lookup is used for C++ virtual functions. A virtual function f defined on
an object o is called by the syntax o.f( . . . ) or p->f( . . . ) if p is a pointer to object o.
When a virtual function is called, the code for that function is located by a sequence
of run-time steps. These steps are similar to the lookup algorithm for Smalltalk, but
simpler because of some optimizations that are made possible by the C++ static
type system. Each object has a pointer to a data structure associated with its class,
called the virtual function table, or vtable for short (sometimes written as vtbl). The
relationship among an object, the vtable of its class, and code for virtual functions is
shown in Figure 12.1 with points and colored points.
Virtual Function on Base Class. Suppose that p is a pointer to the Pt object in
Figure 12.1. When an expression of the form p->move( . . . ) is evaluated, the code
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Objects and Run-Time Efficiency: C++
Point object
Point vtable
Code for move
ColorPoint vtable
Code for move
vptr
x
3
ColorPoint object
vptr
x
5
c
blue
Code for darken
Figure 12.1. Representation of Point and ColoredPoint in C++.
for move must be found and executed. The process begins by following the vtable
pointer in p to the vtable for the Pt class. The vtable for Pt is an array of pointers
to functions. Because move is the first (and only) virtual function of the Pt class, the
compiler can determine that the first location in this array is a pointer to the code for
move. Therefore, the run-time code for finding move simply follows the first pointer
in the Pt vtable and calls the function reached by this pointer. Unlike in Smalltalk,
here there is no need to search the vtable at run time to determine which pointer
is the one for move. The C++ type system lets the compiler determine the type of
the object pointer at compile time, and this allows the compiler to find the relative
position of the virtual function pointer in the vtable at compile time, eliminating the
need for run-time search of the vtable.
Virtual Function on Derived Class. Suppose that cp is a pointer to the ColorPt
object in Figure 12.1. When an expression of the form cp->move( . . . ) is evaluated, the
algorithm for finding the code for move is exactly the same as for p->move( . . . ): The
vtable for ColorPt is an array containing two pointers, one for move and the other for
darken. The compiler can determine at compile time that cp is a pointer to a ColorPt
object and that move is the first virtual function of the class, so the run-time algorithm
follows the first pointer in the vtable without Smalltalk-style run-time search.
Correspondence Between Base- and Derived-Class Vtables. As a consequence of
subtyping, a program may assign a colored point to a pointer to a point and call move
through the base-class pointer, as in
p = cp;
p-> move(. . . );
Because p now points to a colored point, the call p->move( . . . ) should call the move
function from the ColorPt class. However, the compiler does not know at compile
time that p will point to a ColorPt object. In fact, the same statement can be executed
many times (if this is inside a loop, for example) and p may point to a Pt object on
12.3 Classes, Inheritance, and Virtual Functions
some executions and a ColorPt object on others. Therefore, the code that the compiler
produces for finding move when p points to a Pt must also work correctly when p points
to a ColorPt. The important issue is that because move is first in the Pt vtable, move must
also be first in the ColorPt vtable. However, the compiler can easily arrange this, as the
compiler will first compile the base class Pt before compiling the derived-class ColorPt.
In summary, when a derived class is compiled, the virtual functions are ordered in
the same way as the base class. This makes the first part of a derived-class vtable look
like a base class vtable. Because member data can be accessed by inherited functions,
member data in a derived class are also arranged in the same order as member data
in a base class.
Unlike in Smalltalk, here there is no link from ColorPt to Point; the derived-class
vtable contains a copy of the base-class vtable. This causes vtables to be slightly
larger than Smalltalk method dictionaries might be for corresponding programs, but
the space cost is small compared with the savings in running time.
Note that nonvirtual functions do not appear in a vtable. Because nonvirtual
functions cannot be redefined from base class to derived class, the compiler can
determine the location of a nonvirtual function at compile time (just like normal
function calls in C).
12.3.4 Why is C++ Lookup Simpler than Smalltalk Lookup?
At run-time, C++ lookup uses indirection through the vtable of the class, with an
offset (position in the vtable) that is known at compile time. In contrast, Smalltalk
method lookup does a run-time search of one or more method dictionaries. The C++
lookup procedure is much faster than the Smalltalk procedure. However, the C++
lookup procedure would not work for Smalltalk. Let’s find out why.
Smalltalk has no static type system. If a Smalltalk program contains the line
p selector : parameters
sending a message to an object p, then the compiler has no compile-time information
about the class of p. Because any object can be assigned to any Smalltalk pointer,
p could point to any object in the system. The compiler knows that selector must
refer to some method defined in the class of p, but different classes could put the
same selector (method name) in different positions of their method dictionaries. As
a consequence, the compiler must generate code to perform a run-time search for
the method.
The C++ static type system makes all the difference. When a call such as
p->move( . . . ) is compiled, the compiler can determine a static type for the pointer p.
This static type must be a class, and that class must declare or inherit a function called
move. The compiler can examine the class hierarchy to see what location a virtual
function move will occupy in the vtable for this class. A call
p->move(x)
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Objects and Run-Time Efficiency: C++
compiles to the equivalent of the C code
(*(p->vptr[1]))(p,x)
where the index 1 in the array reference vptr[1] indicates that move is first function in
the vtable (represented by the array vptr) for the class of p. The reason why p is passed
as an argument to the function (*(p->vptr[1])) is explained in the next subsection.
Arguments to Member Functions and this
There are several issues related to calls to member function, function parameters,
and the this pointer. Consider the following code, in which one virtual function calls
another:
class A {
public:
virtual int f (int x);
virtual int g(int y);
};
int A::f(int x) { . . . g(i) . . . ;}
int A::g(int y) { . . . f(j) . . . ;}
If virtual function f is redefined in a derived class B, then a call to the inherited g
on class B objects should invoke the new function B::f, not the original function A::f
defined for class A. Therefore, calls to one virtual function inside another must use a
vtable. However, the call to f inside A::g does not have the form p->f( . . . ), and it is
not clear at first glance what object p we would use if we wanted to change the call
from the simple f( . . . ) that appears above to p->f( . . . ).
One way of understanding the solution to this problem is to rewrite the code as
it is compiled. In other words, the preceding function A::g is compiled as if it were
written as
int A::g(A* this, int x) { . . . this->f(j) . . . ;}
with a new first parameter this to the function, called this, and the call f(j) replaced
with this->f(j). Now the call this->f(j) can be compiled in the usual way, by use of
the vtable pointer of this to find the code for f, provided that when g is called, the
appropriate object is passed as the value of this.
The calling sequence for a member function, whether it is virtual or not, passes the
object itself as the this pointer. For example, returning to the Pt and ColorPt example,
in code such as
ColorPt* q = new ColorPt(3,4);
q->darken(5);
12.3 Classes, Inheritance, and Virtual Functions
the call to darken is compiled as if it were a C function call:
(*(q->vptr[2]))(q,5);
This call shows the offset of darken in the vtable (assumed here to be 2) and the
call with the pointer to q passed as the first argument to the compiled code for the
member function.
There are several ways that the this pointer is used. As previously illustrated, the
this pointer is used to call virtual functions on the object. The this pointer is also used
to access data members of the object, whether there are virtual functions or not. The
this pointer is also used to resolve overloading, as described in the next subsection.
Scope Qualifiers
Because some of the calling conventions may be confusing, it is worth saying clearly
how names are interpreted in C++. There are three scope qualifiers. They are ::
(double colons), –> (right arrow, consisting of two ascii characters – and >), and .
(period or dot). These are used to qualify a member name with a class name, a pointer
to an object, or an object name, respectively.
The following rules are for resolving names:
A name outside a function or class, not prefixed by :: and not qualified, refers to
global object, function, enumerator, or type.
Suppose C is a class, p is a pointer to an object of class C, and o is an object of class
C. These might be declared as follows:
Class C : . . . { . . . . }; /* C is a class */
C *p = new C( . . . ); /* p is a pointer to an object of class C */
C o( . . . )
/* o is an object of class C */
Then the following qualified names can be formed with ::, –>, and .:
C::n /* Class name C followed by member name n */
p->n /* pointer p to object of class C followed by member name n */
o.n
/* pointer p to object of class C followed by member name n */
These refer to a member n of class C, or a base class of C if n is not declared in C
but is inherited from a base class.
Nonvirtual and Overloaded Functions
The C++ virtual function mechanism does not affect the way that the address of a
nonvirtual or overloaded function is determined by the C++ compiler. For nonvirtual
functions, the C++ calls work in exactly the same way as in C, except for the way that
the object itself is passed as the this pointer, as described in the preceding subsection.
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Objects and Run-Time Efficiency: C++
There are also some situations in which overloading and virtual function lookup may
interact or be confusing.
Recall that, if a function is not a virtual function, the compiler will know the
address of the function at compile time. (Those familiar with linking may realize
that this is not strictly true for separately compiled program units. However, linkers
effectively make it possible for compilers to be written as if the location of a function
is known at compile time.) Therefore, if a C++ program contains a call f(x), the
compiler can generate code that jumps to an address associated with the function f.
If f is an overloaded function and a program contains a call f(x), then the compiletime type associated with the parameter x will be used to decide, at compile time,
which function code for f will be called when this expression is executed at run time.
The difference between overloading and virtual function lookup is illustrated by
the following code. Here, we have two classes, a parent and a child class, with two
member functions in each class. One function, called printclass, is overloaded. The
other, called printvirtual, is a virtual function that is redefined in the derived class:
class parent {
public:
void printclass() {printf(“parent”);};
virtual void printvirtual() {printf(“parent”);};
};
class child : public parent {
public:
void printclass() {printf(“child”);};
void printvirtual() {printf(“child”);}; };
main() {
parent p; child c; parent *q;
p.printclass(); p.printvirtual(); c.printclass(); c.printvirtual();
q = &p; q->printclass(); q->printvirtual();
q = &c; q->printclass(); q->printvirtual();
}
The program creates two objects, one of each class. When we invoke the member functions of each class directly through the object identifiers c and p, we get
the expected output: The parent class functions print “parent” and the child class
functions print “child.” When we refer to the parent class object through a pointer
of type *parent, then we get the same behavior. However, something else happens
when we refer to the child class object through the pointer of type *parent. The call
q->printclass() always causes the parent class member to be called; the printclass function is not virtual so the type of the pointer q is used. The static type of q is *parent, so
overloading resolution leads to the parent class function. On the other hand, the call
q->printvirtual() will invoke a virtual function. Therefore, the output of this program is
parent parent child child parent parent parent child.
12.4 Subtyping
The call q->printclass() is effectively compiled as a call printclass(q), passing the
object q as the this pointer to printclass. Although the printclass function does not
need the this pointer in order to print a string, the argument is used to resolve
overloading. More specifically, q->printclass() calls the parent class function because
this call is compiled as printclass(q) and the type of the implicit argument q is used
by the compiler to choose which version of the overloaded printclass function to
call.
12.4 SUBTYPING
In principle, subtyping and inheritance are independent concepts. However, subtyping as implemented in C++ occurs only when inhertance is used. In this section, we
look at how subtyping-in-principle might work in C++ and compare this with the
form of subtyping used by the C++ type checker. Although the C++ type checker
is not as flexible as it conceivably could be, there are also some sound and subtle
reasons for some central parts of the C++ design.
12.4.1 Subtyping Principles
Subtyping for Classes. The main principle of subtyping is that, if A <: B, then we
should be able to use an A value when a B is required. For classes, subtyping requires
a superset relation between public members of the class. For example, we can see
that the colored-point class ColorPt defined in Subsection 12.3.2 is a subtype of the
point class Pt defined in Subsection 12.3.1 by a comparison of the public members of
the classes:
Pt:
int getX();
void move(int);
ColorPt: int getX();
int getColor();
void move(int);
void darken(int tint);
The superset relation between sets of public members is guaranteed by C++ inheritance: Because ColorPt inherits from Pt, ColorPt has all of the public members of Pt.
Because Pt is a public base class of ColorPt, the C++ type checker will allow ColorPt
objects to be assigned to Pt class pointers.
Subtyping for Functions. One type of function is substitutable for another if there
is a certain correspondence between the types of arguments and the types of results.
More specifically, if functions f and g return the same kind of result and function f can
be applied to every kind of argument to which function g can be applied, then we can
use function f in place of function g without type error. Similarly, if functions f and
g are applicable to the same type of arguments and the result type of f is a subtype
of the result type of g, then f can be used in place of g without type error. These two
subtyping ideas for functions can be combined in a relatively simple-looking rule
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that is hard for most people to understand and remember:
The function type A → B is a subtype of C → D whenever C <: A and B <: D.
It helps to look at a few examples. Let us assume that circle <: shape and consider
some function types involving circle and shape.
One reasonably straightforward relation is (circle → circle) <: (circle → shape): You
can understand this by assuming you have a program that calls a circle → shape function f:
shape f (circle x) { . . . return(y) ;}
main {
. . . f(circ) . . .
}
The assertion that circle → circle is a subtype of circle → shape means that it is type
safe to replace this f with another function that returns a circle instead of a shape.
In other words, where a program requires an circle → shape function, we would not
create a type error if we used a circle → circle function instead.
When we change the argument type of a function, the subtyping relation is reversed. For example, (shape → circle) <: (circle → circle): The reasoning involved
here is similar to the case we just discussed. If a program uses a function f of type
circle → circle correctly, then it may apply f to a circle argument. However, it would
work just as well to use a shape → circle function g instead, as our assumption
circle <: shape means that g could be applied to any circle argument without causing
a type error.
When these two basic ideas are combined, there are four function types involving
circle and shape. They are ordered as shown in the following illustration, in which
each type shown is a subtype of those above it:
It is worth taking some time to be sure you understand all of the relationships implied
by this diagram.
12.4.2 Public Base Classes
When a derived class B has public base class A, the rules of C++ inheritance guarantee that the type of B objects will be a subtype of the type of A objects. The
12.4 Subtyping
following basic rule guarantees that a derived class defines a subtype of its public base
class:
Standard C++ Inheritance: A derived class may redefine virtual members of
the base class and may add new members that are not defined in the base class.
However, the type and visibility (public, protected or private) of any redefined
member must be exactly the same in the base and the derived classes.
This rule was originally used in most implementations of C++, except for the clause
about maintaining visibility, which is added here to eliminate some awkward cases.
In principle, if the compiler follows systematic rules for setting the order of member data in objects and the order of virtual function pointers in the vtable, it is possible
to define compatible classes of objects without using inheritance. For example, consider the following two classes. The one on the left is the Pt class defined in Subsection
12.3.1, with protected and private members elided for brevity. The class on the right
is equivalent (for most compilers) to the ColorPt class defined in Subsection 12.3.2.
class Point {
public:
int getX();
void move(int);
protected: . . .
private:
...
};
class ColorPoint {
public:
int getX();
int getColor();
void move(int);
void darken(int);
protected: . . .
private:
...
}:
It has exactly the same public, private and protected members, in exactly the same
order, producing exactly the same object layout and exactly the same vtable. However, the C++ type checker will not recognize subtyping between this ColorPt class
and this Pt class because ColorPt does not use Pt as a public base class.
12.4.3 Specializing Types of Public Members
A more permissive rule than the standard C++ inheritance rule stated in Subsection
12.4.2 allows virtual functions to be given subtypes of their original types:
Permissive Inheritance: A derived class may redefine virtual members of the
base class and may add new members that are not defined in the base class.
The derived-class type of any redefined public member must be a subtype of
the type of this member in the base class. The visibility (public, protected, or
private) of each member must be the same in the derived class as in the base
class.
Although early C++ used standard C++ inheritance, contemporary implementations
of C++ use a rule that is halfway between the standard and the permissive inheritance
rules:
Contemporary C++ Inheritance: A derived class may redefine virtual members of the base class and may add new members that are not defined in the
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Objects and Run-Time Efficiency: C++
base class. The visibility (public, protected, or private) of any redefined member must be the same in the base and the derived classes. If a virtual member
function f has type A → B in a base class, then f may be given type A → C in a
derived class, provided C <: B.
This allows the return type to be replaced with a subtype, but does not allow changes
in the argument types of member functions. Because the argument types of functions
are used to resolve overloading, but the return types are not, this inheritance rule
does not interact with the C++ overloading algorithm.
12.4.4 Abstract Base Classes
In many object-oriented programs, it is useful to define general concepts, such as
container, account, shape, or vehicle, that are not implemented themselves, but are
useful as generalizations of some set of implemented classes. For example, in the
Smalltalk container class library discussed in Section 11.7, all of the classes define the
methods of the container interface, but the implemented containers come from more
specialized classes such as dictionary, array, set, or bag. Because subtyping follows
inheritance in C++, there is a specific C++ construct developed to meet the need
for classes that serve an organizational purpose but do not have any implemented
objects of their own.
An abstract class is a class that has at least one pure virtual member function.
A pure virtual member function is one whose implementation is declared empty. It
is not possible to construct objects of an abstract class. The purpose of an abstract
class is to define a common interface for one or more derived classes that are not
abstract.
For example, an abstract-class file can be used to define the operations open and
read that must be implemented in every kind of file class that uses file as a base class:
class file {
public:
void virtual Open() = 0; /* pure virtual member function */
void virtual Read() = 0; /* pure virtual member function */
//..
};
Another example appears in the geometry classes in Appendix B.1.5. The class shape
is abstract, defining the common interface of various kinds of shapes. Two derived
classes of shape are circle and rectangle.
Although it is not possible to create an instance of an abstract class like file or
shape, it is possible to declare a pointer with type file* or shape*. If pointer f has type
file*, then f can be assigned any object from any derived class of file. Another way
that abstract classes are used is as the argument types of functions. If some function
can be applied to any type of file, its argument can be declared to have type file*.
In addition to defining an interface, an abstract class establishes a layout for a
virtual function table (vtable). In particular, if textFile and streamFile have file as a
12.5 Multiple Inheritance
base class, then both classes will have open and read in the same positions in their
respective vtables.
The use of “abstract” in the term “abstract class” is not the same as in “abstract
data type.” In particular, an abstract class is not a class with a stronger form of
information hiding.
12.5 MULTIPLE INHERITANCE
Multiple inheritance is a controversial part of object-oriented programming. Once
we have adopted the idea of using inheritance to build new classes, it seems natural
to allow a class to be implemented by inheriting from more than one base class.
Unfortunately, multiple inheritance brings with it a set of problems that does
not have simple, elegant solutions. For this reason, many language designers tried to
avoid putting multiple inheritance in their language (like Java) or postponed adding
multiple inheritance as long as possible (like Smalltalk and C++).
Some of the most compelling examples of multiple inheritance involve combining
two or more independent kinds of functionality. For example, suppose we are doing
some geometric calculations and want to manage the memory allocated to various
geometric objects in various ways. We might have a class hierarchy of geometric
objects, such as shapes, circles, and rectangles, and a hierarchy of classes of memory
management mechanisms. For example, we might have ordinary C++ heap objects
that require users to call their destructors and garbage-collected objects that are
automatically reclaimed when some function invoking the garbage collector is called.
Another possibility that might be useful for some objects would be to associate a
reference count with each object (maintaining a count of the number of pointers
that refer to this object). This is a simple form of garbage collection that works more
efficiently when it is easy to maintain a count accurately. If we have separate geometry
and memory management hierarchies, then it will be possible to implement many
combinations of shape and memory by multiple inheritance. An example is shown
in Figure 12.2.
Here, reference-counted rectangles are implemented by inheriting the geometric aspects of rectangles from the rectangle class and the mechanism for reference
counting from the reference-counted class.
shape
reference
counted
rectangle
refCounted
Rectangle
Figure 12.2. Multiple inheritance.
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Objects and Run-Time Efficiency: C++
12.5.1 Implementation of Multiple Inheritance
The fundamental property of C++ class implementation is that if class B has class A
as a public base class, then the initial segment of every B object must look like an A
object, and similarly for the B and A virtual function tables. This property makes it
possible to access an inherited A member of a B object in exactly the same way we
would access the same member of an A object. Although it is fairly easy to guarantee
this property with only single inheritance, the implementation of objects and classes
with multiple inheritance is more complex.
The implementation issues are best illustrated by example. Suppose class C is
defined by inheriting from classes A and B, as in the following code:
class A {
public:
int x;
virtual void f();
};
class B {
public:
int y;
virtual void g();
virtual void f();
};
class C: public A, public B {
public:
int z;
virtual void f();
};
C *pc = new C;
B *pb = pc;
A *pa = pc;
In the final three lines of the sample code, we have three pointers to the same object.
However, the three pointers have three different static types.
The representation of the C object created from the preceding example class C and
the values of the associated pointers are illustrated in Figure 12.3. The illustration
shows three pointers to the class C object. Pointers pa and pc point to the top of the
object and pointer pb points to a position δ locations below the top of the object. The
class C object has two virtual function table pointers and three sections of member
data, one for member data declared in the A class, one for member data declared in
the B class, and one for member data declared in the C class. The two virtual function
table pointers point to two different vtables whose use is subsequently explained.
The top vtable, labeled C-as-A, is used when a C object is used as an object of type C
or as an object of type A. The lower vtable, labeled C-as-B, is used when a C object is
used as an object of type B and to call C member functions that are inherited from B
(in this case, function g).
12.5 Multiple Inheritance
C object
pa, pc
δ
pb
C-as-A vtbl
& C::f
vptr
0
A
A data
C-as-B vtbl
vptr
B
B data
& B::g
0
& C::f
δ
C data
Figure 12.3. Object and vtable layout for multiple inheritance.
Figure 12.3 shows three complications associated with the implementation of multiple
inheritance in C++:
There are multiple virtual function tables for class C, not just one.
A pointer of static type B points to a different place in the object than an A or C
pointer.
There are some extra numbers stored in the vtable, shown here to the right of the
address & C::f of a virtual function.
There are two vtables for class C because a C object may be used as an A object or as a
B object. In symbols, we want C <: A and C <: B. For this to work, the C virtual function
table must have an initial segment that looks like an A virtual function table and an
initial segment that looks like a B virtual function table. However, because A and B
are independent classes, there is no reason why A and B should have similar vtables.
The solution is to provide two vtables, one that allows a C object to be used as an
A object, and another that allows a C object to be used as a B object. In principle, if
a derived class has n base classes, the class will have n+1 vtables, one for each base
class and one for the derived class itself. However, in practice, only n vtables are
needed, as the vtable for the derived class can follow the order of the vtable for
one of the base classes, allowing the vtable for the derived class to be the same as
the vtable used for viewing objects of the derived class as objects of one of the base
classes.
There is a similar issue surrounding the arrangement of data members within a
C object. As we can see from Figure 12.3, the pointer pb, referring to a C object as
a B object, points to a different place in the object than pa and pc. When a C object
is used as a B object, the member data must be accessed in exactly the same way as
if the object had been created as a B object. This is accomplished when the member
data inherited from B are arranged in a contiguous part of the object, below the A
data. When a pointer to a C object is converted or cast to type B, the pointer must
be incremented by some amount δ to skip over the A part and point directly to the B
part of the object.
The number δ , which is the numeric difference pb – pa, is stored in the B table, as it
is also used in calling some of the member functions. The offset δ in this vtable is used
in call to pb->f(), as the virtual function f defined in C may refer to A member data
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that are above the pointer pb. Therefore, on the call to inherited member function f,
the this pointer passed to the body of f must be δ less than the pointer pb.
12.5.2 Name Clashes, Diamond Inheritance, and Virtual Base Classes
Some interesting and troubling issues arise in connection with multiple inheritance.
The simplest and most pervasive problem is the problem of name clashes, which
arises when two base classes have members of the same name. A variant of the name
clash problem arises when a class C inherits from two classes, A and B, that inherit the
same member from a common base class, D. In this case, the inherited members of
C with the same name will have the same definition. This eliminates some problems.
However, there is still the issue of deciding whether to place two copies of doubly
inherited member data in a derived class. By default, C++ duplicates members that
are inherited twice. The virtual base-class mechanism provides an alternative way of
sharing common base classes in which doubly inherited members occur only once in
the derived class.
Name Clashes
If class C inherits from classes A and B and A and B have members of the same name,
then there is a name clash. There are three general ways of handling name clashes:
Implicit resolution: The language resolves name conflicts with an arbitrary rule.
Explicit resolution: The programmer must explicitly resolve name conflicts in the
code in some way.
Disallow name clashes: Programs are not allowed to contain name clashes.
There is no systematic best solution to the problem of name clashes. The simplest
solution is to disallow name clashes. However, there are some situations, such as the
diamond inheritance patterns subsequently described, for which this seems unnecessarily restrictive.
Implicit resolution also has its problems. Python and the CLOS, for example,
have mechanisms that give priority to one base class over another. This approach
does not work well in some situations. For example, suppose class C inherits from
classes A and B and suppose that both A and B have a member function named f. If
class C is declared so that A has priority over B, then class C will inherit the member
function f from A and not inherit the member f from B. However, suppose that class B
has another member function g that calls member function f and relies on a particular
definition of f in order to work properly. Then, in effect, inheriting f from class A will
cause the inherited member g from B not to work properly.
C++ allows name clashes and requires explicit resolution in the program. If a data
member m is inherited from two bases classes A and B, then there will be two data
members in the derived class. If a member m is inherited from two bases classes A
and B, then the only way to refer to member m is by a fully qualified name. This is
illustrated in the following C++ code samples.
The following code contains a compile-time error because the call to an inherited
member function is ambiguous:
12.5 Multiple Inheritance
class A {
public:
virtual void f() { . . . }
};
class B {
public:
virtual void f() { . . . }
};
class C : public A, public B { };
...
C* p;
p->f(); // error since call to member function is ambiguous
In C++, it is not an error to inherit from two classes containing the same member
names. The error occurs only when the compiler sees a reference to the doubly
inherited member. The preceding program fragment can be rewritten to avoid the
error, by use of fully qualified names. Classes A and B remain the same; only the
definition of class C needs to be changed to specify which inherited member function
f should be called:
class C : public A, public B {
public:
virtual void f( ) {
A::f( ); // Call member function f from A, not B::f();
}
...
};
Here, class C explicitly defines a member function f. However, the definition of C::f
is simply to call A::f. This makes the definition of C::f unambiguous and has the same
effect as inheriting A::f by some other name resolution rule.
Window (D)
Text Window
(A)
Graphics
Window (B)
Text, Graphics
Window (C)
Figure 12.4. Diamond inheritance.
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Diamond Inheritance
An interesting kind of name clash occurs when two classes share a common base
class. In this situation, a member that is inherited twice will have the same definition in both classes. The complications associated with this situation are commonly
called the diamond inheritance problem, as this situation involves a diamond-shaped
inheritance graph. For example, consider the four classes shown in Figure 12.4.
The classes are labeled A, B, C, and D because these shorter labels fit better in
subsequent illustrations.
Under ordinary multiple inheritance, there are two copies of window members in
a text graphics window. More specifically, the data members of a class C object have
the following form:
D part
inherited from class A
A part
D part
B part
C part
inherited from class B
defined in class C
A class C object has all of these parts because a class C object has parts for both of
the base classes, A and B. However, classes A and B both inherit from D, so A objects
have a copy of each data member inherited from D and B objects have a copy of each
data member inherited from D.
For window classes, it does not make sense for a text and graphics window to have
two copies of each window-class member. In a windowing system with this design,
the window class might contain data members to store the bounding box of the
window, data members to determine the background color and border color of the
window, and so on. The text window class might have additional data members and
member functions to store and display text inside a window, and similarly the graphics
window class would contain declarations and code for displaying graphics inside a
window. If we want a window that can contain graphics and text within the same
window, then C++ multiple inheritance as described so far will not work. Instead,
the C++ implementation of multiple inheritance will produce objects that consist
of two windows, one capable of displaying text and the other capable of displaying
graphics.
The diamond inheritance problem is the basic problem that there is no “right”
way to handle multiple inheritance when two base classes of a class C both have the
same class D as a base class. This is called the diamond inheritance problem because
the classes A, B, C, and D form a diamond shape, as shown in Figure 12.4. The biggest
problem with diamond inheritance is what to do with member data. One option is to
duplicate the D class data members. This can give useful behavior in some cases, but is
undesirable in situations like the window classes in Figure 12.4. If data members are
duplicated, then there is also a naming problem: If you select an inherited D member
12.5 Multiple Inheritance
Class A object
Class B object
Class C object
A part
B part
A part
D part
D part
B part
C part
D part
Figure 12.5. Virtual base class.
by name, which copy should you get? The C++ solution is to require a fully qualified
name.
The alternative to duplicating data members is to put only one copy of each
D member into a C object and have the inherited A and B class member functions
both refer to the shared copy. This also can lead to problems if the A and B classes
may use D data members differently. For example, some A member function may be
written assuming an inherited D class data member is an even number, whereas some
B class member function may set it to an odd number. Diamond inheritance leads to
problems that do not have a simple, generally applicable solution.
Virtual Base Classes
C++ has a mechanism for eliminating multiple copies of duplicated base-class members, called virtual base classes. If D is declared a virtual base class of A and B, and
class C has A and B as base classes, then A, B, and C class objects will have the form
illustrated in Figure 12.5.
The main idea is that if class A has a virtual base class D, then access to D data
members of an A object is by indirection through a pointer in the object. As a result,
the location of the D part does not matter. The D part could immediately follow the
A part of the object or reside someplace else in memory. This indirection through a
pointer allows C objects to have an A part and a B part that share a common D part.
Inheritance with virtual base classes has its complications. One annoyance is that
the choice between virtual and nonvirtual base classes is made once for all uses of a
class. More specifically, suppose that one class, A, has another class, D, as a base class.
Then the designer of A must decide whether D should be a virtual base class or not,
even though this affects only other classes that inherit from A. Moreover, if some
classes that inherit from A need D to be a virtual base class of A and others require the
opposite, then the only solution is to make two versions of the A class, one with D as a
virtual base class and the other with D as a nonvirtual base class. An oddity of virtual
base classes is that it is possible to convert a pointer to an instance of a class that has
a virtual base class to a pointer to an object of that virtual base class. However, the
opposite conversion is not allowed, making this form of type conversion irreversible.
For this reason, some style guides recommend against casts when virtual base classes
are used. This can be a complex guideline to follow, as users of a class library must
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pay careful attention to when virtual base classes are used in the implementation of
the library.
12.6 CHAPTER SUMMARY
The main goal of C++ is to provide object-oriented features in a C-based language
without compromising the efficiency of C. C++ is a very carefully designed language
that has succeeded admirably in spite of difficult design constraints.
Some features of C++ that are not part of the object system are type bool, reference types and pass-by-reference, user-defined overloading, function templates, and
exceptions. C++ also has better static type checking than C.
The focus on C++ efficiency is articulated in the design principle,“If you do not
use a feature, you should not pay for it.” This principle suggests that C programs
should compile as efficiently under the C++ compiler as under the C compiler. It
would violate this principle to implement C integers as objects, for example, and use
dynamic method lookup to find integer functions at run-time as in Smalltalk, as this
would significantly reduce the performance of C integer calculations.
The most significant part of C++ is the set of object-oriented concepts added to
C; these are the main concepts:
Classes, which declare the type associated with all objects constructed from the
class, the data members of each object, and the member functions of the class.
Objects, which consist of private data and public functions for accessing the hidden
data, as in other object-oriented languages.
Dynamic lookup, for member functions that are declared virtual. A virtual function in a derived class (subclass) may be implemented differently from virtual
functions of the same name in a base class (superclass).
Encapsulation, based on programmer designations public, private, and protected
that determine whether data and functions declared in a class are visible outside
the class definition.
Inheritance, using subclassing, in which one class may be defined by inheriting the
data and functions declared in another. C++ allows single inheritance, in which a
class has a single base class (superclass) or multiple inheritance in which a class
has more than one base class.
Subtyping, based on subclassing. For one class to define a subtype of the type
defined by another class, inheritance must be used. However, the programmer
may decide whether inheritance results in a subtype or not by specifying that the
base class is public or not.
At run time, C++ virtual function lookup uses indirection through the virtual function
table (vtable) of the class. The C++ static type system makes it possible to determine
the offset (position of a pointer in the vtable) of virtual function at compile time.
When a call like p->move(. . . ) is compiled, the compiler can determine a static type
for the pointer p. This static type must be a class, and this class must declare or
inherit a function called move. The compiler can examine the class hierarchy to see
what location the virtual function move will occupy in the vtable for the class.
A fundamental property of C++ class implementation is that if class B has class
A as a public base class, then the initial segment of every B object must look like an
Exercises
A object, and similarly for the B and A virtual function tables. This property makes it
possible to access an inherited A member of a B object in exactly the same way we
would access the same member of an A object. Although it is fairly easily to guarantee
this property with only single inheritance, the implementation of objects and classes
with multiple inheritance is more complex.
There are a number of aspects of C++ that programmers occasionally find difficult.
Some of the main problem areas are
casts and conversions that can be complex and unpredictable in certain situations,
objects allocated on the stack and other aspects of object memory management,
overloading, a complex code selection mechanism in C++ that can interact unpredicatably with dynamic lookup (virtual function lookup),
multiple inheritance, which is more complex in C++ than in other languages because of the way objects and virtual function tables are configured and accessed.
These problem areas exist not because of oversights but because the goals of C++,
followed to their logical conclusion, led to a design with these properties.
EXERCISES
12.1 Assignment and Derived Classes
This problem exams the difference between two forms of object assignment. In
C++, local variables are stored on the run-time stack, whereas dynamically allocated data (created with the new keyword) are stored on the heap. A local object
variable allocated on the stack has an object L-value and an R-value. Unlike
many other object-oriented languages, C++ allows object assignment into object
L-values.
A C++ programmer writes the following code:
class Vehicle {
public:
int x;
virtual void f();
void g();
};
class Airplane : public Vehicle {
public:
int y;
virtual void f();
virtual void h();
};
void inHeap() {
Vehicle *b1 = new Vehicle;
// Allocate object on the heap
Airplane *d1 = new Airplane;
// Allocate object on the heap
b1->x = 1;
d1->x = 2;
d1->y = 3;
b1 = d1;
// Assign derived class object to base class pointer
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}
void onStack() {
Vehicle b2;
Airplane d2;
b2 . x = 4;
d2 . x = 5;
d2 . y = 6;
b2 = d2;
}
// Local object on the stack
// Local object on the stack
// Assign derived class object to base class variable
int main() {
inHeap();
onStack();
}
(a) Draw a picture of the stack, heap, and vtables that result after objects b1 and
d1 have been allocated (but before the assignment b1=d1) during the call to
inHeap. Be sure to indicate where the instance variables and vtable pointers of
the two objects are stored before the assignment b1=d1 and to which vtables
the respective vtable pointers point.
(b) Redraw your diagram from (a), showing the changes that result after the
assignment b1=d1. Be sure to clearly indicate where b1’s vtable pointer points
after the assignment b1=d1. Explain why b1’s vtable pointer points where it
does after the assignment b1=d1.
(c) Draw a picture of the stack, heap, and vtables that result after objects b2 and
d2 have been allocated (but before the assignment b2=d2) during the call to
onStack. Be sure to indicate where the instance variables and vtable pointers
of the two objects are stored before the assignment b2=d2 and to which vtables
the respective vtable pointers point.
(d) In C++, assignment to objects (such as b2=d2) is performed by copying into
all the left object’s data members (b2’s in our example) from the right object’s
(d2’s) corresponding data members. If the right object contains data members
not present in the left object, then those data simply are not copied. In b2=d2
this means overwriting all b2’s data members with the corresponding value
from d2.
i. Re-draw your diagram from (c), showing the changes that result after the
assignment b2=d2.
ii. Explain why it is not sensible to copy all of d2’s member data into b2’s
record.
iii. Explain why b2’s vtable pointer is not changed by the assignment.
(e) We have used assignment statements b1=d1 and b2=d2. Why are the opposite
statements d1=b1 and d2=b2 not allowed?
12.2 Function Objects
The C++ Standard Template Library (STL) was discussed in an earlier chapter.
In STL terminology, a function object is any object that can be called as if it
is a function. Any object of a class that defines operator() is a function object. In
Exercises
addition, an ordinary function may be used as a function object, as f() is meaningful
is f is a function and similarly if f is a function pointer.
Here is a C++ template for combining an argument type, return type, and
operator() together into a function object. Every object from every subclass of
FuncObj<A,B>, for any types A and B is a function object, but not all function objects
come from such classes:
template <typename Arg, typename Ret>
class FuncObj {
public:
typedef Arg argType;
typedef Ret retType;
virtual Ret operator()(Arg) = 0;
};
Here are two example classes of function objects. In the first case, the constructor
stores a value in a protected data field so that different function objects from
this class will divide by different integers. In a sense to be explored later in this
problem, instances of DivideBy are similar to closures as they may contain hidden
data:
class DivideBy : public FuncObj<int, double> {
protected:
int divisor;
public:
DivideBy(int d) {
this->divisor = d;
}
double operator()(int x) {
return x/((double)divisor);
}
};
class Truncate : public FuncObj<double, int> {
public:
int operator()(double x) {
return (int) x;
}
};
Because DivideBy uses the FuncObj template as a public base class, DivideBy is a
subtype of FuncObj<int, double>, and similarly for Truncate.
(a) Fill in the blanks to complete the following Compose template. The Compose
constructor takes two function objects f and g and creates a function object
that computes their composition λx. f(g(x)).
You will need to fill in type declarations on lines 9 and 13 and code on
lines 10, 11, and 14. If you do not know the correct C++ syntax, you may use
short English descriptions for partial credit. (We have double-checked the
parentheses to make sure they are correct.)
To give you a better idea of how Compose is supposed to work, you may
want to look at the following sample code that uses Compose to compose
DivideBy and Truncate:
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Objects and Run-Time Efficiency: C++
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
template < typename Ftype , typename Gtype >
class Compose :
public FuncObj < typename Gtype : : argType,
typename Ftype : : retType > {
protected:
Ftype *f;
Gtype *g;
public:
Compose(
f,
g) {
=f;
=g;
}
operator()(
x) {
return (
)((
)(
));
}
};
void main() {
DivideBy *d = new DivideBy(2);
Truncate *t = new Truncate();
Compose<DivideBy, Truncate> *c1
= new Compose<DivideBy, Truncate>(d,t);
Compose<Truncate, DivideBy> *c2
= new Compose<Truncate, DivideBy>(t,d);
cout << (*c1)(100.7) << endl; // Prints 50.0
cout << (*c2)(11) << endl;
// Prints 5
}
(b) Consider code of the following form, in which A, B, C, and D are types that
might not all be different (i.e., we could have A = B = C = D = int or A, B, C, and
D might be four different types):
class F : public FuncObj<A, B> {
...
};
class G : public FuncObj<C, D> {
...
};
F *f = new F( . . . );
G *g = new G( . . . );
Compose<F, G> *h
= new Compose<F, G>(f,g);
cout << (*h)( . . . ) << endl; // call compose function
What will happen if the return type D of g is not the same as the argument
type A of f? If it is possible for an error to occur and possible for an error not
to occur, say which conditions will cause an error and which will not. If it is
possible for an error to occur at compile time or at run time, say when the
error will occur and why.
Exercises
(c) Our Compose template is written assuming that Ftype and Gtype are classes
that have argType and retType type definitions. If you wanted to define a
Compose template that works for all function objects, what arguments would
your template have and why? Your answer should be in the form of “the
types of variables . . . , the arguments of functions . . . , and the return values
of . . . . ” Assume that the code in lines 10, 11, and 14 stays the same. All we
want to change are the template parameters on line 1 and possibly the parts
of the body of the template that refer to template parameters.
The rest of this problem asks about the similarities and differences between
C++ function objects and function closures in languages like Lisp, ML, and
Scheme. In studying scope and activation records, we looked at a function
makeCounter that returns a counter, initialized to some integer value. Here is
makeCounter function in Scheme:
(define makeCounter
(lambda (val)
(let (( counter (lambda (inc) (set! val (+ val inc)) val)) )
counter)))
Here is part of an interactive session in which makeCounter is used:
==> (define c (makeCounter 3))
#<unspecified>
==> (c 2)
5
==> (c 2)
7
The first input defines a counter c, initialized to 3. The second input line adds
2, producing value 5, and the third input line adds 2 again, producing value 7.
A general idea for translating Lisp functions into C++ function objects
is to translate each function f into a class A so that objects from class A are
function objects that behave like f. If f is defined within some nested scope,
then the constructor for A will put the global variables of f into the function
object, so that an instance of A behaves like a closure for f.
Because makeCounter does not have any free variables, we can translate
makeCounter into a class MAKECOUNTER that has a constructor with no parameters:
class MAKECOUNTER {
public:
class COUNTER {
protected:
int val;
public:
COUNTER(int init){val = init;}
int operator()(int inc) {
val = val + inc;
return val;
}
};
MAKECOUNTER(){}
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Objects and Run-Time Efficiency: C++
COUNTER* operator()(int val) {
return new COUNTER(val);
}
};
We can create a MAKECOUNTER function object and use it to create a COUNTER
function object as follows:
MAKECOUNTER *m = new MAKECOUNTER();
MAKECOUNTER : : COUNTER * c = (*m)(3);
cout << (*c)(2) << endl; // Prints 5
(d) Thinking generally about Lisp closures and C++ function objects, describe
one way in which C++ function objects might serve some programming objectives better than Lisp closures.
(e) Thinking generally about Lisp closures and C++ function objects, describe
one way in which Lisp closures might serve some programming objectives
better than C++ function objects.
(f) Do you think this approach sketched in this problem will allow you to translate an arbitrary nesting of Lisp or ML functions into C++ function objects?
You may want to consider the following variation of MAKECOUNTER in which
the counter value val is placed in the outer class instead of in the inner class:
class MAKECOUNTER {
protected:
int val;
public:
class COUNTER {
private:
MAKECOUNTER *mc;
public:
COUNTER(MAKECOUNTER* mc, int init) {
this->mc = mc;
mc->val = init;
}
int operator()(int inc) {
mc->val = mc->val + inc;
return mc->val;
}
};
friend class COUNTER;
MAKECOUNTER(){}
COUNTER* operator()(int val) {
return new COUNTER(this,val);
}
};
Exercises
12.3 Function Subtyping
Assume that A <: B and B <: C. Which of the following subtype relationships
involving the function type B → B hold in principle?
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(B → B)
(B → A)
(B → C)
(C → B)
(A → B)
(C → A)
(A → A)
(C → C)
<:
<:
<:
<:
<:
<:
<:
<:
(B → B)
(B → B)
(B → B)
(B → B)
(B → B)
(B → B)
(B → B)
(B → B)
12.4 Subtyping and Public Data
This question asks you to review the issue of specializing the types of public
member functions and consider the related issue of specializing the types of public
data members. To make this question concrete, we use the example of circles and
colored circles in C++. Colored circles get darker each time they are moved. We
assume there is a class Point of points and class ColPoint of colored points with
appropriate operations. Class Point is a public base class of ColPoint. The basic
definition for circle is:
class Circle {
public:
Circle(Point* c, float r) { center=c; radius=r; }
Point* center;
float radius;
virtual Circle* move(float dx, float dy)
{center->move(dx,dy); return(this);}
};
For each of the following definitions of colored circle, explain why a colored circle
should or should not be considered a subtype of a circle, in principle. If a colored
circle should not be a subtype, give some fragment of code that would be type
correct if we considered colored circles to be a subtype of circles, but would lead
to a type error at run time:
(a) class ColCircle : public Circle {
public:
ColCircle(Point* c, float r, color cl) : Circle(c,r) { col=cl; }
color col;
virtual Circle* move(float dx, float dy)
{center->move(dx,dy); darken(); return(this);}
virtual void darken()
{if (col==green) col=darkgreen;}
};
(b) Suppose we change the definition of ColCircle so that move returns a colored
circle:
class ColCircle : public Circle {
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Objects and Run-Time Efficiency: C++
public:
ColCircle(Point* c, float r, color cl) : Circle(c,r) { col=cl; }
color col;
virtual ColCircle* move(float dx, float dy)
{center->move(dx,dy); darken(); return(this);}
virtual void darken()
{if (col==green) col=darkgreen;}
};
(c) Suppose that instead of representing the color of a circle by a separate data
member, we replace points with colored points and use a colored point for
the center of the circle. One advantage of doing so is that we can use all of
the color operations provided for colored points, as illustrated by the darken
member function below. This could be useful if we wanted to have the same
color operations on a variety of geometric shapes:
class ColCircle : public Circle {
public:
ColCircle(ColPoint* c, float r) : Circle(c,r) { }
ColPoint* center;
virtual ColCircle* move(float dx, float dy)
{center->move(dx,dy); darken(); return(this);}
virtual void darken()
{center->darken();}
};
The important issue is the redefinition of the type of center. Assume that each
ColCircle object has center and radius data fields at the same offset as the center
and radius fields of a Circle object. In the modified version of C++ considered
in this question, the declaration ColPoint* center does not mean that a ColCircle
object has two center data fields.
12.5 Phantom Members
A C++ class may have virtual members that may be redefined in derived classes.
However, there is no way to “undefine” a virtual (or nonvirtual) member. Suppose
we extend C++ by adding another kind of member, called a phantom member,
that is treated as virtual, but only defined in derived classes if an explicit definition
is given. In other words, a “phantom” function is not inherited unless its name is
listed in the derived class. For example, if we have two classes
class A {
...
public:
phantom void f(){. . . }
...
};
class B : public A {
...
Exercises
public:
. . . /* no definition of f */
};
then f would appear in the vtbl for A objects and, if x is an A object, x . f() would
be allowed. However, if f is not declared in B, then f might not need to appear
in the vtbl for B objects and, if x is a B object, x . f() would not be allowed. Is
this consistent with the design of C++, or is there some general property of the
language that would be destroyed? If so, explain what this property is and why it
would be destroyed.
12.6 Subtyping and Visibility
In C++, a virtual function may be given a different access level in a derived class.
This produces some confusing situations. For example, this is legal C++, at least
for some compilers:
class Base {
public:
virtual int f();
};
class Derived: public Base {
private:
virtual int f();
};
This question asks you to explain why this program conflicts with some reasonable
principles, yet somehow does not completely break the C++ type system.
(a) In C++, a derived class D with public base class B is treated as a subtype of
B. Explain why this is generally reasonable, given the definition of subtyping
from class.
(b) Why would it be reasonable for someone to argue that it is incorrect to allow
a public member inherited from a public base class to be redefined as private?
(c) A typical use of subtyping is to apply a function that expects an B argument
to a D when D <: B. For example, here is a simple “toy” program that applies a
function defined for base-class objects to a derived-class object. Explain why
this program compiles and executes for the preceding Base and Derived classes
(assuming we have given an implementation for f):
int g(Base &x) {
return(x.f()+1);
}
int main() {
Base b;
cout << “g(b) = ” << g(b) << endl;
Derived d;
cout << “g(d) = ” << g(d) << endl;
}
(d) Do you think there is a mistake here in the design of C++ ? Briefly explain
why or why not.
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12.7 Private Virtual Functions
At first thought, the idea of a private virtual function might seem silly: Why would
you want to redefine a function that is not visible outside the class? However, there
is a programming idiom that makes use of private virtual functions in a reasonable
way. This idiom is illustrated in the following code, which uses a public function
to call a private virtual function:
class Computer {
public:
void retailPrice(void) {
int p = 2 * manufactureCost();
printf(“$ %d \ n”, p); // Print p
}
private:
virtual int manufactureCost(void) {return 1000;}
};
class IBMComputer: public Computer {
private:
virtual int manufactureCost(void) {return 1500;}
};
int main(void) {
Computer *cPtr = new Computer();
IBMComputer *ibmPtr = new IBMComputer();
Computer *cibmPtr = new IBMComputer();
cPtr->retailPrice();
ibmPtr->retailPrice();
cibmPtr->retailPrice();
return 0;
}
This question asks about how the function calls to retailPrice() are evaluated.
(a) Explain which version of manufactureCost() is used in the call cibmPtr->
retailPrice() and why.
(b) If the virtual keyword is omitted from the declaration of manufactureCost in
the derived class IBMComputer, the preceding code will still compile without
error and execute. Will manufactureCost be implemented as a virtual function
in class IBMComputer? Use your knowledge of how C++ is implemented to
explain why or why not.
(c) It is possible for the private base-class function to be declared public in the
derived class. Does this conflict with subtyping principles? Explain why or
why not in a few words.
12.8 “Like Current” in Eiffel
Eiffel is a statically typed object-oriented programming language designed by
Bertrand Meyer and his collaborators. The language designers did not intend
the language to have any type loopholes. However, there are some problems
Exercises
surrounding an Eiffel type expression called like current. When the words like
current appear as a type in a method of some class, they mean, “the class that
contains this method.” To give an example, the following classes were considered
statically type correct in the language Eiffel.
Class Point
x : int
method equals (pt : like current) : bool
return self.x == pt.x
class ColPoint inherits Point
color : string
method equals (cpt : like current) : bool
return self.x == cpt.x and self.color == cpt.color
In Point, the expression like current means the type Point, whereas in ColPoint, like
current means the type ColPoint. However, the type checker accepts the redefinition
of method equals because the declared parameter type is like current in both cases.
In other words, the declaration of equals in Point says that the argument of p.equals
should be of the same type as p, and the declaration of equals in ColPoint says the
same thing. Therefore the types of equals are considered to match.
(a) Using the basic rules for subtyping objects and functions, explain why ColPoint
should not be considered a subtype of Point “in principle.”
(b) Give a short fragment of code that shows how a type error can occur if we
consider ColPoint to be a subtype of Point.
(c) Why do you think the designers of Eiffel decided to allow subtyping in this
case? In other words, why do you think they wanted like current in the language?
(d) When this error was pointed out (by W. Cook after the language had been in
use for several years), the Eiffel designers decided not to remove like current,
as this would “break” lots of existing code. Instead, they decided to modify
the type checker to perform whole-program analysis. More specifically, the
modified Eiffel type checker examined the whole Eiffel program to see if
there was any statement that was likely to cause a type error.
i. What are some of the disadvantages of whole-program analysis? Do not
just say, “it has to look at the whole program.” Instead, think about trying
to debug a program in a language in which the type checker uses wholeprogram analysis. Are there any situations in which the error messages
would not be as useful as in traditional type checking in which the type of
an expression depends only on the types of its parts?
ii. Suppose you were trying to design a type checker that allows safe uses
of like current. What kind of statements or expressions would your type
checker look for? How would you distinguish a type error from a safe use
of like current?
12.9 Subtyping and Specifications
In the Eiffel programming language, methods can have preconditions and postconditions. These are Boolean expressions that must be true before the method
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is called and true afterwards. For example, the pop method in the following Stack
class has a precondition size > 0. This means that in order for the pop method to
execute properly, the stack should be nonempty beforehand. The postcondition
size’ = size - 1 means that the size after executing pop will be one less than the
size before this method is called. The syntactic convention here is that a “primed
variable,” such as size’, indicates the value of that variable after execution of the
method:
class Stack {
...
int size
...
void pop() pre: size > 0
post: size’ = size - 1 {
...
}
...
}
This question asks you about subtyping when preconditions and postconditions
are considered part of the type of a method.
(a) In an implementation of stacks in which each stack has a maximum size, given
by a constant MAX SIZE, the push method might have the following form:
class FixedStack {
...
int size
...
void push() pre: size < MAX SIZE {
...
}
...
}
whereas without this size restriction, we could have a stack class
class Stack {
...
int size
...
void push() pre: true {
...
}
...
}
with the precondition of push always satisfied. (For simplicity, there are no
postconditions in this example.) If this is the only difference between the
two classes, which one should be considered a subtype of the other? Explain
briefly.
(b) Suppose that we have two classes that are identical except for the preconditions and postconditions on one method:
Exercises
class A {
...
int size
...
void f() pre: PA, post: TA {
...
}
...
}
class B {
...
int size
...
void f() pre: PB, post: TB {
...
}
...
}
What relationships among PA, TA, PB, and TB should hold in order to have
B <: A? Explain briefly.
12.10 C++ Multiple Inheritance and Casts
An important aspect of C++ object and virtual function table (vtbl) layout is that if
class D has class B as a public base class, then the initial segment of every D object
must look like a B object, and similarly for the D and B virtual function tables.
The reason is that this makes it possible to access any B member data or member
function of a D object in exactly the same way we would access the B member data
or member function of a B object. Although this works out fairly easily with only
single inheritance, some effort must be put into the implementation of multiple
inheritance to make access to member data and member functions uniform across
publicly derived classes.
Suppose class C is defined by inheriting from classes A and B:
class A {
public:
int x;
virtual void f();
};
class B {
public:
int y;
virtual void f();
virtual void g();
};
class C : public A, public B {
public:
int z;
virtual void f();
};
C *pc = new C; B *pb = pc; A *pa = pc;
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Objects and Run-Time Efficiency: C++
and pa, pb, and pc are pointers to the same object, but with different types. The
representation of this object of class C and the values of the associated pointers
are illustrated in this chapter.
(a) Explain the steps involved in finding the address of the function code in the
call pc->f(). Be sure to distinguish what happens at compile time from what
happens at run time. Which address is found, &A : : f(), &B : : f(), or &C : : f()?
(b) The steps used to find the function address for pa->f() and to then call it are
the same as for pc->f(). Briefly explain why.
(c) Do you think the steps used to find the function address for and to call pb->f()
have to be the same as the other two, even though the offset is different?
Why or why not?
(d) How could the call pc->g() be implemented?
12.11 Multiple Inheritance and Thunks
Suppose class C is defined by inheriting from classes A and B:
class A {
public:
virtual void g();
int x;
};
class B {
public:
int y;
virtual B* f();
virtual void g();
};
class C : public A, public B {
public:
int z;
virtual C* f();
virtual void g();
};
C *pc = new C; B *pb = pc; A *pa = pc;
and pa, pb, and pc are pointers to the same object, but with different types.
Then, pa and pc will contain the same value, but pb will contain a different
value; it will contain the address of the B part of the C object. The fact that pb
and pc do not contain the same value means that in C++, a cast sometimes has a
run-time cost. (In C this is never the case.)
Note that in our example B : : f and C : : f do not return the same type. Instead,
C : : f returns a C* whereas B : : f returns a B*. That is legal because C is derived from
B, and thus a C* can always be used in place of a B*.
However, there is a problem: When the compiler sees pb->f() it does not know
whether the call will return a B* or a C*. Because the caller is expecting a B*,
the compiler must make sure to return a valid pointer to a B*. The solution is to
have the C-as-B vtable contain a pointer to a thunk. The thunk calls C : : f, and then
adjusts the return value to be a B*, before returning.
Exercises
(a) Draw all of the vtables for the classes in these examples. Show to which
function each vtable slot points.
(b) Does the fact that casts in C++ sometimes have a run-time cost, whereas C
never does, indicate that C++ has not adhered to the principles given in class
for its design? Why or why not?
(c) Because thunks are expensive and because C++ charges programmers only
for features they use, there must be a feature, or combination of features, that
are imposing this cost. What feature or features are these?
12.12 Dispatch on State
One criticism of dynamic dispatch as found in C++ and Java is that it is not flexible
enough. The operations performed by methods of a class usually depend on the
state of the receiver object. For example, we have all seen code similar to the
following file implementation:
class StdFile {
private:
enum { OPEN, CLOSED } state; /* state can only be either OPENor CLOSED */
public:
StdFile() { state = CLOSED; }
void Open() {
if (state == CLOSED) {
/* open file . . . */
state = OPEN;
} else {
error “file already open”;
}
}
void Close() {
if (state == OPEN ) {
/* close file . . . */
state = CLOSED;
} else {
error “file not open”;
}
}
/* initial state is closed */
}
Each method must determine the state of the object (i.e., whether or not the
file is already open) before performing any operations. Because of this, it seems
useful to extend dynamic dispatch to include a way of dispatching not only on the
class of the receiver, but also on the state of the receiver. Several object-oriented
programming languages, including BETA and Cecil, have various mechanisms to
do this. In this problem we examine two ways in which we can extend dynamic
dispatch in C++ to depend on state. First, we present dispatch on three pieces of
information:
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Objects and Run-Time Efficiency: C++
the name of the method being invoked
the type of the receiver object
the explicit state of the receiver object
As an example, the following declares and creates objects of the File class with
the new dispatch mechanism:
class File {
state in { OPEN, CLOSED };
public:
File() { state = CLOSED; }
switch(state) {
/* declare states that a File object may be in */
/* initial state is closed */
case CLOSED: {
void Open() {
/* 1 */
/* open file . . . */
state = OPEN;
}
void Close() {
error “file not open”;
}
}
case OPEN: {
void Open() {
/* 2 */
error “file already open”;
}
void Close() {
/* close file . . . */
state = CLOSED;
}
}
}
}
File* f = new File();
f->Open();
/* calls version 1 */
f->Open();
/* calls version 2 */
...
The idea is that the programmer can provide a different implementation of the
same method for each state that the object can be in.
(a) Describe one advantage of having this new feature, i.e., are there any advantages to writing classes like File over classes like StdFile. Describe one
disadvantage of having this new feature.
(b) For this part of the problem, assume that subclasses cannot add any new
states to the set of states inherited from the base class. Describe an object
representation that allows for efficient method lookup. Method call should
be as fast as virtual method calls in C++, and changing the state of an object
Exercises
should be a constant time operation. (Hint: you may want to have a different
vtable for each state). Is this implementation acceptable according to the C++
design goal of only paying for the features that you use?
(c) What problems arise if subclasses are allowed to extend the set of possible
states? For example, we could now write a class such as
class SharedFile: public File {
state in { OPEN, CLOSED, READONLY };
...
}
/* extend the set of states */
Do not try to solve any of these problems. Just identify several of them.
(d) We may generalize this notion of dispatch based on the state of an object
to dispatch based on any predicate test. For example, consider the following
Stack class:
class Stack {
private:
int n;
int elems[100];
public:
Stack() { n = 0; }
when(n == 0) {
int Pop() {
error “empty”;
}
}
when(n > 0) {
int Pop() {
return elems[––n];
}
}
...
}
Is there an easy way to extend your proposed implementation in part (b) to
handle dispatch on predicate tests? Why or why not?
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13
Portability and Safety: Java
The Java programming language was designed by James Gosling and others at Sun
Microsystems. The language, arising from a project that began in 1990, was originally
called Oak and was intended for use in a device affectionately referred to as a set-top
box. The set-top box was intended to be a small computational device, attached to a
network of some kind, and placed on top of a television set. There are various features
a set-top box might provide. You can imagine some of your own by supposing that a
web browser is displayed on your television set and, instead of a keyboard, you click
on icons by using some buttons on your remote control. You might want to select
a television program or movie or download a small computer simulation that could
be executed on the computational device and displayed on your screen. A television
advertisement for an automobile might allow you to download an interactive visual
tour of the automobile, giving each viewer a personalized simulation of driving the car
down the road. Whatever scenario might appeal to you, the computing environment
would involve graphics, execution of simple programs, and communication between
a remote site and a program executed locally.
At some point in the development of Oak, engineers and managers at Sun
Microsystems realized that there was an immediate need for an Internet-browser
programming language, a language that could be used to write small applications
that could be transmitted over the network and executed under the control of any
standard browser on any standard platform. The need for a standard is a result of
the intrinsic desire of companies and individuals with web sites to be able to reach
as large an audience as possible. In addition to portability, there is also a need for
security so that someone downloading a small application can execute the program
without fear of computer viruses or other hazards.
The Oak language started as a reimplementation of C++. Although language
design was not an end goal of the project, language design became an important
focus of the group. Some of the reasons for designing a new language are given in
this overly dramatic but still informative quote from “The Java Saga” by David Bank
in Hot Wired (December 1995):
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Portability and Safety: Java
JAMES GOSLING
James Gosling was the lead engineer and key architect behind the Java
programming language and platform. He is now back in Sun’s research
laboratories, working on software development tools. His first project at
Sun was the NeWS window system, distributed for Sun workstations in
the 1980s. Before joining Sun, Gosling built a multiprocessor version
of UNIX; the original Andrew window system and toolkit; and several
compilers and mail systems. He is known to many as the author of the
original UNIX ‘Emacs.’
A techie with a sense of humor, Gosling is shown in the right-hand
photograph about to put a pie in the face of a stagehand wearing a
Bill Gates mask, in a picture from his partially politicized web page at
http://java.sun.com/people/jag /.
James Gosling received a B.S. in Computer Science from the University of Calgary, Canada, and a Ph.D. in Computer Science from CarnegieMellon University for a dissertation entitled, “The Algebraic Manipulation
of Constraints.”
Gosling quickly concluded that existing languages weren’t up to the job. C++
had become a near-standard for programmers building specialized applications
where speed is everything . . . But C++ wasn’t reliable enough for what Gosling
had in mind. It was fast, but its interfaces were inconsistent, and programs kept
on breaking. However, in consumer electronics, reliability is more important than
speed. Software interfaces had to be as dependable as a two-pronged plug fitting
into an electrical wall socket. “I came to the conclusion that I needed a new
programming language,” Gosling says.
For a variety of reasons, including a tremendous marketing effort by Sun Microsystems, Java became surprisingly successful a short time after it was released as an
Internet communication language in mid-1995.
The main parts of the Java system are
the Java programming language,
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Portability and Safety: Java
Java compilers and run-time systems (Java virtual machine),
an extensive library, including a Java toolkit for graphic display and other applications, and sample Java applets.
Although the library and toolkit helped with early adoption, we will be primarily interested in the programming language, its implementation, and the way language design and implementation considerations influenced each other. Gosling,
more modest in real life than the preceding quote might suggest, has this to say
about languages that influenced Java: “One of the most important influences on the
design of Java was a much earlier language called Simula. It is the first OO language I ever used (on a CDC 6400!). . . . [and] where the concept of a ‘class’ was
invented.”
13.1 JAVA LANGUAGE OVERVIEW
13.1.1 Java Language Goals
The Java programming language and execution environment were designed with the
following goals in mind:
Portability: It must be easy to transmit programs over the network and have them
run correctly in the receiving environment, regardless of the hardware, operating
system, or web browser used.
Reliability: Because programs will be run remotely by users who did not write the
code, error messages and program crashes should be avoided as much as possible.
Safety: The computing environment receiving a program must be protected from
programmer errors and malicious programming.
Dynamic Linking: Programs are distributed in parts, with separate parts loaded
into the Java run-time environment as needed.
Multithreaded Execution: For concurrent programs to run on a variety of hardware and operating systems, the language must include explicit support and a
standard interface for concurrent programming.
Simplicity and Familiarity: The language should appeal to your average website designer, typically a C programmer or a programmer partially familiar with
C/C++.
Efficiency: This is important, but may be secondary to other considerations.
Generally speaking, the reduced emphasis on efficiency gave the Java designers more
flexibility than the C++ designers had.
13.1.2 Design Decisions
Some of the design goals and global design decisions are listed in Table 13.1, in which
+ indicates that a decision contributes to this goal, − indicates that a decision detracts
from this goal, and +/− indicates that there are advantages and disadvantages of the
decision. Some squares are left blank, indicating that a design decision has little or no
effect on the goal. We can see the relative importance of efficiency in the Java design
process by looking down the rightmost column. This does not mean that efficiency
13.1 Java Language Overview
Table 13.1. Java design decisions
Interpreted
Type safe
Most values are object
Objects by means of pointers
Garbage collection
Concurrency support
Portability
Safety
+
+
+/−
+
+
+
+
+
+/−
+
+
Simplicity
Efficiency
+/−
+
+
+
−
+/−
−
−
−
was sacrificed needlessly, only that relative to other goals, efficiency was not the
primary objective.
Interpreted. The initial and most widely used implementations of Java are based
on interpreted bytecode. This is discussed in more detail in Section 13.4. In brief, Java
programs are compiled to a simplified form of lower-level language. This language,
called Java bytecode, is the form that is usually used when Java programs are sent
across the network as parts of web pages. Java bytecode is executed by an interpreter
called the Java virtual machine (JVM). One advantage of this architecture is that
once a JVM is implemented for a particular hardware and operating system, all Java
programs can be run on that platform without change. In addition to portability,
interpreted bytecode facilitates safe execution, as commands that violate the semantics of the Java language can be recognized just before they are executed. A good
example is array-bounds checking. It is not feasible to tell at compile time whether a
program will access arrays out of bounds. However, the JVM does run-time tests to
make sure that no Java program accesses memory incorrectly through out-of-bounds
array indexing.
Type Safety. There are three levels of type safety in Java. The first is compiletime type checking of Java source code. The Java type checker works like other
conventional type checkers (as in Pascal, C++, and so on), preventing compilation
of programs that do not conform to the Java type discipline. There is no pointer
arithmetic, there are no unchecked type casts, and the language is garbage collected,
proving a greater degree of type safety than C++, for example. The second level of
type safety is provided by type-checking Java bytecode programs before they are executed. The third level is provided by run-time type checks, such as the array-bounds
checks described in the preceding subsection. In addition to safety, the Java type
system simplifies the language to some degree by eliminating constructs that would
complicate the language semantics or run-time system. There are some efficiency
costs associated with run-time checking, however.
Objects and References. In Java, many things are objects but not all things. In
particular, values of certain basic types such as integers, Booleans, and strings are
not objects. This is a compromise between simplicity and efficiency. In particular,
if all integer operations required dynamic method lookup, this would slow down
integer arithmetic considerably. A simplifying decision associated with objects is
that all objects are accessed by pointer, and pointer assignment is the only form of
assignment provided for all objects. This simplifies programs by eliminating some
special cases, but in some situations reduces program efficiency.
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Portability and Safety: Java
All parameters to Java methods are passed by value. When the parameter has a
reference type (including all objects and arrays), though, it is the reference itself that
is copied and passed by value. In effect, this means that values of primitive types are
passed by value and objects are passed by reference.
Garbage Collection. As discussed in Subsection 6.2.1, garbage collection is necessary for complete type safety. Garbage collection also simplifies programming by
eliminating the need for code that determines whether memory can be deallocated,
but has a run-time cost. Java garbage collection takes advantage of concurrency.
Specifically, the Java garbage collector is implemented as a low priority background
thread, allowing garbage collection to occur during times when it might not affect a
user’s perception of running speed.
Dynamic Linking. The classes defined and used in a Java program may be loaded
into the JVM incrementally, as they are needed by the running program. This shortens
the elapsed time between the beginning of transmission of a program across the
network and the beginning of program execution, as the program can begin executing
before all of the related code is transmitted. Moreover, if a program terminates
without needing some classes, these classes never need to be transmitted or loaded
into the virtual machine. Dynamic linking does not have a large effect on the language
design, other than to require clear interfaces that can be used to check one part of a
program under assumptions about the code provided by another part of the program.
Concurrency Support. Java has a concurrency model based on threads, which are
independent concurrent processes. This is a significant part of the language, both because the design is substantial and because of the importance of having standardized
concurrency primitives as part of the Java language. Clearly, if Java programs relied
on operating system specific concurrency mechanisms, Java programs would not be
portable across different operating system platforms.
Simplicity. Although Java has grown over the years, and features like reflection
and inner classes may not seem “simple,” the language is still smaller and simpler in
design than most production-quality general-purpose programming languages. One
way to see the relative simplicity of Java is to list the C++ features that do not appear
in Java. These include the following features:
Structures and unions: Structures are subsumed by objects, and some uses of
unions can be replaced with classes that share a common superclass.
Functions can be replaced with static methods.
Multiple inheritance is complex and most cases can be avoided if the simpler
interface concept of Java is used.
Goto is not necessary.
Operator overloading is complex and deemed unnecessary; Java functions can be
overloaded.
Automatic coercions are complex and deemed unnecessary.
Pointers are the default for objects and are not needed for other types. As a result
a separate pointer type is not needed.
Some of these features appear in C++ primarily because of the C++ design goal of
backward compatibility with C. Others were omitted from Java after some discussion,
because it was decided that complexity of including them was more significant than
13.2 Java Classes and Inheritance
the functionality they would provide. The most significant omissions are multiple
inheritance, automatic conversions, operator overloading, and pointer operations of
the forms found in C and C++.
13.2 JAVA CLASSES AND INHERITANCE
13.2.1 Classes and Objects
Java is written in a C++-like syntax so that programming is more accessible to C
and C++ programmers. This makes the C++ one-dimensional point class used in
Subsection 12.3.1 look similar when translated into Java. Here is an abbreviated
version of the class, with the move method omitted:
class Point {
public int getX() { . . . }
protected void setX (int x) { . . . }
private int x;
Point(int xval) {x = xval;}
};
Like other class-based languages, Java classes declare the data and functions associated with all objects created by this class. When a Java object is created, space is
allocated to store the data fields of the object and the constructor of the class is called
to initialize the data fields. As in C++, the constructor has the same name as the class.
Also following C++, the Point class has public, private, and protected components.
Although public, private, and protected are keywords of Java, these visibility specifications do not mean exactly the same thing in the two languages, as explained in
Subsection 13.2.2.
Java terminology is slightly different from that of Simula, Smalltalk and C++.
Here is a brief summary of the most important terms used to discuss Java:
Class and object have essentially the same meaning as in other class-based objectoriented languages, field: data member
Method: member function, static member: analogous to Smalltalk class field or
class method, this: like C++ this or Smalltalk self, the identifier this in the body
of a Java method refers to the object on which this method was invoked
Native Method: method written in another language, such as C
Package: set of classes in shared name space.
We will look at several characteristics of classes and objects in Java, including static
fields and methods, overloading, finalize methods, main methods, toString methods
used to produce a print representation of an object, and the possibility of defining
native methods. We will discuss the Java run-time representation of objects and the
implementation of method lookup in Section 13.4 in connection with other aspects
of the architecture of the run-time system.
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Initialization. Java guarantees that a constructor is called whenever an object is
created. Because some interesting issues arise with inheritance, this is discussed in
Subsection 13.2.3.
Static Fields and Methods. Java static fields and methods are similar to Smalltalk
class variables and class methods. If a field is declared to be static, then there is one
field for the entire class, instead of one per object. If a method is declared static,
the method may be called without using an object of the class. In particular, static
methods may be called before any objects of the class are created. Static methods can
access only static fields and other static methods; they cannot refer to this because
they are not part of any specific object of the class. Outside a class, a static member
is usually accessed with the class name, as in class name.static method(args), rather
than through an object reference.
Static fields may be initialized with initialization expressions or a static initialization block. Both are illustrated in the following code:
class . . . {
/* --- static variable with initial value --- */
static int x = initial value;
/* --- static initialization block
--- */
static { /* code to be executed once, when class is loaded */
}
}
As indicated in the program comment, the static initialization block of a class is executed once, when the class is loaded. Class loading is discussed in Section 13.4 in
connection with the JVM. There are specific rules governing the order of static initialization, when a class contains both initialization expressions and a static initialization
block. There are also restrictions on the form of static initialization blocks. For example, a static block cannot raise an exception, as it is not certain that a corresponding
handler will be installed at class-loading time.
Overloading. Java overloading is based on the signature of a method, which consists of the method name, the number of parameters, and the type of each parameter.
If two methods of a class (whether both declared in the same class, or both inherited,
or one declared and one inherited) have the same name but different signatures, then
the method name is overloaded. As in other languages, overloading is resolved at
compile time.
Garbage Collection and Finalize:. Because Java is garbage collected, it is not necessary to explicitly free objects. In addition, programmers do not need to worry about
dangling references created by premature deallocation of objects. However, garbage
collection reclaims only the space used by an object. If an object holds access to another sort of resource, such as a lock on shared data, then this must be freed when the
object is no longer accessible. For this reason, Java objects may have finalize methods,
which are called under two conditions, by the garbage collector just before the space
is reclaimed and by the virtual machine when the virtual machine exits. A useful
convention in finalize methods is to call super.finalize, as subsequently illustrated, so
13.2 Java Classes and Inheritance
that any termination code associated with the superclass is also executed:
class . . . {
...
protected void finalize () {
super.finalize();
close(file);
}
};
There is an interesting interaction between finalize methods and the Java exception
mechanism. Any uncaught exceptions raised while a finalize method is executed are
ignored.
A programming problem associated with finalize methods is that the programmer
does not have explicit control over when a finalize method is called. This decision
is left to the run-time system. This can create problems if an object holds a lock
on a shared resource, for example, as the lock may not be freed until the garbage
collector determines that the program needs more space. One solution is to put
operations such as freeing all locks or other resources in a method that is explicitly
called in the program. This works well, as long as all users of the class know the
name of the method and remember to call this method when the object is no longer
needed.
Some other interesting aspects of Java objects and classes are main methods used
to start program execution, toString methods used to produce a print representation
of an object, and the possibility of defining native methods:
main:
A Java application is invoked with the name of the class that drives the
application. This class must have a main method, which must be public, static,
must return void, and must accept a single argument of type String[]. The main
method is called with the program arguments in a string array. Any class with a
main method can be invoked directly as if it were a stand-alone application, which
can be useful for testing.
toString: A class may define a toString method, which is called when a conversion
to string type is needed, as in printing an object.
native methods: A native method is one written in another language, such as C.
Portability and safety are reduced with native methods: Native code cannot be
shipped over the network on demand, and controls incorporated into the JVM
are ineffective because the method is not interpreted by the virtual machine.
The reasons for using native methods are (1) efficiency of native object code
and (2) access to utilities or programs that have already been written in another
language.
13.2.2 Packages and Visibility
Java has four visibility distinctions for fields and methods, three corresponding to
C++ visibility levels and a fourth arising from packages.
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package
class
field
method
Figure 13.1. Java package and
class visibility.
Java packages are an encapsulation mechanism similar to C++ name space that
allows related declarations to be grouped together, with some declarations hidden
from other packages. In a Java program, every field or method belongs to a specific
class and every class is part of a package, as shown in Figure 13.1. A class can belong to
the unnamed default package, or some other package if specified in the file containing
the class.
The visibility distinctions in Java are
public: accessible anywhere the class is visible,
protected: accessible to methods of the class and any subclasses, as well as to
other classes in the same package,
private: accessible only in the class itself,
package: accessible only to code in the same package; not visible to subclasses
in other packages. Members declared without an access modifier have package
visibility.
Put another way, a method can refer to the private members of the class it belongs
to, nonprivate members of all classes in the same package, protected members of
superclasses (including superclasses in a different package), and public members of
all classes in any visible package.
Names declared in another package can be accessed with import, which imports
declarations from another package, or with qualified names of the following form,
which indicate the package containing the name explicitly:
java.lang . String. substring()
package class method
13.2.3 Inheritance
In Java terminology, a subclass inherits from its superclass. The Java inheritance
mechanism is essentially similar to that of Smalltalk, C++, and other class-based
object-oriented languages. The syntax associated with inheritance is similar to C++,
with the keyword extends, as shown in this example ColorPoint class extending the
Point class from Subsection 13.2.1:
13.2 Java Classes and Inheritance
class ColorPoint extends Point {
// Additional fields and methods
private Color c;
protected void setC (Color d) {c = d;}
public Color getC() {return c;}
// Define constructor
ColorPoint(int xval, Color cval) {
super(xval); // call Point constructor
c = cval; } // initialize ColorPoint field
};
Method Overriding and Field Hiding. As in other languages, a class inherits all
the fields and methods of its superclass, except when a field or method of the same
name is declared in the subclass. When a method name in the subclass is the same
as a method name in the superclass, the subclass definition overrides the superclass
method with the same signature. An overriding method must not conflict with the
definition that it overrides by having a different return type. An overridden method
of the superclass can be accessed with the keyword super. For fields, a field declaration
in a subclass hides any superclass field with the same name. A hidden field can be
accessed by use of a qualified name (if it is static) or by use of a field access expression
that contains a cast to a superclass type or the keyword super.
Constructors. Java guarantees that a constructor is called whenever an object is
created. In compiling the constructor of a subclass, the compiler checks to make sure
that the superclass constructor is called. This is done in a specific way that programmers generally want to take into consideration. In particular, if the first statement of
a subclass constructor is not a call to super, then the call super() is inserted automatically by the compiler. This does not always work well, because, if the superclass does
not have a constructor with no arguments, the call super() will not match a declared
constructor, and a compiler error results. An exception to this check occurs if one constructor invokes another. In this case, the first constructor does not need to call the superclass constructor, but the second one must. For example, if the constructor declaration ColorPoint() { ColorPoint(0,blue);} is added to the preceding ColorPoint class, then
this constructor is compiled without inserting a call to the superclass Point constructor.
A slight oddity of Java is that the inheritance conventions for finalize are different
from the conventions for constructors. Although a call to the superclass is required
for constructors, the compiler does not force a call to the superclass finalize method
in a subclass finalize method.
Final Methods and Classes. Java contains an interesting mechanism for restricting
subclasses of a class: A method or an entire class can be declared final. If a method
is declared final, then the method cannot be overridden in any subclass. If a class is
declared final, then the class cannot have any subclasses. The reason for this feature
is that a programmer may wish to define the behavior of all objects of a certain type.
Because subclasses produce subtypes, as discussed in Section 13.3, this requires some
restriction on subclasses. To give an extreme example, the singleton pattern discussed
in Section 10.4 shows how to design a class so that only one object of the class can
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be created. The pattern hides the constructor of the class and makes public only a
function that will call the constructor once during program execution. This pattern
solves the problem of restricting the number of objects of the class, but only if no
subclass overrides the public method with a method that can create more than one
object. If a programmer really wants to enforce the singleton pattern, there must be
a way to keep other programmers from defining subclasses of the singleton class.
The Java class java.lang.System is another example of a final class. This class is
final so that programmers do not override system methods.
In a loose sense, Java final is the opposite of C++ virtual: Java methods can
be overridden until they are marked final, whereas C++ member functions can be
overridden only if they are virtual. The analogy is not exact, though, as a C++ member
function cannot be virtual in one class and nonvirtual in a base class or derived class,
because that would violate the requirement that base- and derived-class vtables must
have the same layout.
Class Object. In principle, every class declared in a Java program extends another
class, as a class without an explicit superclass is interpreted as a subclass of the class
Object. The class Object is the one class that has no superclasses. Class Object contains
the following methods, which can be overridden in derived classes:
GetClass,
which returns the Class object that represents the class of the object.
This can be used to discover the fully qualified name of a class, its members, its
immediate superclass, and any interfaces that it implements.
ToString, which returns a String representation of an object.
equals, which defines a notion of object equality based on value, not reference,
comparison.
hashCode, which returns an integer that can be used to store the object in a hash
table.
clone, which is used to make a duplicate of an object.
Methods wait, notify, and notifyAll used in concurrent programming.
finalize, which is run just before an object is destroyed (discussed in Subsection
13.2.1 in connection with garbage collection),
Because all classes inherit the methods of class Object, every object has these methods.
13.2.4 Abstract Classes and Interfaces
The Java language has an abstract-class mechanism that is similar to C++. As we
discussed in Chapter 12, an abstract class is a class that does not implement all of its
methods and therefore cannot have any instances. Java uses the keyword abstract
instead of the C++ “=0” syntax, as shown in the following code:
abstract class Shape {
...
abstract point center();
abstract void rotate(degrees d);
...
}
13.2 Java Classes and Inheritance
Java also has a “pure abstract” form of class called an interface. An interface is
defined in a manner similar to that of a class, except that all interface members
must be constants or abstract methods. An interface has no direct implementation, but classes may be declared to implement an interface. In addition, an interface may be declared as an extension of another, providing a form of interface
inheritance.
One reason that Java programmers use interfaces instead of pure abstract classes
when a concept is being defined but not implemented is that Java allows a single class
to implement several interfaces, whereas a class can have only one superclass. The
following interfaces and class illustrate this possibility. The Shape interface identifies
some properties of simple geometric shapes, namely, each has a center point and a
rotate method. Drawable similarly identifies properties of objects that can be displayed
on a screen. If circles are geometric shapes that can be displayed on a screen, then the
Circle class can be declared to implement both Shape and Drawable, as subsequently
shown.
Interfaces are often used as the type of argument to a method. For example, if
the windows system has a method for drawing items on a screen, then the argument
type of this method could be Drawable, allowing every object that implements the
Drawable interface to be displayed:
interface Shape {
public Point center();
public void rotate(float degrees);
}
interface Drawable {
public void setColor(Color c);
public void draw();
}
class Circle implements Shape, Drawable {
// does not inherit any implementation
// but must define Shape, Drawable methods
}
Unlike C++ multiple inheritance (discussed in Section 12.5), there is no name clash
problem for Java interfaces. More specifically, suppose that the preceding two interfaces Shape and Circle also both define a Size method. If the two Size methods both
have the same number of arguments and the same argument types, then class Circle
must implement one Size method with this number of arguments and the argument
types given in both interfaces. On the other hand, if the two Size methods have a
different number of arguments or the arguments can be distinguished by type, then
these are considered two different method names and Circle must define an implementation for each one. Because Java method lookup uses the method name and
number and types of arguments to select the method code, the two methods with the
same name will be treated separately at method lookup time.
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13.3 JAVA TYPES AND SUBTYPING
13.3.1 Classification of Types
The Java types are divided into two categories: primitive types and reference types.
The eight primitive types are the boolean type and seven numeric types. The seven
numeric types are the forms of integers byte, short, int, long, and char and the floatingpoint types float and double. The three forms of reference types are class types,
interface types (subsequently discussed), and array types. There is also a special null
type. The values of a reference type are references to objects (which include arrays).
All objects, including arrays, support the methods of class Object.
The subtyping relationships between the main families of types are illustrated in
Figure 13.2, which includes a Shape interface and Circle and Square classes to show
how user-defined classes and interfaces fit into the picture. Other predefined types
such as String, ClassLoader, and Thread occupy positions similar to that of Exception
in this figure. Object[] is the type of arrays of objects, and similarly for array types
Shape[], Circle[], and Square[].
Although it is standard Java terminology to call Object and its subtypes reference
types, this may be slightly confusing. Although C++ distinguishes Object from Object *,
there are no explicit pointer types in Java. Instead, the difference between a pointer
to an object and an object itself is implicit, with pointer dereferencing combined with
operations like method invocation and field access. If T is a reference type, then a
variable x of type T is a reference to T objects; in C++ the variable x would have type
T*. Because there is no explicit way to dereference x to get a value of type T, Java
does not have a separate type for objects not referred to by pointer.
Reference Types
Object
Object[]
Shape
Circle
Throwable
Shape[]
Square
Circle[]
Exception
types
Square[]
Primitive Types
boolean
int
byte
Figure 13.2. Classification of types in Java.
…
float
long
13.3 Java Types and Subtyping
Because every class is a subtype of Object, variables of type Object can refer to
objects or arrays of any type.
13.3.2 Subtyping for Classes and Interfaces
Subtyping for classes is determined by the class hierarchy and the interface mechanism. Specifically, if a class A extends class B, then the type of A objects is a subtype of
the type of B objects. There is no other way for one class to define a subtype of another,
and no private base classes (as in C++) to allow inheritance without subtyping.
A class may be declared to implement one or more interfaces, meaning that any
instance of the class implements all the abstract methods specified in the interface.
This (multiple) interface subtyping allows objects to support (multiple) common
behaviors without sharing any common implementation.
Run-Time Type Conversion. Java does not allow unchecked casts. However, objects of a supertype may be cast to a subtype by a mechanism that includes a run-time
type test. If you want to make a list class in Java, you make it hold objects of type
Object. Objects of any class can then be put onto a list, but, to take them off the list and
use them nontrivially, they must be converted back to their original type (or some
supertype). In Java, type conversion is checked at run time, raising an exception if
the object does not have the designated type.
Implementation. Java uses different bytecode instructions for member lookup by
interface and member lookup by class or subclass. In a high-performance compiler,
lookup by class could be implemented as in C++, with offset known at compile
time. However, lookup by interface cannot, because a class may implement many
interfaces and the interfaces may list members in different orders. This is discussed
in detail in Subsection 13.4.4.
13.3.3 Arrays, Covariance, and Contravariance
For any type T, Java has an array type T[ ] of arrays whose elements have type T.
Although array types are grouped with classes and interfaces, it is not possible to
inherit from an array type. In Java terminology, array types are final.
Array types are subtypes of Object, and therefore arrays support all of the methods
associated with class Object. Like other reference types, an array variable is a pointer
to an array and can be null. It is common to create arrays when an array reference is
declared, as in
Circle[ ] x = new Circle[array size]
However, it is also possible to create array objects “anonymously,” in much the same
way that we can create other Java objects. For example,
new int[ ] {1,2,3, . . . 10}
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is an expression that creates an integer array of length 10, with values 1, 2, 3, . . . , 10.
Because a variable of type T[ ] can be assigned an array of any length, the length of
an array is not part of its static type.
There are some complications surrounding the way that Java array types are
placed in the subtype hierarchy. The most significant decision is that if A <: B then
the Java type checker also uses the subtyping A[ ] <: B[ ]. This introduces a problem
often referred to as the array covariance problem. Consider the following class and
array declarations:
class A { . . . }
class B extends A { . . . }
B[ ] bArray = new B[10]
A[ ] aArray = bArray
// considered OK since B[ ] <: A[ ]
aArray[0] = new A() // allowed but run-time type error; raises
ArrayStoreException
In this code, we have B <: A because class B extends class A. The array reference bArray
refers to an array of B objects, initially all null, and the array reference aArray refers
to the same array. The declaration A[ ] aArray = bArray is allowed by the Java type
checker (although semantically it should not be allowed) because of the Java design
decision that if B <: A then B[ ] <: A[ ]. The problem with allowing the A[ ] array
reference aArray to refer to an array of B objects is illustrated by the last statement.
The assignment aArray[0] = new A() looks perfectly reasonable: aArray is an array with
static type A[ ], suggesting that it is acceptable to assign an A object to any location
in the array. However, because aArray actually refers to an array of B objects, this
assignment would violate the type of bArray. Because this assignment would cause
a typing problem, the Java implementation does not allow the assignment to be
executed. A run-time test will determine that the value being assigned to an array of
B objects is not a B object and the ArrayStoreException exception will be raised.
Although the Java designers considered array covariance advantageous for some
specific purposes (writing some binary copy routines), array covariance in Java leads
to some confusion and many run-time tests. It does not seem to be a successful
language design decision.
13.3.4 Java Exception Class Hierarchy
Java programs may declare, raise, and handle exceptions. The Java exception mechanism has the general features we discussed in Section 8.2. Java exceptions may be the
result of a throw statement in a user program or the result of some error condition
detected by the virtual machine such as an attempt to index outside the bounds of
an array. In Java terminology, an exception is said to be thrown from the point where
it occurred and is said to be caught at the point to which control is transferred. As in
other languages, throwing an exception causes the Java implementation to halt every
expression, statement, method or constructor invocation, initializer, and field initialization expression that has started but not completed execution. This process continues until a handler is found that matches the class of the exception that is thrown.
13.3 Java Types and Subtyping
One interesting aspect of the Java exception mechanism is the way that it is
integrated into the class and type hierarchy. Every Java exception is represented by an
instance of the class Throwable or one of its subclasses. An advantage of representing
exceptions as objects is that an exception object can be used to carry information from
the point at which an exception occurs to the handler that catches it. Subtyping can
also be used in handling an exception: A handler matches the class of an exception if
it explicitly names the class of the exception that is thrown or names some superclass
of the class of the exception.
The Java exception mechanism is designed to work well in multithreaded (concurrent) programs. When an exception is thrown, only the thread in which the throw
occurs is affected. The effect of an exception on the concurrent synchronization
mechanism is that locks are released as synchronized statements and invocations of
synchronized methods complete abruptly. We will return to this topic in Chapter 14.
Java exceptions are caught inside a construct called a try-finally block. Here is
an example outline, showing a block with two handlers, each identified by the catch
keyword. Intuitively, a try-finally block tries to execute a sequence of statements.
If the sequence of statements terminates normally, then that is the end result of
the block. However, if an exception is raised, it may be caught inside the block. If
an exception is raised and caught, then the sequence of statements following the
keyword finally will be executed after the exception handler has finished:
try {
statements
} catch (ex-type1 identifier1) {
statements
} catch (ex-type2 identifier2) {
statements
} finally {
statements
}
Although it may not be apparent why, there is some complication in the JVM associated with try-finally blocks. Specifically, a significant fraction of the complexity
of the Java bytecode verifier is a result of the way finally clauses are implemented
as a form of “local subroutine” (called jsr) in the Java bytecode interpreter. We will
discuss the JVM in Section 13.4.
The classes of exceptions are shown in Figure 13.3. Every exception is, by definition, an object of some subclass of Throwable. The class Throwable is a direct subclass
of Object. Programs can use the preexisting exception classes in throw statements
or define additional exception classes. Additional exceptions classes must be subclasses of Throwable or one of its subclasses. To take advantage of the Java platform’s
compile-time checking for exception handlers, it is typical to define most new exception classes as checked exception classes. These are subclasses of Exception that are
not subclasses of RuntimeException.
The phrases checked exceptions and “unchecked exceptions” refer to compiletime checking of the set of exceptions that may be thrown in a Java program. The
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Portability and Safety: Java
Throwable
Exception
Runtime
Exception
User-defined
Exception
classes
Error
Unchecked exceptions
checked exceptions
Figure 13.3. Java exception classes.
Java compiler checks, at compile time, that a program contains handlers for every
checked exception. We accomplish this by analyzing the checked exceptions that
can be thrown during execution of a method or constructor. These checks are based
on the declared set of exceptions that a Java method could throw, called the throws
clause of the method. For each checked exception that is a possible result of a method
call, the throws clause for the method must mention the class of that exception or
one of the superclasses of the class of that exception. This compile-time checking for
the presence of exception handlers is designed to reduce the number of exceptions
that are not properly handled.
Because Error and RuntimeException exceptions are generally thrown by the Java
run-time system and not by the user code, it is not necessary for a programmer to
declare these exceptions in the throws clause of a method. More specifically, the
run-time exception classes (RuntimeException and its subclasses) are exempt from
compile-time checking because, in the judgment of the designers of the Java programming language, declaring such exceptions would not aid significantly in establishing
the correctness of programs. Many of the operations and constructs of the Java programming language can result in run-time exceptions. The information available to
a compiler and the level of analysis the compiler performs are usually not sufficient
to establish that such run-time exceptions cannot occur, even though this may be
obvious to the programmer.
Ordinary programs do not usually recover from exceptions from the class Error
or its subclasses. The class Error is a separate subclass of Throwable, distinct from
Exception in the class hierarchy, to allow programs to use the idiom
} catch (Exception e) {
13.3 Java Types and Subtyping
to catch all exceptions from which recovery may be possible without catching errors
from which recovery is typically not possible.
13.3.5 Subtype Polymorphism and Generic Programming
In Section 9.4, we discussed some approaches to generic programming based on templates and related mechanisms such as generic modules and packages. The main idea
is that we can define a generic data structure, such as a list or binary search tree,
so that any type of data can be inserted into or retrieved from the data structure,
provided that a few required operations, such as equality test or a linear ordering,
are defined and implemented on the data. Generic algorithms such as sorting, applicable to any list or array of data, can be defined similarly by use of templates
or related mechanisms. Templates and generic modules are useful in generic programming because they provide a form of polymorphism: The same code may be
instantiated and executed on a variety of types of data.
In object-oriented programming, the term polymorphism is often used to refer
to a specific kind of polymorphism that we call subtype polymorphism. In subtype
polymorphism, a single piece of code, typically a set of functions or methods, can
be applied to more than one type of argument. However, the language mechanism
for allowing this is not through implicit type parameters (as in ML type-inference
polymorphism) or explicit type parameters (as in C++ templates or Ada generic
packages), but through subtyping. Specifically, in a language with subtyping, if a
method m will accept any argument of type A, then m may also be applied to any argument from any subtype of A. This works almost as well as parametric polymorphism
(polymorphism based on type parameters), except that typically the compile-time
type checker has less information about the types of arguments. This either leads to
run-time type checking or some sacrifice in type safety. The general phenomenon is
illustrated in the following Java example.
Example 13.1 Java Subtype Polymorphism
Let us consider the problem of defining a general class of stacks. We would like to
be able to build stacks of any type of object. We will implement stacks as linked lists
of stack nodes, each node containing a stack element.
Because Object is a supertype of all reference types (including all references to
objects and arrays), we can let stack cells contain references of type Object. Let us
begin with a subsidiary class definition, the class of stack nodes:
class Node {
Object element; // Any object can be placed in a stack
Node next;
Node (Node n, int e) {
next=n;
element=e;
}
}
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This class is intended to belong to the same package as that of the Stack class. Because
there are no access qualifiers here (public, private, or protected) the fields and methods
are accessible only to methods in the same package.
Here is the outline of a Stack class in which Node objects are used to store elements
of a linked list of stack elements. The methods shown here are empty, which tells
whether a stack is empty, push, which adds a new element to the top of a stack, and
pop, which removes the top node and returns the object stored in that node. The class
is declared public so that it is visible outside the package in which it is defined:
public class Stack {
private Node top=null;
public Stack(){
}
public boolean empty(){
return top == null;
}
public void push(Object val) {
top = new Node(top,val);
}
public Object pop() {
if (top==null) return -1;
Node temp = top;
top = top.next;
return temp.element;
}
}
// top of the stack, starts as empty
// determine whether stack is empty
// push element on top of stack
// Remove first node and return value.
Most of the code here is entirely straightforward. The important limitation, for the
purpose of generic programming, is the type of the node elements and the return
type of pop. Because the argument type of push is Object, any object can be pushed
onto a stack. However, the return type of pop is also Object, meaning that, when an
object is removed from a stack, the static type of the expression removing the object
is Object. Therefore, if a program pushes strings onto a stack and then pops them
off the stack, the Java type checker will not be able to determine that the objects
removed from the stack are strings.
Here is an example of the way that stacks might be used in a Java program:
String s = “Hello”;
Stack st = new Stack();
...
st.push(s);
...
s = (String) st.pop;
13.3 Java Types and Subtyping
In this code fragment, a String object is pushed onto a stack. When the String object
is popped off the stack, the static type of st.pop() is Object, not string. Therefore, to
assign st.pop() to a string variable, it is necessary to cast the Object to String. This cast
generates a run-time test. If the top element on the stack is not a string when this
assignment is executed, the run-time test will throw an exception and prevent the
assignment from occurring.
In comparison, a C++ stack class template may be defined by the following form:
template <typename t> class Stack {
private: Node<t> top;
public: boolean empty() { . . . };
void
push (t* x) { . . . }
t*
pop ( ) { . . . }
};
If Java had a template mechanism similar to this, then we could define a generic stack
class like this,
class Stack<A> {
public Boolean empty(){ . . . }
public void push(A a) { . . . }
public A pop() { . . . }
}
and use generic stacks as follows:
String s = “Hello”;
Stack<String> st = new Stack<String>();
st.push(s);
...
s = st.pop();
There are several important points of comparison between the two styles illustrated
here. In the template-based code, the stack st has type Stack<string> instead of Stack.
As a result, st.pop() has static type String. Two advantages are clarity and efficiency.
The programmer’s intent is clearer, as the stack st is explicitly declared to be a stack
of strings. The efficiency advantage is that there is no need for a cast and run-time
test.
There have been several research projects aimed at adding a template mechanism to Java. Although the basic goals of such a mechanism seem clear, there are
a number of details that need to be addressed. One issue is implementation. One
approach is to simply translate Java with templates into Java without templates. This
allows programmers to write code with templates and provides a relatively simple
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implementation. However, the result of the translation is code that contains many
run-time-checked type conversions, so it may not be as efficient as other implementation techniques. An alternative is to compile a generic class such as Stack<t>
into a form of class file that has type parameters and load class Stack<String> by
instantiating this class file at class-load time. This produces more efficient code,
but might involve loading more classes. Because class loading is slow, the overall
program may run more slowly in some cases. There is also the problem of instantiating class templates for primitive types such as boolean, int, float, and so on. The
first implementation method will not work for primitive types, as these are stored
differently from object types. The second method, instantiating class files at load
time, can be made to work, although handling primitive types adds some complexity to the mechanism. In either case, there are some additional subtleties in
the design of the type-checking mechanism for generic classes that we have not
discussed.
13.4 JAVA SYSTEM ARCHITECTURE
13.4.1 Java Virtual Machine
There are several implementations of Java. This section discusses the implementation architecture used in the Sun Java compiler and the JVM. The biggest difference
between this architecture and some others is that some optimized implementations
generate native code for the underlying hardware instead of interpreted code. However, because the “just-in-time” or (JIT), compilers that generate native code on the
fly are significantly more complicated than the standard JVM architecture, we use the
JVM architecture as the basis for our discussion and analysis of Java implementation
issues.
The Java compiler produces a form of machine code called bytecode. The origin
of the name bytecode, which is a standard term that predates Java, is that bytecodes
are instructions for a virtual machine that may be 1 byte long. Many compilers for
high-level languages actually produce bytecode instead of native machine code. For
example, the influential UCSD Pascal compiler produces a form of bytecode called
P-code, which is then interpreted by a virtual machine. Common ML and Smalltalk
implementations also generate virtual machine bytecode.
Figure 13.4 shows the main parts of the JVM and some of the steps involved in
compiling and executing Java source code. The main parts of the JVM are the class
loader, the bytecode verifier, the linker, and the bytecode interpreter.
The top of Figure 13.4 shows a Java source code file A.java compiled to produce a
class file A.class. The class file is in a specific format, containing bytecode instructions
and some auxiliary data structures, including a symbol table called a constant pool.
The class loader reads the class file and arranges the bytecode instructions in a
form suitable for the verifier. The Java bytecode language contains type information,
allowing the verifier to check that bytecode is type correct before it is executed by the
virtual machine. The linker resolves interfile references, and the bytecode interpreter
executes the bytecode instructions.
The box labeled B.class in Figure 13.4 is meant to show that Java classes are
loaded incrementally, as needed during program execution, and may be loaded over
13.4 Java System Architecture
A.java
Java
Compiler
A.class
Java Virtual Machine
Loader
B.class
Network
Verifier
Linker
Bytecode Interpreter
Figure 13.4. JVM.
the network as well as from the local file system. More specifically, if class A has a
main method, then class A may be loaded as a main program and executed. If some
method of class A refers to class B, then class B will be loaded when this method is
called. The class loader then attempts to find the appropriate class file and load it
into the virtual machine so that program execution may continue.
The JVM may run several threads (processes) concurrently; it terminates when
either all nondaemon threads terminate or some thread runs the exit method of the
class Runtime or class System.
13.4.2 Class Loader
The Java run-time system loads classes, as they are needed, searching compiled bytecodes for any class that is referenced but not already loaded. There is a default loading
mechanism, which we can replace by defining an alternative ClassLoader object. One
reason to define an alternative ClassLoader is to obtain bytecodes from some remote
source and load them at run time.
We define a class loader by extending the abstract ClassLoader class and implementing its loadClass method. Here is a possible form of a class-loader definition:
class Load My Classes extends ClassLoader {
private Hashtable loaded classes = new Hashtable();
public Class loadClass(string name, boolean resolve)
throws ClassNotFoundException {
/* Code to try to load class “name” and add to “loaded classes”.
If boolean arg “resolve” is true, then invoke “resolveClass”
method to ensure that all classes referred to by class are
loaded. Raise exception if class cannot be found and loaded. */
}
...
}
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When loadClass finds the bytecodes of the class to be loaded, it may invoke the
inherited defineClass method, which takes a byte array, a start position, and a number
of bytes. This method installs this sequence of bytes as the implementation of the
desired class.
The return type of method loadClass is Class. Classes are objects, and we may be
able to find the class loader for a given class by invoking the getClassLoader method
of the class. If the class has no class loader, the method returns null.
13.4.3 Java Linker, Verifier, and Type Discipline
The linker takes a binary form of class or interface as input and adds it to the run-time
state of the JVM.
Each class is verified before it is linked and made available for execution. Java
bytecode verification is a specific process that checks to make sure that every class
has certain properties:
Every instruction must have a valid operation code.
Every branch instruction must branch to the start of some other instruction, rather
than the middle of some instruction.
Every method has a structurally correct signature.
Every instruction obeys the Java type discipline.
If an error occurs in verification, then an instance of the class VerifyError will be
thrown at the point in the Java program that caused the class to be verified.
Linking involves creating the static fields of a class or interface and initializing
them to the standard default (typically 0). Names are also resolved, which involves
checking symbolic references and replacing them with direct references that can be
processed more efficiently if the reference is used repeatedly.
Classes are not explicitly unloaded. If no instance of class is reachable, the class
may be removed as part of garbage collection.
13.4.4 Bytecode Interpreter and Method Lookup
The bytecode interpreter executes the Java bytecode and performs run-time tests
such as array-bound tests to make sure that every array access is within the declared
bounds of the array. The basic run-time architecture of the bytecode interpreter is
similar to the simple machine architecture outlined in Section 7.1, with a program
counter, instruction area, stack, and heap.
A running Java program consists of one or more sequential threads. When a new
thread is created, it is given a program counter and a stack. The program counter
indicates the next instruction to execute. The Java stack is composed of activation
records, each storing the local variables, parameters, return value, and intermediate
calculations for a Java method invocation. The JVM has no registers to hold intermediate data values; these are stored on the Java stack. This approach was taken by
the Java designers to keep the JVM instruction set compact and to facilitate implementation on a variety of underlying machine architectures. When an object or array
is created, it is stored on the heap. All threads running within a single JVM share the
same heap.
13.4 Java System Architecture
A Java activation record has three parts: local variable area, operand stack, and
data area. The sizes of the local variable area and operand stack, which are measured
in words, depend on the needs of the method and are determined at compile time. The
local variable area of a Java activation record is organized as an array of words, with
different types of data occupying different numbers of words. The operand stack is
used for intermediate calculations and passing parameters to other methods. Instead
of accessing data from memory by using a memory address, most Java bytecode
instructions operate by pushing, popping, or replacing values from the top of this
stack-within-a-stack. An add instruction with two operands, for example, pops its
two operands off the operand stack and pushes their sum onto the operand stack.
This kind of architecture leads to shorter instruction codes because an instruction
with several operands needs to identify only the operation – it is not necessary to
give an address or register number for the operands because the operands are always
the top few data items on the stack. The data area of an activation record stores data
used to support constant pool resolution (which we discuss in the next paragraph),
normal method return, and exception dispatch.
Compiled Java bytecode instructions are stored in a file format that includes a data
structure called the constant pool. The constant pool is a table of symbolic names,
such as class names, field names, and methods names. When a bytecode instruction
refers to a field, for example, the reference will actually be a number, representing
an index into the constant pool. In the instruction to get a field, for example, the
instruction may contain the number 27, indicating the 27th symbolic name in the constant pool. This approach stores symbolic names from the source code in the bytecode
file, but saves space by storing each symbolic name only once.
When executing programs, the bytecode interpreter makes the following run-time
tests to prevent type errors and preserve the integrity of the run-time system:
All casts are checked to make sure they are type safe.
All array references are checked to make sure the array index is within the array
bounds.
References are tested to make sure they are not null before they are dereferenced.
In addition, the absence of pointer arithmetic (see Section 5.2) and the use of automatic garbage collection (programmers cannot explicitly free allocated memory)
contribute to type-safe execution of Java programs.
When instructions that access a field or method are executed, there is often a
search for the appropriate location in an object template or method lookup table,
by use of the symbolic name of the field or method from the constant pool. This is
inefficient if access through the constant pool occurs frequently. Search through the
constant pool is also an execution bottleneck for concurrent programs, as this may
require a lock on the constant pool and locking the constant pool restricts execution
of other threads.
The JVM implementation optimizes the search for object fields and methods by
modifying bytecode during program execution. More specifically, bytecode instructions that refer to the constant pool are modified to an equivalent so-called quick
bytecodes, which refers to the absolute address of a field or method. For example,
the bytecode instruction
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getfield #18 <Field Obj var>
pushes the value of an object field onto the operand stack. (The object whose field
is retrieved is the object that is on top of the operand stack before the operation is
performed.) When the getfield instruction is executed, the constant pool is searched
for the symbolic field name that matches the string stored at position #18 in the
constant pool. If this field is found, and it is located 6 bytes below the first location
of the object, then the preceding instruction is replaced with the following quick
version,
getfield quick 6
which uses the calculated offset of the field from the top of the object. If the program
passes through this instruction again, as when the instruction appears inside a loop,
for example, the quick instruction is used. You do not need to understand all of the
parts of the getfield instruction – the important issue here is the way that modifying
a bytecode instruction the first time it is executed can make the program run more
quickly if this instruction is executed again.
Finding a Virtual Method by Class
There are four different bytecodes for invoking a method:
invokevirtual:
used when a superclass of the object is known at compile time,
used when only an interface of the object is known at compile
invokeinterface:
time,
invokestatic:
used to invoke static methods,
used in some special cases that will not be discussed.
invokespecial:
In simplest terms, invokevirtual is used in situations that resemble C++ virtual function calls, as Java is a statically typed programming language and all subclasses of a
class can be implemented with the same method table (vtable) order. Invokeinterface
similarly corresponds to Smalltalk method lookup, as an interface implemented by
the class of an object does not determine the relative position of the method in a
method table. We will look at invokevirtual before considering properties of invokeinterface.
Like getfield, invokevirtual involves searching for a symbolic name the first time
the instruction is executed. More specifically, suppose we have a Java source code
declaring an object reference x and invoking a method of x:
Object x;
...
x.equals(“test”);
13.4 Java System Architecture
For concreteness, and because this affects the way the code is compiled, let us assume
that x is the first local variable in this block. The call x.equals(“test”) is compiled into
something like the following sequence of bytecodes, in which the phrases to the right
of “;” are comments:
aload 1
; push local variable 1 (which is ’x’) onto the operand stack
ldc “test” ; push the string “test” onto the operand stack
invokevirtual java/lang/Object/equals(Ljava/lang/Object;)Z
The string java/lang/Object/equals(Ljava/lang/Object;)Z is called a method specification. This string is not literally located in the bytecode, but is stored in the constant
pool, as previously described.
When invokevirtual is executed, the virtual machine looks at method specification and related information in the class file and determines how many arguments
the method requires. For our example equals method, there is one argument. The
argument and the object reference, in this case the number 1 indicating the first local
variable in the current scope, are popped off the operand stack. Using the information stored in the object reference (local variable x), the virtual machine retrieves the
Java class for the object, searches the list of methods defined by that class and then
its superclasses, looking for a method matching the method specification. When the
correct method is located, the method is called in the same way that function calls
are executed in most block-structured programming languages.
Bytecode Rewriting for Invokevirtual . Although this method lookup process will
find and correctly call a method, it does not take advantage of the static type information computed by the Java compiler at compile time. More specifically, the Java
compiler can arrange every subclass method table in the same way as the superclass
table, just as the C++ compiler does. By doing this, the compiler can guarantee that
each method is located at the same relative position in all method tables associated
with all subclasses of any class. Therefore, once method lookup is used to find a
method in a method table, the offset of this method will be the same for all future
executions of this instruction.
Figure 13.5 shows how bytecode can be modified so that on subsequent executions
of an invokevirtual instruction, the correct method can be found immediately within
the method table (called “mtable” in the illustration) of the object.
The part of Figure 13.5 above the dotted line shows the invokevirtual command
in the instruction stream, followed by a pointer to the method specification in the
constant pool. After the method is found, the instruction stream can be modified
as shown below the dotted line. In the modified instruction stream, invokevirtual is
replaced with the “quick” version invokevirtual quick and the method specification
pointer is replaced with the offset of the method in the method table. When invokevirtual quick is executed, the virtual machine will use the object reference on the
operand stack to locate the class and method table and then use the mtable offset in
the instruction stream to find the appropriate method without further search.
It is important to realize that if the method invocation is in a loop, for example,
then different objects may be on the operand stack for different executions of the
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Bytecode
invokevirtual
Constant pool
“Object.eq()”
inv_virt_quick
mtable offset
Figure 13.5. Optimizing invokevirtual by rewriting bytecode.
method call. As a result, the class of the object may change from one call to the next.
However, the offset of the method in the method table will be the same on each call,
as each class is a subclass of the static type of the object reference in the Java source
code.
A short summary of this optimization is that the first time a method call instruction
is executed, Smalltalk-style method search is used to find the position of the method
in a method table. After the search succeeds, the code is changed so that if this method
call instruction is executed again, C++-style lookup will find the method quickly.
Finding a Virtual Method by Interface
The situation is similar for finding a method when an interface of the object is known
at compile time. The difference is that the optimization involving invokevirtual quick
will not work correctly. Therefore, another technique is used.
Suppose a compiled Java program contains a method declaration
void add2(Incrementable x) { x.inc(); x.inc(); }
where Incrementable is an interface guaranteeing that the argument x will have an
inc method when add2 is called at run time. There may be several classes that implement the Incrementable interface. For example, there could be classes IntCounter and
FloatCounter of the following form:
interface Incrementable {
public void inc();
}
class IntCounter implements Incrementable {
public void add(int);
public void inc();
public int value();
}
class FloatCounter implements Incrementable {
public void inc();
13.4 Java System Architecture
public void add(float);
public float value();
}
Because the classes implementing Incrementable might be declared in different packages, neither referring to the other, there is no reason to believe that the method inc
is located at the same offset in the two method tables. (In fact, it is easy to construct
a set of interfaces and classes for which there is no way of making all classes that
implement each interface compatible in this way.)
When the add2 method is compiled, the Java compiler generates code of the
following form (with method specification from the constant pool written in the
code for clarity):
aload 1 ; push local variable 1 (i.e. x) onto the stack
invokeinterface package/inc()Z 1
When invokeinterface is executed, the class of the object is found and the method table
for this class is searched to find the method with the given method specification. When
the method is located, the method is called as for invokevirtual.
Because the offset of the method in the method table may be different on the
next execution of this instruction, it is not correct to rewrite invokeinterface to invokevirtual quick. However, there is some possibility that the next execution of this
instruction will be for an object from the same class, so it seems wasteful to discard
the offset and other information. Figure 13.6 shows the bytecode modification used
for invokeinterface.
The part of Figure 13.6 above the dotted line shows the initial instruction sequence, with the invokeinterface instruction followed by a pointer to the method
specification in the constant pool and an unused empty location. After execution of invokeinterface, the instruction is replaced with invokeinterface quick and a
pointer to the class and method is stored below the method specification. When
Bytecode
Constant pool
invokeinterface
“Incr...inc()”
inv_int_quick
“Incr...inc()”
Figure 13.6. Optimizing invokeinterface by rewriting bytecode
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invokinterface quick is executed on another pass through this instruction sequence, the
class of the object operand is compared with the cached class. If the object is from the
same class, then the same method is used. If the object is from a different class, then,
because the method specification is still available in the instruction sequence, the
same search process can be followed as if the instruction were invokeinterface
again.
Efficiency Summary. In Java, the method call overhead is high the first time an
instruction is executed and potentially much lower on subsequent executions. On
the first lookup that uses invokevirtual there are two indirections (one to look up the
constant table of the class this method is in and one to look up the method table of
the class of the object), two array indexes (in the constant table and the method list),
and the cost of rewriting some bytecode. On subsequent calls, there is one indirection
and one array index. For invokeinterface, there is a search through the method table
the first time an instruction is executed. On subsequent calls, there is either one array
index or another search through a method table if the class is changed.
13.5 SECURITY FEATURES
Computer security involves preventing unauthorized use of computational resources.
Some computer security topics are network security, which provides methods for
preserving the secrecy and integrity of information transmitted on the network, and
operating system security, which is concerned with preventing an attacker from gaining access to a system. In this section, we look at ways that programming languages
can help and hinder our efforts to build secure systems, with particular attention to
how Java is designed to improve the security of systems that use Java.
One way that attackers may try to gain access to a system is to send data over
the network to a program that processes network input. For example, many attacks
against the e-mail program Sendmail have been devised over the years. These attacks
typically take advantage of some error or oversight in a particular implementation of
Sendmail. Sendmail is a target because this program accepts network connections and
runs with superuser privileges. In Subsection 13.5.1 we will look at buffer overflow
attacks, which are an important example of the kind of attack that is often mounted
against network programs.
Because Java code is sometimes transmitted over the network, a computer user
may run code that comes from another site, such as a web page. Mobile code, a
general term for code transmitted over the network before it is executed, provides
another way for an attacker to try to gain access to a system or cause damage to a
system. The two main mechanisms for managing risks raised by mobile code are called
sandboxing and code signing. The idea behind sandboxing, which was invented before
the development of Java, is to give a program a restricted execution environment. This
restricted environment, called a sandbox, may contain certain functions (analogous
to certain toys you might give a child to play with in a sandbox), but the program
cannot call functions or perform actions that are outside the sandbox.
Code signing is used to determine which person or company produced a class file.
With digital signatures, it is possible for the producer of code to sign it in a way that
any recipient can verify the signature. The JVM uses digital signatures to identify the
producer of a file that contains Java bytecode. The user of a computer system can
13.5 Security Features
specify a set of trusted producers and allow code from these producers to execute on
their computer. With the Java security manager, it is also possible to assign different
access permissions to different code producers.
After looking at buffer overflow, we will discuss some of the main features of the
Java sandbox. If you want to learn more about the Java security model, you can read
one of the books on Java security or find additional information on the web.
13.5.1 Buffer Overflow Attack
In a buffer overflow attack, an attacker sends network messages that cause a program
to read data into a buffer in memory. If the program does not check the number of
bytes placed in the buffer, then the attacker may be able to write past the allocated
memory. By writing over the return address that is stored in the activation record of a
function, the attacker may cause the program to misbehave and “return” by jumping
to instructions chosen by the attacker. This is a powerful form of attack. The vast
majority of security advisories issued by the Computer Emergency Response Team
(CERT) over the past decade or two have been security problems related to buffer
overflow.
Java code is not vulnerable to buffer overflow attacks because the JVM checks
array bounds at run time. This prevents a function from writing more data into an
array than the array can hold. To show how important this feature is for Java security,
we look at how a buffer overflow attack may be designed against C code.
Here is a very simple C function that illustrates the principle, taken from an article
called “Smashing the Stack for Fun and Profit,” written by someone called Aleph
One (Phrack 49(14), 1996):
void f (char *str) {
char buffer[16];
...
strcpy(buffer,str);
}
This function copies a character string, passed by pointer, into the array buffer on
the run-time stack. The C function strcpy() copies the contents of *str into buffer[ ]
until a null character is found in the string. Because the function does not check
the length of the string, careful C programmers would consider the use of strcpy()
a coding error. The similar function strncpy(), which has an additional parameter n
and copies at most n characters, should be used instead.
If the function f is called from this main program,
void main() {
char large string[256];
int i;
for( i = 0; i < 255; i++)
large string[i] = ’A’;
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large string[255]=0;
f(large string);
}
then the call to strcpy in function f will write over locations on the stack. Because
buffer[ ] is 16 bytes long and large string is 256 bytes, the copy will write 240 bytes beyond the end of the array. This will change the return address of f, which is sitting in
the activation record for f a few memory locations away from buffer[ ]. Because
large string contains the character ’A’, with hex character value 0x41, the return
address of f is set to 0x41414141. Because this address is outside of the process
address space, a segmentation fault will occur when function f tries to “return” to
location 0x41414141. However, if we change ’A’ to a value that is the address of
a meaningful instruction, the buffer overflow will cause function f to jump to that
location.
Several steps are needed to turn this simple idea into an attack. First, the attacker
needs to choose a function that reads data that the attacker can control. Next, the
attacker must figure out what data will cause the function to misbehave in a way that
is of interest to the attacker. On some computer architectures, an attacker can fill a
buffer with code the attacker wants to execute and then cause the function to jump
into the buffer. This allows the attacker to execute any instructions.
13.5.2 The Java Sandbox
The JVM executes Java bytecode in a restricted environment called the Java sandbox.
The term sandbox is metaphorical – there is no literal data structure or code called
the sandbox. The word sandbox is used to indicate that, when bytecode is executed,
some operations that can be written in the Java language might not be allowed to
proceed, just the way that, when a child plays in a sandbox, an adult supervisor may
give the child only those toys that the supervisor considers safe.
The JVM restricts the operation of a Java bytecode program by using four interrelated mechanisms: the class loader, the Java bytecode verifier, run-time checks
performed by the JVM, and the actions of the security manager. Although the class
loader, bytecode verifier, and virtual machine run-time checks perform general functions related to type correctness and preserving the integrity of the run-time system,
the security manager is a specific object designed specifically for security functions.
We take a quick look at how each of these mechanisms contributes to Java security.
If you are interested in learning more, you may want to read Securing Java by Gary
McGraw and Ed Felten (Wiley, 1999) or other books on Java security or the JVM.
Class Loader
The class loader, discussed in Subsection 13.4.2, contributes to the Java sandbox in
three ways:
The class-loader architecture separates trusted class libraries from untrusted
packages by making it possible to load each with different class loaders.
The class-loader architecture provides separate name spaces for classes loaded
by different class loaders.
13.5 Security Features
The class loader places code into categories (called protection domains) that let
the security manager restrict the actions that specific code will be allowed to take.
The Bytecode Verifier and Virtual Machine Run-Time Tests
The Java bytecode verifier checks Java bytecode programs before they are executed,
as discussed in Subsection 13.4.3. Complimentary run-time checks, such as arraybounds checks, are performed by the JVM. Together, these checks provide the following guarantees:
No Stack Overflow or Underflow: The verifier examines the way that bytecode
manipulates data on the operand stack and guarantees that no method will overflow the operand stack allocated to it.
All Methods are Called with Parameters of the Correct Type: This type-correctness
guarantee prevents the kind of type-confusion attacks discussed in Subsection
13.5.3.
No Illegal Data Conversions (Casts) Occur: For example, treating an integer as a
pointer is not allowed. This is also essential to prevent the kind of type-confusion
attacks discussed in Subsection 13.5.3.
Private, Public, Protected, and Default Accesses must be Legal: In other words,
no improper access to restricted classes, interfaces, variables, and methods will
be allowed.
The Security Manager
The security manager is a single Java object that answers at run time. The job of the
security manager is to keep track of which code is allowed to do which dangerous
operations. The security manager does its job by examining the protection domain
associated with a class. Each protection domain has two attributes – a signer and a
location. The signer is the person or organization that signed the code before it was
loaded into the virtual machine. This will be null if the code is not signed by anyone.
The location is the URL where the Java classes reside. These attributes are used
to determine which operations the code is allowed to perform. A standard security
manager will disallow most operations when they are requested by untrusted code
and may allow trusted code to do whatever it wants.
A running virtual machine can have only one security manager installed at a time.
In addition, once a security manager has been installed, it cannot be uninstalled without restarting the virtual machine. Java-enabled applications such as web browsers
install a security manager as part of their initialization, thus locking in the security
manager before any potentially untrusted code has a chance to run.
The security manager does not observe the operation of other running bytecode.
Instead, the security manager uses the system policy to answer questions about access
permissions. It is up to library code that has access to important resources to consult
the security manager. In outline, the standard Java library uses the security manager
according to the following steps:
If a Java program makes a call to a potentially dangerous operation in the Java
API, the associated Java API code asks the security manager whether the operation should be allowed.
If the operation is permitted, the call to the security manager returns normally
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(without throwing an exception) and the Java API performs the requested operation.
If the operation is not permitted, the security manager throws a SecurityException.
This exception propagates back to the Java program.
One limitation of this mechanism is that a hostile applet or other untrusted code
may catch a security exception by using a try-finally block. Therefore, although the
security manager may prevent an action by raising an exception, this exception might
not be noticed by the user of a system under attack.
13.5.3 Security and Type Safety
Type safety is the cornerstone of Java security, in the sense that all of the sandbox
mechanisms rely on type safety. To understand the importance of type safety, we
discuss some of the ways that type errors can allow code to perform arbitrary and
potentially dangerous actions. We begin with a C example, as C has many type loopholes. It is then shown how similar problems could occur in Java code if it were
possible to create two pointers with different types that point to the same object.
One way for a C/C++ program to call an arbitrary function is through a function
pointer. For example, suppose a C/C++ program contains lines of the following form:
int (*fp)() /* variable “fp” is a function pointer
...
fp = addr; /* assign fp an address stored in an integer variable
(*fp)(n); /* call the function at this address
*/
*/
*/
Using casts, code like this can call a function at virtually any address. Therefore, if
a program may cast integers to function pointers, it is impossible to tell in advance
which functions the program calls.
Similar type confusion between object types would also allow a program to perform dangerous actions. Here is an example from Securing Java by McGraw and
Felten. Suppose that it is possible to create a Java program with two pointers to the
same object o, one pointer with type T and one with type U. Let us further suppose
that classes T and U are defined as follows:
class T {
SecurityManager x;
}
class U {
MyObject x;
}
where MyObject is some class that we can write in whatever way we want. Here is
a program that uses the two pointers to object o to change the security manager,
even though the fields of the security manager are declared private and should not
13.6 Java Summary
be subject to alteration by arbitrary code:
T t = (the pointer to object o with type T);
U u = (the pointer to object o with type U);
t.x = System.getSecurity(); // the Security Manager
MyObject m = u.x;
The first two lines of this code segment create two references to object o with types
T and U, respectively. The next line assigns the security manager to the field x of
object o. This assignment is allowed because the type of field x is SecurityManager.
The final line then gives us a reference m to the security manager, where the type of
m is MyObject, a class that we can declare in any way we wish. By changing the fields
of m, the program can change the private fields of the security manager.
Although this example shows how type confusion can be used to corrupt the
security manager, the basic tactic may be used to corrupt almost any part of the
running Java system.
An early type-confusion attack on Java was made possible by a buggy class loader
and linker. The incorrect system aborted class loading if an exception was raised but
did not remove the class name and associated class methods. More specifically, the
first step in class loading was to add the class name and constructor to the table of
defined references, then load the byte code for the class. If an exception was raised
during the second step, the first step was not undone. This made it possible to then
successfully load another class with a different interface and to use the first class
name for objects of the second class.
13.6 JAVA SUMMARY
The Java programming language and execution environment were designed with
portability, reliability, and safety in mind. These goals took higher priority than efficiency, although many engineers and implementers have worked hard to make Java
implementations run efficiently.
Java programs are compiled from Java source code (in the Java programming
language) to Java bytecode, an interpreted language that is executed on a simple
stack-based virtual machine. The run-time environment provides dynamic linking,
allowing classes to be loaded into the virtual machine as they are needed. The Java
language contains specific concurrency primitives (discussed in Chapter 14), and the
virtual machine runs multiple threads simultaneously.
Java is a modern general-purpose object-oriented language that is used in many
academic, educational, and commercial projects. We studied the object-oriented features of Java, the Java Virtual Machine and implementation of method lookup, and
Java security features.
Objects and Classes
A Java object has fields, which are data members of the object, and methods, which
are member functions. All objects are allocated on the run-time heap (not the runtime stack), accessible through pointers, and garbage collected.
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Every Java object is an instance of a class. Classes can have static (class) fields
that are initialized when the class is loaded into the virtual machine. A constructor
of the class is used to create and initalize objects, and objects may have a finalize
method that is called when the garbage collector reclaims the space occupied by the
object.
Dynamic Lookup
Method lookup in Java is more efficient than in Smalltalk, because the static type
system gives the compiler more information, but often is less efficient than in C++.
When instructions that access a field or method are executed the first time, there
is a search for the appropriate location in an object template or method lookup
table. After the first execution of a reference inside a program, the virtual machine
optimizes the search for object fields and methods by modifying bytecode. Bytecode
instructions that refer to the constant pool are modified to an equivalent so-called
quick bytecode, which refers to the absolute address of a field or method.
Encapsulation
Java has four visibility distinctions for fields and methods:
Public: accessible anywhere the class is visible.
Protected: accessible to methods of the class and any subclasses as well as to other
classes in the same package.
Private: accessible only in the class itself.
Package: accessible only to code in the same package; not visible to subclasses
in other packages. Members declared without an access modifier have package
visibility.
Names declared in another package can be accessed by import or by qualified names.
Java type checking, applied when source code is compiled and again when bytecode is loaded into the virtual machine, together with run-time tests performed by
the bytecode interpreter, guarantees that private fields are truly private and similarly
for other visibility levels.
Inheritance
In Java terminology, a subclass inherits from its superclass. The Java inheritance
mechanism is essentially similar to that of Smalltalk, C++, and other class-based
object-oriented languages. Java provides single inheritance for classes and multiple
inheritance for interfaces.
Java individual methods of a class or an entire class can be declared final. If a
method is declared final, then the method cannot be overridden in any subclass. If
a class is declared final, then the class cannot have any subclasses. This gives the
designer of a class the ability to fix the implementation of a method (or the entire
class) by keeping designers of subclass from overriding the method.
In principle, every class declared in a Java program inherits from another class.
The class Object is the one class that has no superclasses.
13.6 Java Summary
Subtyping
Subtyping for classes is determined by the class hierarchy and the interface mechanism. Specifically, if class A extends class B, then the type of A objects is a subtype of
the type of B objects. There is no other way for one class to define a subtype of another,
and no private base classes (as in C++) to allow inheritance without subtyping.
A class may be declared to implement one or more interfaces, meaning that any
instance of the class implements all the abstract methods specified in the interface.
This (multiple) interface subtyping allows objects to support (multiple) common
behaviors without sharing any common implementation.
The Java types are divided into two categories, called primitive types and reference types. The primitive types include boolean and seven numeric types. The three
forms of reference types are class types, interface types, and array types. All class,
interface, and array types are considered subtypes of Object. Java exception types are
subclasses of Throwable, which is a subtype of Object.
Virtual Machine Architecture
The main parts of the Java Virtual Machine are the class loader, the bytecode verifier,
the linker, and the bytecode interpreter. A Java source code file is compiled to produce a class file. Class files are in a specific format that contains bytecode instructions
and some auxiliary data structures, including a symbol table called a constant pool.
The class loader reads the class file and arranges the bytecode instructions in a form
suitable for the verifier. The Java bytecode language contains type information, allowing the verifier to checks that bytecode is type correct before it is executed by the
virtual machine. The linker resolves interfile references, and the bytecode interpreter
executes the bytecode instructions.
When instructions that access a field or method are executed, there is a search
for the appropriate location, based on the symbolic name of the field or method
in the constant pool. The JVM implementation optimizes this search by modifying
bytecode during program execution. Bytecode instructions that initially refer to the
constant pool are modified to an equivalent so-called quick bytecode, which refers
to the absolute address after the name has been resolved. Different degrees of optimization are possible for method invocation when the class is known and method
invocation when the interface is guaranteed to be the same for all executions of the
instruction.
Security
Java security is provided by type-correct compilation and execution and by access
control through the security manager. The JVM restricts the operation of a Java
bytecode program by using four interrelated mechanisms: the class loader, the Java
bytecode verifier, run-time checks performed by the JVM, and the actions of the
security manager. Whereas the class loader, bytecode verifier, and virtual machine
run-time checks perform general functions related to type correctness and preserving
the integrity of the run-time system, the security manager is a specific object designed
specifically for security functions. The techniques used to restrict execution of a
bytecode program are collectively called the Java sandbox.
Code signing is used to determine which person or company produced a class file.
The security manager uses the producer of a class and the URL from which it was
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obtained to decide whether it can perform potentially hazardous operations such as
creating, opening, or writing into a file.
Without type safety, Java programs could be susceptible to the buffer overflow
attacks and could carry out the type-confusion attacks we discussed in Section 13.5.
EXERCISES
13.1 Initializing Static Fields
Why do static fields of a class have to be initialized when the class is loaded? Why
can’t we initialize static fields when the program starts? Give an example of what
goes wrong if, instead of static fields being initialized too early, they are initialized
too late.
13.2 Java final and finalize
Java has keywords final and finalize.
(a) Describe one situation in which you would want to mark a class final, and
another in which you would want a final method but not a final class.
(b) Describe the similarity and differences between Java final and the C++ use of
nonvirtual in similar situations.
(c) Why is Java finalize (the other keyword!) a useful feature?
13.3 Subtyping and Exceptions
In Java, a method that can throw an exception (other than from a subclass of Error
or RuntimeException) must either catch the exception or specify the types of possible
exceptions with a throws clause in the method declaration. For example, a method
declaration might have the form
public void f(int x) throws Exception1, Exception2
meaning that a call to f may either terminate normally or raise one of the listed
exceptions (without catching them internally).
Assuming that the type of the method f in B is a subtype of the method f in A,
class declarations of the following form are type correct in principle:
class A {
...
public Returntype1 f(Argtype1 x) . . .
}
class B extends A {
...
public Returntype2 f(Argtype2 x) . . .
}
This example of function subtyping Argtype2 → Returntype2 <: Argtype1 → Returntype1 requires Returntype2 <: Returntype1 and Argtype1 <: Argtype2.
Suppose we keep the argument and return types the same, but vary the set of
exceptions, as in the following code:
class A {
...
public Returntype f(Argtype x) throws Exception1, Exception2, . . .
Exercises
}
class B extends A {
...
public Returntype f(Argtype x) throws Exception1, Exception2, . . .
}
(a) What relation between the two sets of exceptions is required for the subclass
B to be a subtype of class A? Do the sets have to be the same? Or would it be
alright for one to be a subset of the other? Explain briefly. In this part, do not
worry about subtyping of exception types – we are concerned with only the
sets of types.
(b) Now suppose that we allow for the possibility that the exceptions in one set
could be subtypes of exceptions in the other set. What is the relation we require
for class B to be a subtype of class A?
13.4 Java Interfaces and Multiple Inheritance
In C++, a derived class may have multiple base classes. In contrast, a Java derived
class may only have one base class but may implement more than one interface.
This question asks you to compare these two language designs.
(a) Draw a C++ class hierarchy with multiple inheritance using the following
classes:
Pizza, for a class containing all kinds of pizza,
Meat, for pizza that has meat topping,
Vet, for pizza that has vegetable topping,
Sausage, for pizza that has sausage topping,
Ham, for pizza that has ham topping,
Pineapple, for pizza that has pineapple topping,
Mushroom, for pizza that has mushroom topping,
Hawaiian, for pizza that has ham and pineapple topping.
For simplicity, treat sausage and ham as meats and pineapple and mushroom
as vegetables.
(b) If you were to implement these classes in C++ for some kind of pizzamanufacturing robot, what kind of potential conflicts associated with multiple
inheritance might you have to resolve?
(c) If you were to represent this hierarchy in Java, which would you define as
interfaces and which as classes? Write your answer by carefully redrawing
your picture, identifying which are classes and which are interfaces. If your
program creates objects of each type, you may need to add some additional
classes. Include these in your drawing.
(d) Give an advantage of C++ multiple inheritance over Java classes and interfaces
and one advantage of the Java design over C++.
13.5 Array Covariance in Java
As discussed in this chapter, Java array types are covariant with respect to the types
of array elements (i.e., if B <: A, then B[ ] <: A[ ]). This can be useful for creating
functions that operate on many types of arrays. For example, the following function
takes in an array and swaps the first two elements in the array.
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Portability and Safety: Java
1:
2:
3:
4:
5:
6:
7:
public swapper (Object[] swappee){
if (swappee.length > 1){
Object temp = swappee[0];
swappee[0] = swappee[1];
swappee[1] = temp;
}
}
This function can be used to swap the first two elements of an array of objects of
any type. The function works as is and does not produce any type errors at compile
time or run time.
(a) Suppose a is declared by Shape[] a to be an array of shapes, where Shape is some
class. Explain why the principle if B <: A, then B[ ] <: A[ ] allows the type checker
to accept the call swapper(a) at compile time.
(b) Suppose that Shape[] a is as in part (a). Explain why the call swapper(a) and
execution of the body of swapper will not cause a type error or exception at run
time.
(c) Java may insert run-time checks to determine that all the objects are of the
correct type. What run-time checks are inserted in the compiled code for the
swapper function and where? List the line number(s) and the check that occur
on that line.
(d) A friend of yours is aghast at the design of Java array subtyping. In his brilliance,
he suggests that Java arrays should follow contravariance instead of covariance
(i.e., if B <: A, then A[ ] <: B[ ]). He states that this would eliminate the need
for run-time type checks. Write three lines or fewer of code that will compile
fine under your friend’s new type system, but will cause a run-time type error
(assuming no run-time type tests accompany his rule). You may assume you
have two classes, A and B, that B is a subtype of A, and that B contains a method,
foo, not found in A. Here are two declarations that you can assume before your
three lines of code:
B b[];
A a[] = new A[10];
(e) Your friend, now discouraged about his first idea, decides that covariance in
Java is all right after all. However, he thinks that he can get rid of the need
for run-time type tests through sophisticated compile-time analysis. Explain
in a sentence or two why he will not be able to succeed. You may write a
few lines of code similar to those in part (d) if it helps you make your point
clearly.
13.6 Java Bytecode Analysis
One property of a Java program that is checked by the verifier is that each object
must be properly initialized before it is used. This property is fairly difficult to
check. One relatively simple part of the analysis, however, is to guarantee that each
subclass constructor must call the superclass constructor. The reason for this check
is to guarantee that the inherited parts of every object will be initialized properly. If
we were designing our own bytecode verifier, there are two ways we might consider
designing this check:
Exercises
(i) The verifier can analyze the bytecode program to make sure that on every execution of a subclass constructor, there is some call to a superclass constructor.
(ii) The verifier can check that the first few bytecode instructions of a subclass
constructor contain a call to the superclass constructor, before any loop or
jump inside the subclass constructor.
In design (i), the verifier should accept every bytecode program that satisfies this
condition and reject every bytecode program that allows some subclass constructor
to complete without calling the superclass constructor. In design (ii), some subclass
constructors that would be acceptable according to condition (i) will be rejected by
the bytecode verifier. However, it may be possible to design the Java source code
compiler so that every correct Java source code program is compiled to bytecode
that meets the condition described in design (ii).
(a) If you were writing a Java compiler and another person on your team was
writing the bytecode verifier, which design would you prefer? Explain briefly.
(b) If you were writing a Java compiler and your manager told you that the standard
verifier used design (ii) instead of (i), could you still write a decent compiler?
Explain briefly.
(c) If you were writing a bytecode verifier and your manager offered to double
your salary if you satisfied design condition (i) instead of (ii) but would fire
you if you failed, would you accept the offer? Explain in one sentence.
13.7 Exceptions, Memory Management, and Concurrency
This question asks you to compare properties of exceptions in C++ and Java.
(a) In C++, objects may be reside in the activation records that are deallocated
when an exception is thrown. As these activation records are deallocated, all
of the destructors of these stack objects are called. Explain why this is a useful
language mechanism.
(b) In Java, objects are allocated on the heap instead of the stack. However, an
activation record that is deallocated when an exception is raised may contain a pointer to an object. If you were designing Java, would you try to call
the finalize methods of objects that are accessible in this way? Why or why
not?
(c) Briefly explain one programming situation in which the C++ treatment of
objects and exceptions is more convenient than Java and one situation in which
Java is more convenient than C++.
(d) In languages that allow programs to contain multiple threads, several threads
may be created between the point where an exception handler is established
and the point where an exception is thrown. In the spirit of trying to abort any
computation that is started between these two points, a programming language
might try to abort all such threads when an exception is raised. In other words,
there are two possible language designs:
raising an exception affects only the current thread, or
raising an exception aborts all threads that were started between the point
where the exception handler is established and the point where the exception
is thrown.
Which design would be easier to implement? Explain briefly.
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Portability and Safety: Java
(e) Briefly explain one programming situation in which you would like raising an
exception to abort all threads that were started between the point where the
exception handler is established and the point where the exception is thrown.
Can you think of a programming situation in which you would prefer not to
have these threads terminated?
13.8 Adding Pointers to Java
Java does not have general pointer types. More specifically, Java has primitive types
(Booleans, integers, floating-point numbers . . . ) and reference types (objects and
arrays). If a variable has a primitive type, then the variable is associated with a
location and a value of that type is stored in that location. When a variable of
primitive type is assigned a new value, a new value is copied into the location. In
contrast, variables of reference type are implemented as pointers. When a variable
referring to an object is assigned a value, the location associated with the variable
will contain a pointer to the appropriate object.
Imagine that you were part of the Java design team and you believe strongly
in pointers. You want to add pointers to Java, so that, for every type A, there is
a type A* of pointers to values of type A. Gosling is strongly opposed to adding
an address-of operator (like & in C), but you think there is a useful way of adding
pointers without adding address-of.
One way of designing a pointer type for Java is to consider A* equivalent to the
following class:
class A* {
private A data=null;
public void assign(A x) {data=x;};
public A deref(){return data;}
A*(){};
};
Intuitively, a pointer is an object with two methods, one assigning a value to
the pointer and the other dereferencing a pointer to get the object it points to.
One pointer, p, can be assigned the object reached by another, q, by writing
p.assign(q.deref()). The constructor A* does not do anything because the initialization clause sets every new pointer by using the null reference.
(a) If A is a reference type, do A* objects seem like pointers to you? More specifically, suppose A is a Java class with method m that has a side effect on the
object and consider the following code:
A x = new A( . . . );
A* p = new A*();
p.assign(x);
(p.deref()).m();
Here, pointer p points to the object named by x and p is used to invoke a method.
Does this modify the object named by x? Answer in one or two sentences.
(b) What if A is a primitive type, such as int? Do A* objects seem like pointers to
you? [Hint: Think about the code in part (a).] Answer in one or two sentences.
(c) If A <: B, should A* <: B*? Answer this question by comparing the type of
A*::assign with the type of B*::assign and comparing the type of A*::deref with
the type of B*::deref.
Exercises
(d) If Java had templates, then you could define a pointer template Ptr, with Ptr A
defined as A* above. One of the issues that arises in adding templates to Java
is subtyping for templates. From the Ptr example, do you think it is correct to
assume that, for every class template Template, if A <: B then Template A <:
Template B ? Explain briefly.
13.9 Stack Inspection
One component of the Java security mechanism is called stack inspection. This
problem asks you some general questions about activation records and the runtime stack then asks about an implementation of stack inspection that is similar to
the one used in Netscape 3.0. Some of this problem is based on the book Securing
Java, by Gary McGraw and Ed Felten (Wiley, 1999).
Parts of this problem ask about the following functions, written in a Java-like
pseudocode. In the stack used in this problem, activation records contain the usual
data (local variables, arguments, control and access links, etc.) plus a privilege flag.
The privilege flag is part of our security implementation and will be subsequently
discussed. For now, SetPrivilegeFlag() sets the privilege flag for the current activation
record:
void url.open(string url) {
int urlType = GetUrlType(url); // Gets the type of URL
SetPrivilegeFlag();
if (urlType == LOCAL FILE)
file.open(url);
}
void file.open(string filename) {
if (CheckPrivileges())
{
// Open the file
}
else
{
throw SecurityException;
}
}
void foo() {
try {
url.open(“confidential.data”);
} catch (SecurityException) {
System.out.println(“Curses, foiled again!\n”);
}
// Send file contents to evil competitor corporation
}
(a) Assume that the URL confidential.data is indeed of type LOCAL FILE and that
sys.main calls foo(). Fill in the missing data in the following illustration of the
activation records on the run-time stack just before the call to CheckPrivileges().
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Portability and Safety: Java
For convenience, ignore the activation records created by calls to GetUrlType()
and SetPrivilegeFlag(). (They would have been destroyed by this point anyway.)
Activation Records
(1)
Principal
Principal
(2)
(3)
control link
access link
(2)
(1)
•
(3)
(3)
•
(4)
(2)
•
(5)
(1)
url.open
(4)
control link
access link
file.open
(5)
control link
access link
foo
(6)
sys.main
(7)
foo
(8)
url.open
(9)
file.open
control link
access link
privilege flag
control link
access link
privilege flag
control link
access link
privilege flag
url
Compiled Code
( ),
•
code for url.open
( ),
•
code for file.open
( ),
•
code for foo
NOT SET
(
(
)
)
(
(
)
)
(
(
)
)
“
control link
access link
privilege flag
url
Closures
SYSTEM
UNTRUSTED
“
”
”
(b) As part of stack inspection, each activation record is classified as either SYSTEM
or UNTRUSTED. Functions that come from system packages are marked SYSTEM.
All other functions (including user code and functions coming across the network) are marked UNTRUSTED. UNTRUSTED activation records are not allowed
to set the privilege flag.
Effectively, every package has a global variable Principal that indicates
whether the package is SYSTEM or UNTRUSTED. Packages that come across the
network have this variable set to UNTRUSTED automatically on transfer. Activation records are classified as SYSTEM or UNTRUSTED based on the value of
Principal, which is determined according to static scoping rules.
List all activation records (by number) that are marked SYSTEM and list all
activation records (by number) that are marked UNTRUSTED.
(c) CheckPrivileges() uses a dynamic scoping approach to decide whether the function corresponding to the current activation record is allowed to perform privileged operations. The algorithm looks at all activation records on the stack,
from most recent on up, until one of the following occurs:
It finds an activation record with the privilege flag set. In this case it returns
TRUE.
Exercises
It finds an activation record marked UNTRUSTED. In this case it returns FALSE.
(Remember that it is not possible to set the privilege flag of an untrusted
activation record.)
It runs out of activation records to look at. In this case it returns FALSE.
What will CheckPrivileges() return for the preceding stack [resulting from the
call to foo() from sys.main]? Please answer True or False.
(d) Is there a security problem in this code? (That is, will something undesirable
or “evil” occur when this code is run?)
(e) Suppose that CheckPrivileges() returned FALSE and thus a SecurityException was
thrown. Which activation records from part (a) will be popped off the stack
before the handler is found? List the numbers of the records.
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PART 4
Concurrency and Logic Programming
14
Concurrent and Distributed Programming
A concurrent program defines two or more sequences of actions that may be executed
simultaneously. Concurrent programs may be executed in two general ways:
Multiprogramming. A single physical processor may run several processes simultaneously by interleaving the steps of one process with steps of another. Each
individual process will proceed sequentially, but actions of one process may occur
between two adjacent steps of another.
Multiprocessing. Two or more processors may share memory or be connected by
a network, allowing processes on one processor to interact with processes running
simultaneously on another.
Concurrency is important for a number of reasons. Concurrency allows different
tasks to proceed at different speeds. For example, multiprogramming allows one
program to do useful work while another is waiting for input. This makes more efficient use of a single processor. Concurrency also provides programming concepts that
are important in user interfaces, such as window systems that display independent
windows simultaneously and for networked systems that need to send and receive
data to other computers at different times. Multiprocessing makes more raw processing power available to solve a computational problem and introduces additional
issues such as unreliability in network communication and the possibility of one processor proceeding while another crashes. Interaction between sequential program
segments, whether they are on the same processor or different processors, raises
significant programming challenges.
Concurrent programming languages provide control and communication abstractions for writing concurrent programs. In this chapter, we look at some general issues
in concurrent programming and three language examples: the actor model, Concurrent ML, and Java concurrency. Some constructs we consider are more appropriate
for multiprogramming, some for multiprocessing, and some are useful for both.
Historically, many concurrent systems have been written in languages that do
not support concurrency. For example, concurrent processes used to implement
a network router can be written in a language like C, with system calls used in
place of concurrency constructs. Concurrently executed sequential programs can
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Concurrent and Distributed Programming
TONY HOARE
A buoyant, inquisitive person with a twinkle in his eye, Charles Anthony
Richard Hoare began his career in computing in 1960 as a programmer
for Elliott Brothers, a small scientific computer manufacturer. He became
Professor of Computer Science at the Queen’s University in Belfast in
1968 and moved to Oxford University in 1977. Professor Hoare retired
from Oxford at the mandatory age in 1999 but continues to be active in
computing at Microsoft Research Labs in Cambridge, U.K.
Midway through his career, Hoare received the ACM Turing Award in
1980, “For his fundamental contributions to the definition and design of
programming languages.” At the time he was recognized for his work on
methods for reasoning about programs, clever algorithms such as Quicksort, data structuring techniques, and the study of monitors. In 1980, he
was also beginning his study of concurrency through the development of
Communicating Sequential Processes (CSP). Along with Milner’s Calculus of Communicating Systems (CCS), CSP has been a highly influential
system for specifying and analyzing certain types of concurrent systems.
The standard method for proving properties of imperative programs by
using loop invariants, defined in his 1969 journal article, is now commonly called Hoare Logic.
The picture on this page shows Tony Hoare and his wife Jill standing
outside Buckingham Palace, holding the medal he received when he
was knighted by Queen Elizabeth II.
communicate by reading and writing to a shared structure or by using abstractions
provided by the operating system. Programs can send or receive data from a Unix
pipe, for example, by function calls that are handled by the operating system. C
programs can also create processes and terminate them by using system calls.
There are several reasons why a concurrent programming language can be a better
problem-solving tool for concurrent programming than a sequential language. One is
that a concurrent programming language can provide abstractions and control structures that are designed specifically for concurrent programming. We look at several
14.1 Basic Concepts in Concurrency
examples in this chapter. Another is that a programming language may provide
“lighter-weight” processes. Operating systems generally give each process its own
address space, with a separate run-time stack, program counter, and heap. Although
this protects one process from another, there are significant costs associated with
setting up a process and in switching the flow of control from one process to another. Programming language implementations may provide lightweight processes
that run in the same operating system address space but are protected by programming language properties. With lightweight processes, there is less cost associated
with concurrency, allowing a programmer to use concurrency more freely. Finally,
there is the issue of portability across operating systems. Although it is possible to
do concurrent programming with C and Unix system calls, for example, programs
written for one version of Unix may not run correctly under other versions of Unix
and may not be easy to port to other operating systems.
14.1 BASIC CONCEPTS IN CONCURRENCY
Before looking at some specific language designs, we discuss some of the basic issues
in concurrent programming and some of the traditional operating system mechanisms
used to allow concurrent processes to cooperate effectively.
A sequence of actions may be called a process, thread, or task. Although some
authors use these terms interchangeably, they can have different connotations. The
word process is often used to refer to an operating system process, which generally
runs in its own address space. Threads, in specific languages like Concurrent ML
and Java, may run under control of the language run-time system, sharing the same
operating system address space. Some authors define thread to mean “lightweight
process,” which means a process that does not run in a separate operating system
address space. The term process is used in the rest of Section 14.1, as many of the
mechanisms we discuss are used in operating systems. In discussing Concurrent ML
and Java, we use the term thread because this is the standard term for these languages
and because threads defined in a Concurrent ML or Java program are lightweight
processes that all run in the same operating system address space.
14.1.1 Execution Order and Nondeterminism
Concurrent programs may have many possible execution orders. An elementary
and historical concurrent programming construct is the cobegin/coend form used in
Concurrent Pascal. This language was developed by P. Brinch Hansen at Caltech in
the 1970s. Here is an example Concurrent Pascal program that has several execution
orders:
x := 0;
cobegin
begin x := 1; x := x+1 end;
begin x := 2; x := x+1 end;
coend;
print(x);
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This program executes sequentially, as usual in Pascal, except that, within the cobegin/
coend statement, all of the independent blocks are executed concurrently. Concurrent
blocks may interact by reading or assigning to global variables. The cobegin/coend
statement terminates when all of the concurrent blocks inside it terminate. After the
cobegin/coend statement terminates, the program continues sequentially with the
statement following coend.
Here is a diagram showing how our sample Concurrent Pascal program can be
executed on a single processor. Each assignment statement is executed atomically,
meaning that even though an assignment statement might be carried out with a sequence of more elementary machine-language steps, once the processor starts executing an assignment statement, the processor continues and completes the assignment
before allowing another process to execute. In other words, only one assignment
statement runs at a time. However, the order between assignments may be different
on different executions of the program. Each arrow in this figure illustrates a constraint on execution order: The statement at the left end of the arrow must finish
before the statement on the right can start. However, there is no necessary ordering
between statements that are not connected by a sequence of arrows. In particular,
the assignment x:=1 could be executed before or after the assignment x:=2. Here are
some possible execution orders for this program:
x:=0; x:=1; x:=2; x:=x+1; x:=x+1; print(x) // Output 4
x:=0; x:=2; x:=1; x:=x+1; x:=x+1; print(x) // Output 3
x:=0; x:=1; x:=x+1; x:=2; x:=x+1; print(x) // Output 3
x:=0; x:=2; x:=x+1; x:=1; x:=x+1; print(x) // Output 2
This program illustrates one of the main difficulties in designing and testing concurrent systems: nondeterminism. A program is deterministic if, for each sequence
of program inputs, there one sequence of program actions and resulting outputs. In
contrast, a program is nondeterministic if there is more than one possible sequence
of actions corresponding to a single input sequence. Our sample Concurrent Pascal
program is nondeterministic because the order of execution of the statements inside
the cobegin/coend construct is not determined. There are several possible execution
orders and, if the program is run several times on the same computer, it is possible
for the program to produce different outputs on different runs.
Nondeterminism makes it difficult to design and debug programs. In a complex
software system, there may be many possible execution orders. It is difficult for
program designers to think carefully about millions of possible execution orders
and identify errors that may occur in only a small number of them. It also may be
impractical to test program behavior under all possible execution orders. Even if
there is some way to cause the system to try all possible orders (a difficult task for
14.1 Basic Concepts in Concurrency
distributed systems involving many computers connected by a network), there could
be so many orders that it would take years to try them all.
One example that illustrates the complexity of nondeterminism is the design of
cache coherency protocols for multiprocessors. In a modern shared-memory multiprocessor, several processors share common memory. Because access to shared
memory takes time and can be a bottleneck when one processor needs to wait for
another process, each processor maintains a local cache. Each cache is a small amount
of memory that duplicates values stored in main memory. The job of the cache coherence protocol is to maintain the processor caches and to guarantee some form of
memory consistency, e.g., the value returned by every load/store sequence must be
consistent with what could have happened if there were no caches and all load/store
instructions operated directly on shared memory in some order. Cache coherence
protocols are notoriously difficult to design and many careful designs contain flaws
that surface only under certain conditions. Because of their subtlety, many cache
coherence protocols are analyzed by automated tools that are designed to detect
errors in nondeterministic systems. In Subsection 14.4.3 we look at the Java virtual
machine memory model, which involves some of the same issues as cache coherency
protocols for multiprocessors.
14.1.2 Communication, Coordination, and Atomicity
Every programming language for explicit concurrency provides some mechanism
to initiate and terminate independent sequential processes. The programming languages we consider in this chapter also contain mechanisms for some or all of the
following general purposes:
communication between processes, achieved by mechanisms such as buffered or
synchronous communication channels, broadcast, or shared variables or objects,
coordination between processes, which may explicitly or implicitly cause one
process to wait for another process before continuing,
atomicity, which affects both interaction between processes and the handling of
error conditions.
When we compare concurrent programming languages, we always look at the way
processes are defined, how they communicate and coordinate activities, and how
programmers may define atomic actions within the language (if the language provides
explicit support for atomicity).
The most elementary form of interprocess communication, as illustrated in the
Concurrent Pascal example in Subsection 14.1.1, is through shared variables. Processes may also communicate through shared data structures or files. Another form
of interprocess communication is called message passing. Here are some of the main
distinctions among various forms of message-passing mechanisms:
Buffering. If communication is buffered, then every data item that is sent remains
available until it is received. In unbuffered communication, a data item sent before
the receiver is ready to accept that it may be lost.
Synchronicity. In synchronous communication, the sender cannot transmit a data
item unless the receiver is ready to receive it. With asynchronous communication,
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the sending process may transmit a data item and continue executing even if the
receiver is not ready to receive the data.
Message Order. A communication mechanism may preserve transmission order
or it may not. If a mechanism preserves transmission order, then a sequence of
messages will be received in the order that they are sent.
Coordination mechanisms allow one process to wait for another or notify a waiting
process that it may proceed. Concurrent Pascal cobegin/coend provides a rudimentary form of process coordination as all processes started at the same cobegin must
finish before the statement following coend may proceed. Locking and semaphores,
discussed in Subsections 14.1.3 and 14.1.4, provide more sophisticated forms of concurrency control.
An action is atomic if every execution will either complete successfully or terminate in a state that is indistinguishable from the state in which the action was
initiated. A nonatomic action may involve intermediate states that are observable
by other processes. A nonatomic action may also halt in error before the entire action
is complete, leaving some trace of its partial progress. Generally speaking, any concurrent programming language must provide some atomic actions, because, without
some guarantee of atomicity, it is extremely difficult to predict the behavior of any
program.
14.1.3 Mutual Exclusion and Locking
There are many programming situations in which two or more processes share some
data structure, and some coordination is needed to maintain consistency of the data
structure. Here is a simple example that illustrates some of the issues.
Example 14.1
Suppose we have a system that allows people to sign up for mailing lists. This system
may involve a number of lists of people, and some of these lists may be accessed by
more than one process. Here is a procedure to add a person to a list:
procedure sign up(person)
begin
n := n + 1;
list[n] := person;
end;
This procedure uses an array to store the names in the list and an integer variable
to indicate the position of the last person added to the list. There may be other
procedures to read names from the list or remove them.
The sign up procedure works correctly if only one process adds to the list at a
time, but not if two processes add to the list concurrently. For example, consider the
following program section:
14.1 Basic Concepts in Concurrency
cobegin
sign up(fred);
sign up(bill);
end;
As discussed in Subsection 14.1.1, there are several possible execution orders for the
assignments to number and list. The ordering between assignments is shown in this
illustration:
One possible execution order increments n twice, then writes names fred and bill into
the same location. This is not correct: Instead of adding both names to the list, we
end up with an empty location, and only one of the two names in the list. The state
resulting from the sequence of assignments
n := n+1; n := n+1; list[n] := fred; list[n] := bill;
is called an inconsistent state because it is not consistent with the intended behavior
of the operations on lists. An invariant for list and integer variable n is that for every
integer i up to n there are data stored in list[i]. This invariant is preserved by sign up,
if it is called sequentially. However, if it is called concurrently, the data structure
invariant may be destroyed.
Mutual Exclusion
Example 14.1 shows that, for certain operations, it is important to restrict concurrency
so that only one process proceeds at a time. This leads to the notion of critical section.
A critical section is a section of a program that accesses shared resources. Because
a process in a critical section may disrupt other processes, it may be important to
allow only one process into a critical section at a time. Mechanisms for preventing
conflicting actions in independent processes are called mutual-exclusion mechanisms.
Mutual-exclusion mechanisms may be designed to satisfy some or all of the following criteria:
Mutual exclusion: Only one process at a time may be in its critical section.
Progress: If no processes are in their critical section and some process wants to
enter a critical section, it becomes possible for one waiting process to enter its
critical section.
Bounded waiting: If one process is waiting, there must be a bound on the number
of times that other processes are allowed to enter the critical section before this
process is allowed to enter. In other words, no waiting process should have to
wait indefinitely.
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It is also desirable to ensure that, if one process halts in a critical section, other
processes will eventually be able to enter the critical section.
Locks and Busy Waiting
A general approach to mutual exclusion is shown in the following pseudocode, which
uses the sign up procedure from Example 14.1. The main idea is that each process
executes some kind of wait action before calling sign up and executes some kind of
signal action afterward. The wait action should check to see if any other process is
calling sign up or executing any other procedure that might interfere with sign up. If
so, the action should make the process wait, only allowing the call to sign up to occur
when this is acceptable. The signal action allows a waiting process to proceed.
<initialze concurrency control>
cobegin
begin
<wait>
// wait for access
sign up(fred); // critical section
<signal>
// allow waiting process to proceed
end;
begin
<wait>
// wait for access
sign up(bill); // critical section
<signal>
// allow waiting process to proceed
end;
end;
Although the ideas behind wait and signal actions may seem straightforward, implementing these actions is tricky. In fact, in the early days of concurrent programming,
several incorrect implementations were proposed before a good solution was finally
discovered. We can understand some of the subtlety by examining a plausible but
imperfect implementation.
Example 14.2
Let us try to implement wait and signal by using an ordinary integer variable as
a “lock.” Each process will test the variable before it enters its critical section. If
another process is already in its critical section, then the lock will indicate that the
process must wait. If the value of the lock indicates that the process may enter, then
the process sets the lock to keep other processes out. This approach is shown in the
following code:
lock := 0;
cobegin
begin
while lock=1 do end;
// loop doing nothing until lock is 0
14.1 Basic Concepts in Concurrency
lock:=1;
sign up(fred);
lock:=0;
end;
begin
while lock=1 do end;
lock:=1;
sign up(bill);
lock := 0;
end;
// set lock to enter critical section
// critical section
// release lock
// loop doing nothing until lock is 0
// set lock to enter critical section
// critical section
// release lock
end;
Each process (each of the two blocks) tests the lock before entering the critical
section (i.e., before calling sign up). If the lock is 1, indicating that another process
is using sign up, then the process loops until the lock is 0. When the lock is 0, the
process enters the critical section, setting lock:=1 on entry and lock:=0 on exit. Using
a loop to wait for some condition is called busy waiting.
The problem with using a shared variable for mutual exclusion is that the operation that reads the value of the variable is different from the operation that sets
the variable. With the method shown in this code, it is possible for one process to
test the variable and see that the lock is open, but before the process can lock other
processes out, another process can also see that the lock is open and enter its critical
section. This allows two processes to call sign up at the same time.
A fundamental idea is concurrency control is atomic test-and-set. An atomic testand-set, sometimes called TSL for Test and Set Lock, copies the value of the lock
variable and sets it to a nonzero (locked) value, all in one atomic step. Although the
value of the lock variable is being tested, no other process can enter its critical section.
Most modern processors provide some form of test-and-set in their instruction set.
Deadlock
Deadlock occurs if a process can never continue because the state of another process.
Deadlock can easily occur if there are two processes and two locks. Suppose that
Process 1 first sets Lock 1 and then wait for Lock 2. If Process 2 first sets Lock 2 and
then waits for Lock 1, then both processes are waiting for the lock held by the other
process. In this situation, neither process can proceed and deadlock has occurred.
One technique that prevents deadlock is called two-phase locking. In two-phase
locking, a process is viewed as a sequence of independent tasks. For each task, the
process must first acquire all locks that are needed (or could be needed) to perform
the task. Before proceeding from one task to the next, a process must release all
locks. Thus there are two phases in the use of locks, a locking phase in which all locks
are acquired and a release phase in which all locks are released. To avoid deadlock in
the locking phase, all processes agree on an ordering of the locks and acquire locks
in that order. You may find more information about locking policies and deadlock
in books on operating systems and database transactions.
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14.1.4 Semaphores
Semaphores, first proposed by Edsger W. Dijkstra in 1968, provide a way for an operating system to guarantee mutual exclusion without busy waiting. There are several
different detailed implementations of semaphores. All use a counter or Boolean flag
and require the operating system to execute the entire wait or signal procedure atomically. Atomicity prevents individual statements of one wait procedure from being
interleaved with individual statements of another wa