# An Introduction to Constraint (Logic) Programming Using ECLiPSe

ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog An Introduction to Constraint (Logic) Programming Using ECLi PSe Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Stefano Benedettini DEIS - Dipartimento di Elettronica Informatica e Sistemistica University Bologna, Second Faculty of Engineering Artificial Intelligence 2010 Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Topics ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Constraint Logic Programming (CLP) ECLi PSe constraint solver Search methods for CLP (hints) Objectives ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Re-discover Prolog as a constraint solver Familiarize with ECLi PSe and its extensions to standard Prolog Get a grasp of Constraint Logic Programming Learn how to model problems in ECLi PSe Use constraint libraries . . . have fun with the solver Objectives ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Re-discover Prolog as a constraint solver Familiarize with ECLi PSe and its extensions to standard Prolog Get a grasp of Constraint Logic Programming Learn how to model problems in ECLi PSe Use constraint libraries . . . have fun with the solver ECLi PSe CLP Method Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Laboratory lessons will follow a hands-on approach During lessons you are supposed to experiment with tools and techniques Few novel theoretical concepts will be introduced Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries In order to make the most out of these lessons you should: Have a fair theoretical background (i.e., study stuff you saw during classes) Explore new technologies on your own (documentations, tutorials, websites) Be productive as soon as possible ECLi PSe CLP Method Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Laboratory lessons will follow a hands-on approach During lessons you are supposed to experiment with tools and techniques Few novel theoretical concepts will be introduced In order to make the most out of these lessons you should: Have a fair theoretical background (i.e., study stuff you saw during classes) Explore new technologies on your own (documentations, tutorials, websites) Be productive as soon as possible . . . anything else? Method ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Laboratory lessons will follow a hands-on approach During lessons you are supposed to experiment with tools and techniques Few novel theoretical concepts will be introduced Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries In order to make the most out of these lessons you should: Have a fair theoretical background (i.e., study stuff you saw during classes) Explore new technologies on your own (documentations, tutorials, websites) Be productive as soon as possible Ah, of course! You should have fun! Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries ECLi PSe CLP Premises Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries During this course a good understanding of Prolog will be assumed You all followed Viroli’s course so you’ll probably know Prolog better than me anyway. . . We will use fancy stuff like custom operators and structures We won’t use fancier stuff like metaprogramming Though you’re free to explore by yourselves ECLi PSe CLP Tools of the Trade Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs At this point any Prolog implementation will do But we will abandon the usual Prolog fairly soon You’d better start with ECLi PSe right away Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries http://eclipse-clp.org/ Resources for ECLi PSe ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 ECLi PSe main site http://eclipse-clp.org/ 2 http://eclipse-clp.org/examples Keep your friends close but your documentation closer: Syntactic Facilities Iteration Constructs 3 Modules http://eclipse-clp.org/doc (we will use that in due time) Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries 4 Introductory book on CP using ECLi PSe : Constraint Logic Programming Using ECLi PSe ECLi PSe CLP Stefano Benedettini Resources for Constraint Programming in General Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries CS problems and models in various languages: http://www.csplib.org/ Gecode: THE C++ library for CP (not for the faint of heart) Guido Tack’s Ph.D. dissertation (for a challenge) Java library: http://jacop.osolpro.com/ Python library (proprietary): http://www.eveutilities.com/products/emma Python library (GPL): http://labix.org/python-constraint ECLi PSe CLP Stefano Benedettini Resources for Constraint Programming in General Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Comet language: http://dynadec.com/support/downloads/ (keep an eye on this) On-line guide to CP by Roman Bartak ...and much more! A blog on CP (serious stuff) Mapping CSPs in Prolog ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog What we would like to do Setup variables and domains State constraints Specify a search strategy Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries What Prolog has to offer Logic variables and closed world assumption Predicates can be seen as very basic form of constraint Backtracking for free What Prolog lacks Constraint propagation ⇒ Efficiency! Constraints and variables are not first-class citizens Constraints in Prolog ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Any predicate is really a constraint Example member(X, [X|_]). member(X, [_|T]) :- member(X, T). Iteration Constructs Modules Constraint Satisfaction with ECLi PSe p(X, X). p(X, 3). The Interval Constraints Library: ic :- member(X, [1, 2, 3, 4]), X > 2. Search and Optimization with i e ECL PS yields X = 3, X = 4, while Other Constraint Libraries :- member(X, [1, 2, 3, 4]), member(Y, [2, 3, 4, 5]), p(X, Y). yields (X, Y) = (1, 3), (2, 2), (2, 3), (3, 3), (3, 3), (4, 3), (4, 4) ECLi PSe CLP Do Constraints in Prolog Really Work. . . ? Stefano Benedettini Scope and Motivations Example Prolog as a Constraint Solver How about these? ECLi PSe Extensions to Prolog :- X > 3, X < 2. % clearly inconsistent :- 4 is X + 3. % clearly X = 1 Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Both raise an “Instantiation fault” error Non-logical predicates and closed world assumption are Prolog greatest weaknesses Variables in arithmetic predicates must be instantiated (else error) For which values the second query is satisfied? For none, because you never told Prolog that 1 + 3 is true ECLi PSe CLP Do Constraints in Prolog Really Work. . . ? Stefano Benedettini Scope and Motivations Example Prolog as a Constraint Solver How about these? ECLi PSe Extensions to Prolog :- X > 3, X < 2. % clearly inconsistent :- 4 is X + 3. % clearly X = 1 Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Both raise an “Instantiation fault” error Non-logical predicates and closed world assumption are Prolog greatest weaknesses Variables in arithmetic predicates must be instantiated (else error) For which values the second query is satisfied? For none, because you never told Prolog that 1 + 3 is true . . . The Answer Is: “So So” ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Consequences Constraints can only be tested for satisfiability There is no “constraint propagation” (domain reduction) Prolog natively supports only generate and test! Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries CSP in Prolog solve_problem(Variables) :declare_domains(Variables), search(Variables), test_constraints(Variables). First I generate a complete assignment to the decision variables Then I test satisfiability ECLi PSe CLP Stefano Benedettini The Simplest Prolog Constraint Solver That Can Possibly Work Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities % Define domains for a list of variables domain([], _, []). domain([H|T], D, [H dom D|Rest]) :- domain(T, D, Rest ). Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries % Define integral domains for a list of variables ints(Xs, Min, Max, Vs) :int_range(Min, Max, R), domain(Xs, R, Vs). % Simple labeling label([]). label([V dom D|T]) :- member(V, D), label(T). Some Constraints ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities % In constraint X in D :- member(X, D). % Not in constraint _ not_in []. X not_in [H|T] :- not(X in [H|T]). Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries % All different all_different([]). all_different([H|T]) :all_different(T), H not_in T. % Ordered ordered([_]). ordered([X, Y|Z]) :- X < Y, ordered([Y|Z]). Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries The ECLi PSe Programming Environment ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe supports a number of extensions that facilitate Prolog programming: ECLi PSe Extensions to Prolog 1 Structures and arrays Syntactic Facilities 2 “Function-like” predicates 3 Iteration constructs 4 Modules Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries For more information, please refer to the User Manual ECLi PSe is a mature piece of software. We won’t have time to cover everything. You are encouraged to explore by yourselves. Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries “Function-like” Predicates ECLi PSe CLP Stefano Benedettini Scope and Motivations Unofficial name to refer to a syntactic facility that allows you to use custom predicates in expression context “Expression Context” is wherever ECLi PSe expects arithmetic operations that are subsequently evaluated Usually indicated with metavariable Expr in documentation Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries You can use a n-ary predicate p(X1 , . . . , Xn ) as a function that expects X1 , . . . , Xn−1 as arguments and returns Xn as result Example p(_, 3). summation([], 0). summation([H|T], S) :- summation(T, S1), S is S1 + H. :- 10 is 7 + p(ciao). :- 19 < summation([1, 2, 3]) + 15. Structure Notation ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities An easy way to declare and access arguments of a Prolog structure Declare a structure using: :- local struct(functorName(fieldNames...)). Can refer to fields by name instead of index Use fieldName of structName to obtain field index Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Example Declaring and using a structure :- local struct(student(name, surname, exam, mark)). :- student{name:john, surname:doe} = student(N, SN, _ , _), N = john, SN = doe. :- S = student{name:jane}, arg(4, S, 30), S = student (jane, _, _, M), M = 30. Enumerations ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries ECLi PSe does not support directly enumerations, but they can be emulated via structures Encode each value of your enumeration with an integer Declare a structure whose field names are the enumeration values Use operator of to map a value to an integer You should never instantiate a structure of that kind The inverse mapping is not as easy Example A color example :- local struct(colors(black, white, red, green, blue )). :- X = black of colors, X = 1 :- X is (white of colors) + (red of colors), X = 5 % useless but still Arrays ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Arrays in ECLi PSe are really a syntactic convenience to indicate structures of structures Functor name is irrelevant matrix(row(1, 2, 3), row(4, 5, 6)) is a 2 × 3 array and so is a(b(1, 2, 3), c(4, 5, 6)) dim(A, D) unifies a new array A with functor name [] and its dimensions D Use subscript/3 to access individual elements or array slices Usual C-like subscription operator available in expression context provides same behaviour as subscript/3 Example :- dim(A, [2, 3]), A = []([](_, _, _), [](_, _, _)) :- dim([]([](1, 2), [](3, 4)), D), D = [2, 2] :- dim(A, [2, 3]), subscript(A, [2, 1..3], R1), R2 is A[2, 1..3], R1 = R2 Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries ECLi PSe CLP Why Iteration in Prolog? Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Although every form of iteration can be expressed by a recursion, this bring some disadvantages: Code bloat: every iteration requires you to write an additional predicate Readability: those predicates are coded far away in your code and impact readability Non-locality: if your iteration has to access many local variables, you have to write a predicate with that many parameters Iteration in ECLi PSe is as powerful as recursion and often more concise ECLi PSe CLP Iteration Constructs Generalities Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries All iteration constructs have the form (IterationSpecifiers do Body) IterationSpecifiers is a comma-separated sequence of iteration specifiers Each specifier introduces an iteration variable visible only in Body Body can access only variables provided by iteration specifiers (but see param) Each iteration step, iterations variables in Body are unified with iteration values All iteration specifiers step in parallel and the whole loop stops when one of the specifiers completes Iteration Constructs: foreach ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs (foreach(E, List) do Body) executes goal Body iteratively unifying E to each element in list List Has invertible semantics: can be used to scan a list or to construct a list Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Example :- (foreach(E, [1, 2, 3]) do writeln(E)). % prints 1 2 3 :- (foreach(X, L), foreach(E, [10, 20, 30]) do X is E // 10), L = [1, 2, 3]. ECLi PSe CLP Iteration Constructs: param and count Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog param(Variables...) makes Variables visible inside iteration body count(Index, MinExpr, MaxExpr) unifies Index to each integer in [MinExpr , MaxExpr ] Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Example multiply(L, K, Out) :(foreach(E, L), foreach(X, Out), param(K) do X is K * E). my_length(L, X) :- % same as length/2 builtin (foreach(_, L), count(_, 1, X) do true). :- (count(I, 1, 4) do writeln(I)). % prints 1 2 3 4 Iteration Constructs: foreacharg and foreachelem ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries foreacharg/2 iterates on a structure/array arguments foreachelem/2 works only for arrays (functor name must be []). Flattens the array and iterate on its elements Both support an optional third argument that is unified to element index Example :- M = []([](1, 2, 3), [](4, 5, 6)), (foreachelem(E, M) do writeln(E)), % prints 1 2 3 4 5 6 (foreacharg(Row, M) do writeln(Row)), % prints [](1, 2, 3) [](4, 5, 6) (foreachelem(E, M, [I, J]) do writeln(I-J-E)), % prints 1-1-1 1-2-2 1-3-3 2-1-4 2-2-5 2-3-6 (foreacharg(R, M, I) do writeln(I-R)). % prints 1-[](1, 2, 3) 2-[](4, 5, 6) ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Iteration Constructs: for and multifor for(Index, MinExpr, MaxExpr) work like its imperative counterpart multifor(IndexList, MinList, MaxList) concisely describes multiply nested fors Both constructs support an optional fourth argument to provide an iteration step Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Example evens(L) :(for(I, 2, 100, 2), foreach(I, L) do true). mutliplication_table(T) :dim(T, [10, 10]), (multifor([X, Y], [1, 1], [10, 10]), % also multifor ([X, Y], 1, 10) foreachelem(E, T) do E is X * Y ). Iteration Constructs: fromto ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Most general construct (fromto(Init, In, Out, Last) do Body) introduces local variables In and Out in Body and behaves as follows: 1 2 3 Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries 4 First Init = In Executes iteration body After each iteration checks whether Last = Out and breaks if unification succeeds Otherwise unifies a fresh In variable with Out and repeats from step 2 Example filter_even(L, F) :(foreach(E, L), fromto(L, In, Out, []) do (0 is E rem 2 -> In = [E|T], Out = T; In = Out) ). reverse(L, R) :(foreach(E, L), fromto([], In, [E|In], R) do true). Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries ECLi PSe Modules in a Nutshell ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Advantages Group functionally related predicates in the same place Usually one module for each file Control naming access Modules form a namespace structure Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Using modules use lib(ModuleName) to compile an ECLi PSe library and import exported names use use_module(ModuleName) to compile a module and import its exported names use : operator to disambiguate a name: ModuleName:(predicate) ECLi PSe Modules in a Nutshell ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Writing modules use directive module(ModuleName) at the top of a source file to define a module export directive states which predicates to export Example Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS :- module(my_little_module). :- export print_matrix/1. print_row(Row) :(foreachelem(E, Row) do write(E), write(" ")), nl. Other Constraint Libraries print_matrix(M) :(foreacharg(Row, M), param(M) do print_row(Row)). print_matrix is available when this module is imported but print_row is not. Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Overcoming Prolog Limitations ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries The Disease Variables in arithmetic constraints must be instantiated Constraint predicates are mostly used for testing and cannot instantiate variables except the simplest like or, and, etc. . . No constraint propagation The Cure Every goal (i.e., constraint) that is not fully instantiated is suspended and put into the constraint store Propagation is applied to variable domains at different strength Before the Cure ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Example ordered_list([_]). ordered_list([X, Y|T]) :- X < Y, ordered_list([Y|T]). Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries :- ordered_list([1, 2, 3, 4, 5]). % works as expected :- ordered_list([1, X, 3, Y, 5]). % we suppose X, Y be integral variables The second goal raises an instantiation error. After the Cure ECLi PSe CLP Stefano Benedettini Scope and Motivations Example Prolog as a Constraint Solver :- lib(ic). % load a solver library ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs ordered_list([_]). ordered_list([X, Y|T]) :- X #< Y, ordered_list([Y|T]) . Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries :- ordered_list([1, X, 3, Y, 5]). :- ordered_list([1, X, Y, 5]). In the first case ECLi PSe enforces variable integrality and correctly propagates the constraints 1 < X < 3 and 3 < Y < 5 thereby inferring X = 2, Y = 4. In the second case we have a delayed goal, that is a goal that cannot be solved at the moment. ECLi PSe propagates inequalities and infers X ∈ 2, 3, Y ∈ 3, 4. How to Write a Problem Model ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries CSP in ECLi PSe solve_problem(Variables) :declare_domains(Variables), setup_constraints(Variables), search(Variables). How to Write a Problem Model ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries CSP in ECLi PSe solve_problem(Variables) :declare_domains(Variables), setup_constraints(Variables), search(Variables). This part comprises model definition. How to Write a Problem Model ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries CSP in ECLi PSe solve_problem(Variables) :declare_domains(Variables), setup_constraints(Variables), search(Variables). This is the actual search. How to Write a Problem Model ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries CSP in ECLi PSe solve_problem(Variables) :declare_domains(Variables), setup_constraints(Variables), search(Variables). Modeling and search are independent phases in the resolution of a problem. How to Write a Problem Model ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog CSP in ECLi PSe solve_problem(Variables) :declare_domains(Variables), setup_constraints(Variables), search(Variables). Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Problem model and choice points Do not leave choice points during constraint setup! If you do, the solver will miss an opportunity do constraint propagation You are actually splitting a CSP into multiple CSPs, making choices before propagation, not after But it doesn’t mean you can’t use Prolog non determinism An Overview of the Libraries ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries We’ll mostly use only the more recent ic library and briefly touch its derivatives: ic_global very useful global constraints ic_sets supports set variables ic_symbolic handles (not so) conveniently symbolic domains Don’t use old fd libraries: stick with ic! Basic searching strategies and B&B optimization Many more constraints available (look them up in the documentation): Bin packing Knapsack Scheduling Generalized Propagation and Constraint Handling Rules (CHR) will be barely mentioned Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries ECLi PSe CLP Introduction Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Powerful hybrid solver that works on (intervals of) integers and reals Variable domains are unions of disjoint intervals It enforces bound consistency: Weaker than arc-consistency Iteratively “squeezes” a variable domain: x , y ∈ [0, 5] ∩ Z, x < y ⇒ x ∈ [0, 4], y ∈ [1, 5] Cannot propagate “holes” Handles linear, arithmetic and non-linear constraints Issues with real variables: How do we cope with limited precision? How do we represent “holes” in a real domain? Does this make sense: x ∈ [0, 1], x 6= 0.5? Variable Domains ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Constrains integer as well real domains with ::, $:: and #:: operators: ListOfVariables :: RangeSpecification specify either an integer or a real domain depending on the range specification $:: enforces real domains #:: enforces integer domain I suggest to employ only #:: or $:: to clearly state variable type These operators work also with multidimensional arrays See documentation for further details Search and Optimization with i e ECL PS Other Constraint Libraries Example :- Var #:: [1..5, 7, 9..12]. :- [X, Y] #:: 1..5. :- dim(A, [3, 2]), A :: 0..1.0. Arithmetic Constraints ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries ic support every arithmetic and comparison operator of standard Prolog qualified use: ic:(X op Y) where op is one of >, <, =<, =:=, =\=, +, -, *, . . . Prepend $ to comparison predicates to avoid qualification: $>, $<, $=<, $=, $\=, . . . Prepend # to comparison predicates to also enforce integrality: #>, #<, #=<, #=, #\=, . . . New variables spawn into existence as soon they are used Example :- [X, Y] :: 1..5.0, Z #= X + 1, ic:(Z < Y). % 1 :- length(V, 4), V :: 0..5, (fromto(V, [X, Y|T], [Y|T ], [_]) do ic:(X < Y)). % 2 1 X ∈ [1, 3] ∩ N, Y ∈ [2, 5], Z ∈ [2, 4] ∩ N 2 ∀i ∈ [1, 4] xi ∈ N, x1 ∈ [0, 2], x2 ∈ [1, 3], x3 ∈ [2, 4], x4 ∈ [3, 5] Boolean Constraints ECLi PSe CLP Stefano Benedettini Scope and Motivations ic supports basic boolean operators: Prolog as a Constraint Solver or, and, xor, => all infix and neg (for negation) They automatically enforce a boolean domain, that is {0, 1} ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Example How to solve a SAT without really trying :- X and (Y or Z) and (X or neg Y) :- X and neg X 1 Infers X = 1 2 Infers inconsistency Reified Constraints ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries How can we conditionally enforce a constraint? How can I express if x ∈ [1, 3] then y < z? Remember that we cannot leave choice points so, for instance, operator -> is out of question Reified constraints come to the rescue Domain, comparison and boolean constraints all have a reified version If the original constraint is op(X, Y), its reified version is op(X, Y, B) where B is a boolean variable If B = 1 the constraint holds, otherwise holds its negation Thanks to “function-like predicate” facility we can “chain” reified constraints in expression context :- #<(X, Y, B1), #::(X, 1..5, B2), B1 or B2. becomes :- (X #< Y) or (X #:: 1..5). ECLi PSe CLP Global Constraints and Propagation Strength Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Most of constraints in ic are binary Also alldifferent is nothing more than manually setting pairwise inequalities We need a better way to model global constraints than separating them in binary constraints Propagation strength is lower That is we cannot rule out as many values as a we could Enter ic_global Lots of high level constraints: Minimum/maximum of a list occurrences a stronger alldifferent ordered (useful for symmetry breaking) Also check ac_eq in ic Provides an arc-consistent (and not only bound consistent) version of X #= Y + C (C is an integer constant) An Example with Global Constraints ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities :- length(V, 100), V #:: 1..99, ic:alldifferent(V). :- length(V, 100), V #:: 1..99, ic_global: alldifferent(V). :- length(V, 5), V #:: 1..5, ordered(<, V). :- [X, Y] #:: 1..10, X #\= 7, X #= Y + 2. :- [X, Y] #:: 1..10, X #\= 7, ac_eq(X, Y, 2). Iteration Constructs Modules Constraint Satisfaction with ECLi PSe 1 Clearly inconsistent but no propagation is performed 2 Immediately returns “No” as answer The Interval Constraints Library: ic 3 Correctly infers, Xi = i, i = 1, . . . , 5 Search and Optimization with i e ECL PS 4 X ∈ [3, 6] ∪ [8, 10] and Y ∈ [1, 8] but cannot propagate the “hole” at X = 7 5 X ∈ [3, 6] ∪ [8, 10] and Y ∈ [1, 4] ∪ [6, 8]: now it is arc consistent Other Constraint Libraries Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Labeling ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe Propagation is insufficient to prove satisfiability in all but the simplest CSPs In ECLi PSe this translate into having delayed goals The simples strategy consists in: 1 2 3 Recursively considering all unassigned variables in some order Try all values in its domain in ascending order Return a feasible assignment or backtrack and try another value this is called labeling The Interval Constraints Library: ic Example Search and Optimization with i e ECL PS Pythagorean Triplets Other Constraint Libraries :- V = [A, B, C], V #:: 1..100, ordered(<, V), A^2 + B^2 #= C^2, findall(V, labeling(V), L). There are 52 unique Pythagorean triplets of integers in [1, 100] Note the use of ordered to remove symmetries Fine Tuning Your Search ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog ECLi PSe provides a general mechanism to customize a search in the form of predicate ic:search You must specify: 1 Syntactic Facilities Iteration Constructs 2 Variable selection strategy Value selection strategy Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries Some heuristics are already defined: Variable selection: input order, (anti) first-fail, most-constrai ... Value selection: ascending order, starting from max/middle/min value, . . . Search strategy can have a huge impact on efficiency Optimization by Branch-and-Bound ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog ECLi PSe comes also with a ready-made B&B implementation to tackle constraint optimization problems You must also provide an objective function f ECLi PSe B&B works as follows (for minimization problems): 1 Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries 2 Generates a feasible solution s and impose the constraint f (s ′ ) < f (s) for the remainder of the search Restarts or continues the search Example Which is the largest integer n < 1000 such that the cube of the sum of its digits equals the number itself? :- [X, Y, Z] #:: 0..9, N #= 100 * X + 10 * Y + Z, C #= (-N), (X + Y + Z)^3 #= N, minimize(labeling([X, Y, Z]), C). Optimization by Branch-and-Bound ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog ECLi PSe comes also with a ready-made B&B implementation to tackle constraint optimization problems You must also provide an objective function f ECLi PSe B&B works as follows (for minimization problems): 1 Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries 2 Generates a feasible solution s and impose the constraint f (s ′ ) < f (s) for the remainder of the search Restarts or continues the search Example Convert a maximization into a minimization problem. :- [X, Y, Z] #:: 0..9, N #= 100 * X + 10 * Y + Z, C #= (-N), (X + Y + Z)^3 #= N, minimize(labeling([X, Y, Z]), C). Optimization by Branch-and-Bound ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog ECLi PSe comes also with a ready-made B&B implementation to tackle constraint optimization problems You must also provide an objective function f ECLi PSe B&B works as follows (for minimization problems): 1 Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries 2 Generates a feasible solution s and impose the constraint f (s ′ ) < f (s) for the remainder of the search Restarts or continues the search Example Which is the largest integer n < 1000 such that the cube of the sum of its digits equals the number itself? It’s 512. :- [X, Y, Z] #:: 0..9, N #= 100 * X + 10 * Y + Z, C #= (-N), (X + Y + Z)^3 #= N, minimize(labeling([X, Y, Z]), C). Outline ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog 1 Prolog as a Constraint Solver 2 ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules 3 Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with ECLi PSe Other Constraint Libraries Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Other Constraint Libraries ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS Symbolic Domains With ic_symbolic Sometimes it’s more natural to work with a finite domain other than (an interval of) integers Colors Days of the week ic_symbolic permits this kind of symbolic domains This library is implemented on top of ic Every value in a symbolic domain is mapped to an integer Define a new symbolic domain with domain declaration :- local domain(domainName(values...)). Example Other Constraint Libraries :- local domain(weekday(mo, tu, we, th, fr, sa, su)). :- X &:: weekday, X &\= tu, X &< fr. Note the implicit ordering of values. Set Variables With ic_sets ECLi PSe CLP Stefano Benedettini Scope and Motivations Prolog as a Constraint Solver ECLi PSe Extensions to Prolog Syntactic Facilities Iteration Constructs Modules Constraint Satisfaction with ECLi PSe The Interval Constraints Library: ic Search and Optimization with i e ECL PS This library manipulates integer set variables A set is simply a sorted list of integers without repetitions Much like ic, it provides bound consistency on set variable domains But what is a “range of sets”? A set variable σ is a triple: hSlb , Sub , Ci Slb is a set containing all the elements which are certainly in σ Sub is a set containing all the elements that might be in σ (obviously Slb ⊆ Sub ) C is an integer variable equal to |σ| See also the concept of Lattice Other Constraint Libraries Example :- X :: [1]..[1, 2, 3], insetdomain(X, _, _, _). Finds: h{1, 2, 3}, {1, 2}, {1, 3}, {1}i

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