Generating E cient, Terminating Logic Programs

Generating E cient, Terminating Logic Programs
Generating Ecient, Terminating Logic
Programs
Jonathan C. Martin1 and Andy King2
1
Department of Electronics and Computer Science, University of Southampton,
Southampton, SO9 5NH, UK. [email protected]
2 Computing Laboratory, University of Kent at Canterbury,
Canterbury, CT2 7NF, UK. [email protected]
Abstract. The objective of control generation in logic programming is
to automatically derive a computation rule for a program that is ecient
and yet does not compromise program correctness. Progress in solving
this important problem has been slow and, to date, only partial solutions
have been proposed where the generated programs are either incorrect
or inecient. We show how the control generation problem can be tackled with a simple automatic transformation that relies on information
about the depths of SLD-trees. To prove correctness of our transform
we introduce the notion of a semi delay recurrent program which generalises previous ideas in the termination literature for reasoning about
logic programs with dynamic selection rules.
1 Introduction
A logic program can be considered as consisting of a logic component and a
control component [8]. Although the meaning of the program is largely dened
by its logical specication, choosing the right control mechanism is crucial in
obtaining a correct and ecient program. In recent years, one of the most popular
ways of dening control is via suspension mechanisms which delay the selection of
an atom in a goal until some condition is satised. Such mechanisms include the
block declarations of SICStus Prolog [7] and the DELAY declarations of Godel [6].
These mechanisms are used to dene dynamic selection rules with the two main
aims of enhancing performance through coroutining and ensuring termination.
In practise, however, these two aims are not complementary and it is often the
case that termination, and hence program correctness, is sacriced for eciency.
Consider, for instance, the Append program given below (in Godel style
syntax) with its standard DELAY declaration which delays the selection of an
Append/3 atom until either the rst or third argument is instantiated to a nonvariable term.
Append([], x, x).
Append([ujx], y, [ujz])
Append(x, y, z).
DELAY Append(x, , z) UNTIL Nonvar(x) _ Nonvar(z).
Interestingly, although it is intended to assist termination the delay declaration is not sucient to ensure that all Append/3 goals terminate. The goal
Append([xjxs], ys, xs), for example, satises the condition in the declaration
and yet is non-terminating [14].
Termination can only be guaranteed by strengthening the condition in the
delay declaration. This is where the trade o between eciency, termination and
completeness takes place. The stronger the condition, the more goals suspend.
Although termination may eventually be assured, it may be at the expense of
not resolving goals which have nite derivations. Also the stronger the delay
condition, the more time consuming it usually is to check. Thus one of the main
problems in generating control of this form is nding suitable conditions which
are inexpensive to check and guarantee termination and completeness. We will
refer to this as the local termination issue, to contrast it with another global
aspect of the termination problem which we will examine shortly.
1.1 Local Termination
There have been several attempts at solving the local termination problem. We
will examine each of these in the context of the Append program above, though
each technique has wider applicability.
Linearity In the case of single literal goals, one additional condition sucient
for termination is that the goal is linear, that is, no variable occurs more than
once in the goal [10]. Although this restriction would prevent the looping Append/3 call above from proceeding, it would also unfortunately delay many other
goals with nite derivations such as Append([x, x], ys, zs). In addition, the
time complexity for checking linearity is quadratic in the number of variables in
the goal.
Rigidity An alternative approach by Marchiori and Teusink [11] and Mesnard
[13] proposes delaying Append/3 goals until the rst or third argument is a list
of determinate length. Termination is obtained for a large class of goals, but at
a price. Checking such a condition requires the complete traversal of the list and
the condition must be checked on every call to the predicate3 . Naish argues that
this approach can be \... expensive to implement and ... can delay the detection
of failure in a sequential system and restrict parallelism in a stream and-parallel
system" [14].
Modes Naish goes on to solve the problem with the use of modes. Termination can be guaranteed with the above DELAY declaration if the modes of the
Append/3 calls are acyclic, or more generally cycle bounded [14]. This restriction essentially stops the output feeding back into the input. Although modes
form a good basis for solving the local termination problem, they have not been
shown to be satisfactory for reasoning about another termination problem, that
of speculative output bindings.
3
In [13] the check is, in fact, only performed on the initial call, but there is no justication for this optimisation given in the paper. For non-structurally recursive
predicates, e.g. Quicksort/2 of Sect. 1.2, such an optimisation is usually not possible.
1.2 Global Termination
Even when nite derivations exist, delay conditions alone are not, in general,
sucient to ensure termination. Innite computations may arise as a result of
speculative output bindings [14], which can occur due to the dynamic selection
of atoms. There are several problems associated with speculative output bindings (see [14] for a discussion of these). Here we are only interested in the aect
that they have on termination, and this is what we call the global termination
issue. To illustrate the problem caused by speculative output bindings consider
the Quicksort program shown below. This is an example of a well known program
whose termination behaviour can be unsatisfactory. With the given delay declarations, the program can be shown to terminate in forward mode, that is for
queries of the form Quicksort(x, y) where x is bound and y is uninstantiated.
In reverse mode, however, where y is bound and x is uninstantiated, the program does not always terminate. More precisely, a query such as Quicksort(x,
[1,2,3]) will terminate existentially, i.e. produce a solution, but not universally,
i.e. produce all solutions. In fact, experimentation with the Godel and SICStus
implementations indicates that when the list of elements is not strictly increasing, e.g. in Quicksort(x, [1,1]) and Quicksort(x, [2,1]), the program does
not even existentially terminate! This is illustrative of the subtle problems that
dynamic selection rules pose in reasoning about termination, and which suggest
that control should ideally be automated to avoid them.
Quicksort([], []).
Quicksort([xjxs], ys)
Partition(xs, x, l, b) ^ Quicksort(l, ls) ^
Quicksort(b, bs) ^ Append(ls, [xjbs], ys).
DELAY Quicksort(x, y) UNTIL Nonvar(x) _ Nonvar(y).
Partition([], , [], []).
Partition([xjxs], y, [xjls], bs)
Partition([xjxs], y, ls, [xjbs])
x y ^ Partition(xs, y, ls, bs).
x > y ^ Partition(xs, y, ls, bs).
DELAY Partition(x, , y, z) UNTIL Nonvar(x) _ (Nonvar(y) ^ Nonvar(z)).
To improve matters, the delay conditions can be strengthened in the manner
prescribed by Naish or Marchiori and Teusink. In general, however, no matter
how strong the delay conditions are, they are not always sucient to ensure
termination, even though a terminating computation exists. To see why, consider
augmenting the Quicksort program with the clause
Append(x, [ jx], x) False.
The declarative semantics of the program are completely unchanged by the addition of this clause and one would hope that the new program would produce
exactly the same set of answers as the original. This will not be the case, however, if this clause is selected before all other Append/3 clauses. Consider the
query Quicksort(x, [1,2,3]). Following resolution with the second clause of
Quicksort/2, the only atom which can be selected is Append(ls, [xjbs], [1,2,3]).
When this unies with the above clause, both ls and bs are immediately bound
to the term [1,2,3]. As a result of these speculative output bindings the previously suspended calls Quicksort(l, ls) and Quicksort(b, bs) will be woken before the
computation reaches the call to False. The net result is an innite computation
due to recurring goals of the form Quicksort(x, [1,2,3]).
The problem here is that the output bindings are made before it is known
that the goal will fail and no matter how stringent the conditions are on the
Quicksort/2 goals, loops of this kind cannot generally be avoided. The reason for
this is that a delay condition only measures a local property of a goal without
regard for the computation as a whole. The conditions can ensure that goals are
bounded, but are unable to ensure that the bounds are decreasing.
Local Computation Rule To remedy this, Marchiori and Teusink [11] propose
the use of a local computation rule. Such a rule only selects atoms from those that
are most recently introduced in a derivation. This ensures that any atom selected
from a goal, is completely resolved before any other atom in the goal is selected.
The eect in the above example is that the call to False would be selected and
the Append/3 goal fully resolved before the calls to Quicksort/2 are woken. This
prevents an innite loop. The main disadvantage of local computation rules is
that they do not allow any form of coroutining. This is clearly a very severe
restriction.
Delayed Output Unication A similar solution proposed by Naish [14] is
that of delaying output unication. In the example above, assuming a left-toright computation rule, the extra Append/3 clause would be rewritten as
Append(x, y, z) False ^ y = [ jx] ^ z = x.
The intended eect of such a transformation is that no output bindings should
be made until the computation is known to succeed. This has parallels with the
local computation rule and also restricts coroutining.
Constraints Mesnard uses interargument relationships compiled as constraints
to guarantee that the bounds on goals decrease [13]. For example, solving the
constraint jysjlength = jlsjlength + 1 + jbsjlength before selecting the atom Append(ls,
[xjbs], ys) ensures that bs and ls are only bound to lists with lengths less than
that of ys. This is enough to guarantee termination, but is expensive to check as
it requires calculating the lengths of all three arguments of Append/3.
1.3 Our Contribution
In summary, we see that the most promising approaches to control generation,
while guaranteeing termination and completeness, produce programs which are
inecient, either directly due to expensive checks which must be performed at
run-time or indirectly by restricting coroutining.
In this paper we present an elegant solution to the above problems. To solve
the local termination problem, we use delay declarations in the spirit of [11] combined with a novel program transformation which overcomes the ineciencies
of their approach. Simultaneously, the transformation inexpensively solves the
global termination problem without grossly restricting coroutining. The transformation is simple and is easy to automate. Transformed programs are guaranteed
to terminate and are also ecient. Hence, the technique forms a sound basis for
automatically generating ecient, terminating logic programs from logical specications.
The technique is based on the following idea. If the maximum depth of the
SLD-tree needed to solve a given query can be determined, then by only searching
to that depth the query will be completely solved, i.e. all answers (if any) will be
obtained, in a nite number of steps. We rst present the technique through an
example. Then we formalise the transformation and prove termination for the
transformed programs.
2 Example
We demonstrate our program transformation on the Quicksort program from the
introduction. The transformed program is shown below. Termination is guaranteed for all queries Quicksort(x, y). Furthermore when x or y is a ground list
of integers, the computation does not ounder and if it succeeds then the set of
answers produced is complete with respect to the declarative semantics.
Quicksort(x, y) SetDepth Q(x, y, d) ^ Quicksort 1(x, y, d).
DELAY Quicksort 1( , , d) UNTIL Ground(d).
Quicksort 1(x, y, d)
Quicksort 2(x, y, d).
Quicksort 2([], [], d) d 0.
Quicksort 2([xjxs], ys, d + 1)
Partition(xs, x, l, b)
d 0 ^ Partition(xs, x, l, b) ^ Quicksort 2(l, ls, d) ^
Quicksort 2(b, bs, d) ^ Append(ls, [xjbs], ys).
SetDepth P(xs, l, b, d) ^ Partition 1(xs, x, l, b, d).
DELAY Partition 1( , , , , d) UNTIL Ground(d).
Partition 1(xs, x, l, b, d)
Partition 2(xs, x, l, b, d).
Partition 2([], , [], [], d) d 0.
Partition 2([xjxs], y, [xjls], bs, d + 1)
Partition 2([xjxs], y, ls, [xjbs], d + 1)
Append(x, y, z)
d 0 ^ x y ^ Partition 2(xs, y, ls, bs, d).
d 0 ^ x > y ^ Partition 2(xs, y, ls, bs, d).
SetDepth A(x, z, d) ^ Append 1(x, y, z, d).
DELAY Append 1( , , , d) UNTIL Ground(d).
Append 1(x, y, z, d)
Append 2(x, y, z, d).
Append 2([], x, x, d) d 0.
Append 2([ujx], y, [ujz], d + 1)
d 0 ^ Append 2(x, y, z, d).
The predicate SetDepth Q(x, y, d) calculates the lengths of the lists x and
y, delaying until one of the lists is found to be of determinate length, at which
point the variable d is instantiated to this length. Only then can the call to
Quicksort 1/3 proceed. The purpose of this last argument is to ensure niteness of
the subsequent computation. More precisely, d is an upper bound on the number
of calls to the recursive clause of Quicksort 2/3 in any successful derivation.
Thus by failing any derivation where the number of such calls has exceeded this
bound (using the constraint d 0), termination is guaranteed without losing
completeness. The predicates SetDepth P/4 and SetDepth A/3 are dened in a
similar way.
2.1 Local and Global Control
The local control problem is solved in the rst instance with a rigidity check
in the style of [11]. This ensures that the initial goal is bounded. Boundedness
of subsequent goals, however, is enforced by the depth parameter and further
rigidity checks on these depth bounded goals are redundant. This allows, for
example, the call Quicksort 2(l, ls, d) to proceed, without fear of an innite
computation, even if both l and ls are uninstantiated, providing d is ground. A
huge improvement in performance is possible by eliminating these checks. The
global control problem is also neatly solved. By restricting the search space to
be nite, even though speculative output bindings may still occur, they cannot
lead to innite derivations.
2.2 A Simple Optimisation
Even though many of the rigidity checks have now been removed, the eciency
of the program is still unsatisfactory. This is due to the rigidity checks which
are performed on each call to Append/3 and Partition/4. It is easy to show that
the depths of these subcomputations are bounded by the same depth parameter occurring in Quicksort 2/3. Hence, we can replace the atoms Partition(xs,
x, l, b) and Append(ls, [xjbs], ys) in the body of Quicksort 2/3 with the atoms
Partition 2(xs, x, l, b, d) and Append 2(ls, [xjbs], ys, d).
This optimised version of the program is quite ecient. The only rigidity
checks that are performed are those on the initial input, exactly at the point
where they are needed to guarantee termination. Following the initial call to
Quicksort 2/3 the program runs completely without delays and the only other
overhead is the decrementation of the depth parameter and some trivial boundedness checks. The net result is that, with the Bristol Godel implementation,
the program actually runs faster on average than the original program with the
Nonvar delay declarations!
2.3 Coroutining
Notice in particular how the global termination problem is overcome without
reducing the potential for coroutining. Simply knowing the maximum depth of
any potentially successful branch of the SLD-tree allows us to force any derivations along this branch which extend beyond this depth to fail without losing
completeness. These forced failures keep the computation tree nite but do not
restrict the way in which the tree is searched. The addition of the failing Append/3 clause from the introduction (which would appear here as an Append 2/3
clause) cannot eect the termination of the algorithm, even if the same coroutining behaviour of the original program is used. Of course, we need to constrain
the computation rule such that
1. the test d 0 is always selected before any other atoms in the body of the
clause with a subterm d, and
2. the depth parameter is ground on each recursive call (or for any atom with
a subterm d in the optimised version)
but this is not nearly as restrictive as using the local computation rule. Indeed,
using the standard left-to-right selection rule these conditions will clearly be
satised in the above program.
2.4 Termination and Eciency
With termination guaranteed, the programmer is now free to concentrate on
the program's performance. Notice for the program above that the order of the
goals in the body of Quicksort 2 is critical to the eciency of the algorithm. For
the best performance, they must be arranged so that the computation is data
driven. In fact, by dening SetDepth Q/3 by
SetDepth Q(x, y, d)
Length(x, d) ^
Length(y, d).
the computation will be data driven in both forward and reverse modes with the
ordering of the goals as above. This dependence on the ordering can be reduced
by introducing the typical delay declarations used for this program. These declarations do not eect the terminating nature of the algorithm, in that they will
not cause the algorithm to loop, though they may possibly reduce previously
successful or failing derivations to oundering ones. They are inserted solely
to improve the performance through coroutining. Alternatively, one may seek to
optimise the performance for dierent modes through multiple specialisation, for
example. The important point is that with this approach the trade-o between
termination and performance is signicantly reduced. In seeking an ecient algorithm, correctness does not have to be compromised.
3 Preliminaries
Terms, atoms and formulae are dened in the usual way [9]. A program P is a
set of clauses of the form 8(a w) where a is an atom and w is either absent or
a conjunction of atoms. We denote by body(a w) the set of atoms appearing
in w. Given a program P, then P denotes the alphabet of predicate symbols in
P. We denote by var(o) the set of variables in a syntactic object o. A grounding
substitution for a syntactic object o is a substitution in which each variable in
o is bound to a ground term. We denote by rel(A) the predicate symbol of the
atom A. We denote a tuple of elements hd1; : : :; dni by d and write di 2 d if di is
the ith element of the tuple d. If the atom p(t1 ; : : :; tn) is denoted by p(t), then
the atom p(t1 ; : : :; tn; d) is denoted by p(t; d). Finally, we denote the minimal
model of a program P by M(P).
4 The Transformation
Our aim is to develop a program transformation which is able to derive correct
and ecient programs from logical specications. We divide the development
into three stages where we consider termination, completeness and eciency
respectively.
4.1 Termination
To prove termination of the transformed programs we will need to introduce a
new program class which subsumes that of delay recurrent programs introduced
in [11]. Its introduction is motivated by an overly restrictive condition imposed
in the denition of delay recurrency. By removing this unnecessary condition
we obtain the new class of programs which we call semi delay recurrent. Our
transformed programs will lie within this class. The following notions, due to
Bezem [5], will be needed.
Denition1 level mapping [5]. Let P be a program. A level mapping for P
is a function j:j : BP 7! IN from the Herbrand base to the natural numbers. 2
A level mapping is only dened for ground atoms. The next denition lifts
the mapping to non-ground atoms and goals.
Denition2 bounded atom and goal [5]. An atom A is bounded wrt a level
mapping j:j if j:j is bounded on the set [A] of variable free instances of A. If A
is bounded then j[A]j denotes the maximum that j:j takes on A. A goal G =
A1 ; : : :; An is bounded if every Ai is bounded. If G is bounded then j[G]j denotes
the (nite) multiset consisting of the natural numbers j[A1]j; : : :; j[An]j.
2
Level mappings are used to prove termination in the following way. Let G =
G0 ; G1; G2; : : : be the goals in a refutation of G and j:j a level mapping. Given
that G is bounded wrt j:j and j[Gi]j > j[Gi+1]j for all i, we can deduce that
the sequence G = G0; G1; G2; : : : is nite by the well-foundedness of the natural
numbers. To prove the goal ordering property, that j[Gi]j > j[Gi+1]j for all i
and for all possible refutations of G, one must examine the clauses and the
computation rule used. Various classes of program have been identied, where
this property is satised for a given computation rule [1, 2, 5, 11]. Bezem, for
example, introduced the class of recurrent programs [5], where the goal ordering
property is always satised, regardless of the computation rule.
Denition3 recurrency [5]. Let P be a denite logic program and j:j a level
mapping for P. A clause H B1 ; : : :; Bn is recurrent (wrt j:j) if for every
grounding substitution , jHj > jBij for all i 2 [1; n]. P is recurrent (wrt j:j)
if every clause in P is recurrent (wrt j:j).
2
One problem with recurrency, as noted in [3], is that it does not intuitively
relate to the principal cause of non-termination in a logic program { recursion.
The denition requires that level mappings decrease from clause heads to clause
bodies irrespective of the recursive relation between the two. This relation is
formalised in the following denition.
Denition4 predicate dependency. Given p dened by a program P, we
say that p 2 p directly depends on q 2 p if there is a statement in P with
head p(t) and a body atom q(t ). The depends on relation is dened as the reexive, transitive closure of the directly depends on relation. p and q are mutually
dependent, written p ' q, if p depends on q and q depends on p.
2
0
Notice that there is a well-founded ordering among the predicates of a program
induced by the depends on relation. We write p = q whenever p depends on q but
q does not depend on p, i.e. p calls q as a subprogram. By abuse of terminology
we will say that two atoms are mutually dependent (with each other) if they
have mutually dependent predicate symbols.
Apt and Pedreschi [3] observed that while it is necessary for the level mapping to decrease between the head p(t) of a clause and each body atom q(t )
with p ' q, a strict decrease is not required for the other atoms in the body.
They introduced the notion of semi-recurrent program which exploited this observation. Their denition still insisted, however, that the level of the head was
at least greater or equal to the level of all body atoms, whereas in fact it does
not matter if the level of non-mutually dependent atoms is greater than in the
head provided that these atoms are bounded whenever they are selected.
Marchiori and Teusink [11] noticed that boundedness of atoms could be enforced by using delay declarations but did not fully exploit this fact combined
with the above observation in dening delay recurrency, a version of recurrency
for programs using dynamic selection rules. Their denition required a decrease
in the level mapping from the head to the non-mutually dependent atoms when
in fact boundedness was already guaranteed by the delay declarations.
We generalise their denition here by removing this restriction. The new
denition will prove useful for dening a large class of terminating programs
which permit coroutining. We rst need the following two denitions from [11].
0
Denition5 direct cover [11]. Let j:j be a level mapping and c : H B a
clause. Let A 2 body(c) and C body(c) such that A 62 C. Then C is a direct
cover for A wrt j:j in c, if there exists a substitution such that A is bounded
wrt j:j and dom() var(H; C). A direct cover C for A is minimal if no proper
subset of C is a direct cover for A.
2
Denition6 cover [11]. Let j:j be a level mapping and c : H B a clause.
Let A 2 body(c) and C body(c). Then C is a cover for A wrt j:j in c, if (A; C)
is an element of the least set S such that
1. (A; ;) 2 S whenever the empty set is the minimal direct cover for A wrt j:j
in c, and
2. (A; C) 2 S whenever A 62 C, and C is of the form
fA1 ; : : :; Ak g [ D1 [ : : : [ Dk
such that fA1 ; : : :; Ak g is a minimal direct cover of A in c, and for i 2
[1; k]; (Ai; Di) 2 S.
2
Intuitively, a cover of an atom A in a clause is a subset of the body atoms
which must be (partially) resolved in order for A to become bounded wrt some
level mapping. Where possible, we will assume in the following that the level
mapping is xed for a given program. The following denition generalises that
of a delay recurrent program in [11].
Denition7 semi delay recurrency. Let j:j be a level mapping and I an interpretation for a program P. A clause c : H
B1 ; : : :; Bn : is semi delay
recurrent wrt j:j and I if
1. I is a model for c and
2. if rel(H) ' rel(Bi ), then for every cover C for Bi and for every grounding
substitution for c such that I j= C, we have that jHj > jBi j.
A program P is semi delay recurrent wrt j:j and I if every clause is semi delay
recurrent wrt j:j and I.
2
Note that delay recurrency is not equivalent to semi delay recurrency. Every
delay recurrent program is semi delay recurrent, but the converse is not true.
Example 1. The following program is semi delay recurrent, but not delay recurrent.
P([xjy]) Append( , , ) ^ P(y).
2
Due to the possibility of speculative output bindings, in order to be sure that
the condition I j= C holds, each atom in C must be completely resolved. In [11]
local selection rules are used to ensure this property. A local selection rule only
selects the most recently introduced atoms in a derivation and thus completely
resolves sub-computations before proceeding with the main computation.
Notice, however, that for semi delay recurrency, it is only necessary for the
covers of those atoms which are mutually dependent with the head of the clause
to be resolved completely. This means that following the resolution of these
covers, an arbitrary amount of coroutining may take place amongst the remaining
atoms of the clause. To formalise a selection rule based on this idea we introduce
the notion of covers and covered atoms in a goal.
Denition8 covers and covered atoms in a goal. Let G = A ; : : :; An
be a goal and suppose that the atom Ai is resolved with the semi delay recurrent
clause c : H B giving 2 mgu(H; Ai ). If A 2 body(B) and rel(A) ' rel(H),
1
then A is a covered atom in G and C is a cover of A in G where C is a
cover of A in c and G is the resolvent of G.
2
0
0
0
Denition9 semi local selection rule. A semi local selection rule only se-
lects a covered atom in a goal if one of its covers in a previous goal has been
completely resolved.
2
A semi local selection rule ensures that before selecting a covered atom A, we
rst fully resolve a cover of A. Before giving the main result of our construction,
we need the following denition taken from [11].
Denition10 safe delay declaration [11]. A delay declaration for a predicate p is safe wrt j:j if for every atom A with predicate symbol p, if A satises
its delay declaration, then A is bounded wrt j:j.
2
Theorem11. Let P be a program with a delay declaration for each predicate
in P. Let j:j be a level mapping and I an interpretation. Suppose that
1. P is semi delay recurrent wrt j:j and I
2. The delay declarations for P are safe wrt j:j
Then every SLD-derivation for a query Q, using a semi-local selection rule is
nite.
2
We are now able to develop a program transformation based on the above
result. We begin by transforming a given program into one which is semi delay
recurrent, but with equivalent declarative semantics. Then by adding safe delay
declarations we can obtain a program which terminates for all queries using a
semi-local selection rule.
Denition12 semi delay recurrent transform sdr. The transform sdr is dened as follows.
p 2 P
) p 2 sdr(P ) ^ psdr 2 sdr(P ) where psdr 62 P
) 8(psdr (t; ) ) 2 sdr(P)
8(p(t) ) 2 P
c = 8(p(t) w) 2 P ) 8(psdr (t; d) d = c(d) ^ w ) 2 sdr(P)
0
where w is obtained from w by replacing each atom in w of the form qi (s) with
i (s; d ) if p ' qi , d is a tuple such that d 2 d if p ' qi and is a function
qsdr
i
i
c
with the property that c (d) > di 8di 2 d. The variables d and di ; 8i are domain
variables over IN. Finally for each p 2 P we introduce the auxiliary clause
8(p(t) pdepth (t; d) ^ psdr (t; d)) 2 sdr(P)
where t is a tuple of variables.
2
0
Lemma 13 semi delay recurrency. If for each p 2 P , the clauses dening
pdepth are semi delay recurrent wrt M(sdr(P)) and jj:jj, then the program sdr(P)
is semi delay recurrent wrt M(sdr(P)) and the level mapping j:j dened by
jpsdr(t; d)j = d
jp(t)j = 0
jpdepth(t)j = jjpdepth(t)jj
for all p 2 P .
2
By Theorem 11 and Lemma 13 we can obtain a program which terminates
for all queries under a semi-local computation rule by adding for each predicate,
a delay declaration which is safe wrt the level mapping dened in Lemma 13.
Note also that d = (d) is the only atom in the body of each non-auxiliary clause
which will be a covering atom in a goal. This means that after its resolution, an
arbitrary amount of coroutining may take place between the atoms in w .
Example 2. The program of Section 2 is obtained by applying the above transform, with (d) = d + 1, to the Quicksort program of Section 1 and adding
safe delay declarations. Notice that the number of suspension checks performed
has been minimised by introducing an auxiliary clause p1 (t) p2 (t) for each
predicate p.
2
0
4.2 Completeness
Having obtained a terminating program, we need to prove that the declarative semantics of the transformed program coincide with those of the original
program. In this way, under the assumption that the transformed program is
deadlock free [12], we can guarantee that all computed answers of this program
are complete wrt the declarative semantics of the original program. We have the
following result.
Lemma 14 equivalence. If M(P) j= p(t) and d 2 fd j M(sdr(P)) j= p(t; d)g
implies M(sdr(P)) j= pdepth (t; d) then for all p 2 P
p(t) 2 M(P) , p(t) 2 M(sdr(P))
The problem then is to dene pdepth for each p 2 P such that the above
equivalence result holds. Our novel solution to this problem uses information
about the success set of the program. Suppose we can deduce, for example, that
for a given goal G, all computed answers for G can be found in an SLD-tree
of xed depth, then we can compute the SLD-tree to that depth and no more,
and be sure that we have found all answers for G. In reality, the granularity is
ner, relying not on the depth of the SLD-tree as a whole but rather on the
lengths of individual branches. More precisely, for each predicate p we nd an
upper bound on the number of calls to p. It will often be the case that this
bound relates to the input arguments of the predicate. We thus use interargument relationships to capture this relation. Essentially, we dene pdepth as the
interargument relationship of the predicate padr.
Denition15 interargument relationship. Given p 2 P , a norm j:j and
a model M for p=n, an interargument relationship for p=n wrt S is a relation
I INn, such that if M j= p(t) then p(jtj) 2 I.
2
Interargument relationships can be automatically deduced using, for example, the analysis described in [4].
Example 3. The analysis in [4] can be used to deduce the argument size relations
IQuicksort =3 = fhx; y; di j x = y; d = xg, IAppend =4 = fhx; y; z; di j z = x + y; d =
xg and IPartition =5 = fhw; x; y; z; di j w = y + z; d = wg. These relations can be
used to derive the denitions of SetDepth Q/3, SetDepth A/3 and SetDepth P/4
for the program sdr(Quicksort) in Section 2.
2
Example 4. Given the following predicate Split from the program Mergesort
abs
abs
abs
Split([], [], []).
Split([xjxs], [xjo], e)
Split(xs, e, o).
the argument size relation ISplit =3 = fhx; y; z; di j d = x; d 2y; d 2z + 1g
can be derived. From this we can derive a program which terminates for all
queries Split(x, y, z) where either x, y or z is a list of determinate length and
the remaining two arguments are (optionally) unbound. We know of no other
technique in the literature which can prove termination of these queries. The
majority of approaches can only reason about the decrease in the level mapping
of successive goals in a derivation. For the level mappings jSplit(t1 ; t2; t3)j1 = jt1j
and jSplit(t1 ; t2; t3)j2 = jt2j the decrease only occurs on every second goal. A
similar problem which our approach can also deal with occurs in [13].
abs
4.3 Eciency
We now give a brief appraisal of our approach from a performance perspective.
In theory, the rigidity checks should not incur much more overhead than the
original delay declarations. For example, checking rigidity of the rst argument
of the query Append([1,2,3], y, z) requires three Nonvar tests - exactly the
same number that would be required if the query were executed using the conventional delay declarations. There are additional costs due to unication and
the calculation of the depth bound, but these costs could be minimised through
careful implementation. We have naively implemented and tested some sample
programs and some of the preliminary results are given below. The experiments
have been carried out in SICStus Prolog [7] on a Sparc 4.
Program Goal Length Time(s) for P [fGg Time(s) for sdr(P) [fGg
P
G
of list L one solution all solutions one solution all solutions
8-queens qn( )
0.4
6.8
0.3
5.3
permsort ps(L, ) 10
6.8
1
0.7
0.7
permsort ps( , L) 8
1.7
10.5
2.6
10.8
quicksort qs(L, ) 4000
3.7
4.5
4.8
6.0
quicksort qs( , L) 8
12ms
1
6ms
83.0
The main overhead is due to the rigidity checks and our implementation in
this respect is rather naive and could be improved. Even in our experimental
implementation this overhead only reaches a maximum factor of about three for
the simplest programs, e.g. Append. The power of our approach, however, lies in
its scalability and it is here where we believe the most impressive performance
gains are to be made. Preliminary tests indicate that the most benet is obtained
from larger programs where only one rigidity test is performed at the beginning
of the program and the rest of the computation is bounded by the depth bounds.
Then our programs can outperform the original ones with the delay declarations,
particularly as the amount of backtracking or coroutining increases.
5 Conclusion
The aim of control generation is to automatically derive a computation rule for
a program that is ecient but does not compromise program correctness. In our
approach to this problem we have transformed a program into a semantically
equivalent one, introduced delay declarations and dened a exible computation rule which ensures that all queries for the transformed program terminate.
Furthermore, we have shown that the answers computed by the transformed program are complete with respect to the declarative semantics. This is signicant.
Beyond the theoretical aspects of the work, we have demonstrated its practicality. In particular, we have shown how transformed programs can be easily
implemented in a standard logic programming language and how such a program
can be optimised to reduce the number of costly rigidity checks needed to ensure termination, dramatically improving its performance. Furthermore, we have
seen how the termination problems caused by speculative output bindings can
be eliminated without the use of a local computation rule or other costly overhead. The coroutining behaviour which is then possible contributes signicantly
to the eciency of the generated code.
In terms of correctness, we have only considered termination and completeness in this work, though other correctness issues also need investigating. We
believe the connection between acyclic modes and rigid terms may provide a
solution in our approach to the occur check problem, since the check is never
needed for acyclic moded goals. Also, the example of Section 2.2 illustrates how
the problem of deadlock freedom may be handled.
The eciency issues also require further investigation. We have separated to
some extent the issues of termination and performance and it is not now clear
what role extra delay declarations might play in improving the performance of
the transformed programs, or even whether other techniques such as multiple
specialisation would be more appropriate.
Acknowledgements
The authors would like to thank Elena Marchiori for providing useful literature
and clarifying their understanding of delay recurrency.
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