M. Gabbrielli, G.M. Dore, and G. Levi. Observable Semantics for Constraint Logic Programming. Journal of Logic and Computation, 5(2):133-- 171,1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....This reflects on the fact that the observables are taken to be a downward closed set of constraints. If the meaning of a program contains a certain constraint, then it also contains all the constraints entailed by it. Other (more refined) notions of observables for clp have been investigateded in [15, 13, 14]. Of particular interest is the notion corresponding to the so called S semantics ( 12] This notion, in fact, is based on the idea of collecting only the exact outputs of a clp program, and therefore arises quite naturally when regarding clp as a standard programming language. In [8] we have ....

M. Gabbrielli, G.M. Dore, and G. Levi. Observable Semantics for Constraint Logic Programming. Journal of Logic and Computation, 5(2):133-- 171,1995.

....the arithmetic solver employed in the CLP(R) system [7] deals only with linear constraints, whereas non linear ones are delayed. The languages truly implemented by these real systems are not captured by standard semantics constructions such as the original one of [5] and its refinements [4]. These semantics, indeed, refer to the ideal domain of computation, thus pretending that the system is endowed with a complete solver. The idea of [6] consists in parameterizing the semantics construction with respect to the constraint system and two functions modeling the inferential power of ....

....the fact that X = Y # Z (being non linear) is delayed until it becomes linear, namely, until either Y or Z are constrained to take a single value. If we are interested in complete constraint solvers only, then the semantics defined in [6] boils down to the CLP version of the s semantics [4]. In this case the AND compositionality problem has been deeply studied, even with respect to a much more concrete semantics, yielding stronger results [3] ....

M. Gabbrielli, G. M. Dore, and G. Levi. Observable semantics for constraint logic programs. J. Logic Computation, 5(2):133--171, 1995.

....order to build an e ective xpoint semantics for rst order LO programs. Ad hoc notions like the existential regions of [AJ99] nd a natural counterpart as elements of the non ground interpretations of LO programs, a notion inspired by the more advanced semantics of Constraint Logic Programming [GDL95,JM94] Furthermore, we have shown that the algorithms used for (Timed) Petri Nets can be extended in order to capture the richer speci cation language LO. The main interest of the new semantics is that it gives us a way to evaluate bottom up rst order LO programs, i.e. an alternative ....

M. Gabbrielli, M. G. Dore and G. Levi. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation, 5(2): 133-171, 1995.

....evaluation can be used to automatically check properties (speci ed in temporal logic like CTL) of the original program. Finally, a formal de nition of the bottom up semantics can be useful for studying equivalence, compositionality and abstract interpretation (as for traditional logic programs [7, 15]) The reason we selected LO in this preliminary work is that we were looking for a relatively simple linear logic language with a uniform proof presentation, state based computations and aspects of concurrency. Operationally, LO programs behave like a set of multiset rewriting rules. In practice, ....

....consider a formulation of LO with . and . Following the semantic framework of (constraint) logic programming [15, 21], we formulate the bottom up evaluation procedure in two steps. We rst de ne what one could call a ground semantics via a xpoint operator TP de ned over an extended notion of Herbrand interpretation consisting of multisets of atomic formulas. This way, we capture the uniformity of ....

M. Gabbrielli, M. G. Dore and G. Levi. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation, 5(2): 133-171, 1995.

....p(think; think; Turn1 ; Turn 0 2 ) Fig. 3. The CLP program PS F starv for the concurrent system S in Figure 1. If P contains also constrained facts p(x) c, then these are always contained in SP (F ) The SP operator has been introduced to study the non ground semantics of CLP programs in [GDL95] where also its connection to the ground semantics is investigated: the set of ground instances of a xpoint of the SP operator is the corresponding xpoint of the TP operator, formally lfp(T P ) lfp(S P ) D and gfp(TP ) gfp(SP ) D . Thus, Theorem 2 leads to the characterization of CTL ....

....Rule 1 is sound we simply have to show that the set of instances of the constrained fact p(x) D are contained in the least xpoint of P 0 F 0 . Let us recall that, for any P , the TP operator is monotonic and continuous over the lattice of sets of ground facts ordered by set inclusion [JM94,GDL95] From Tarski theorem, it follows that then lfp(T P ) S n=0 TP n , where TP 0 = and TP n 1 = TP (TP n ) Based on this observation, to prove that the instances of p(x) D are contained in lfp(T P 0 F0 ) we need the following lemma. F n ] N TP 0 F0 n ; for all n 0: We ....

M. Gabbrielli, M. G. Dore, and G. Levi. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation, 2(5):133-171, 1995.

....substitution that makes the answer constraint true is a correct answer substitution. This semantics is not oriented to a particular collection of built in relations R i , but rather to a general constraint logic programming scheme. The definitions and notations used here are similar to those in [16]. The difference comes out in bounded quantification. Definition 4.1 (Constraint) A constraint is a formula of the form 9x 1 : x n (A 1 : Am ) where n 0; m 0 and A i , for i 2 f1; mg, are atoms B i (t 1 ; t k i ) When n; m = 0 the constraint is denoted by true. A ....

....was to provide a support for arrays, and in particular for their parallel processing. General constraints over finite sets are considered in [27] The semantics of constraint logic programs were described in many papers (see the recent survey [25] Our presentation is very similar to that of [16], except for the handling of bounded quantifiers and a slightly different treatment of bound variables in constraints. The operational semantics for finite domains that is similar to our semantics is introduced in the form of cardinality operators of [41, 40] Our paper is different from the ....

M. Gabbrielli, G.M. Dore, and G. Levi. Observable semantics for constraint logic programs. Journal of Logic and Computation, 5(2):133--171, 1995.

....] 2 F . SP (F ) fp(x) c 1 c 2 j p(x) c 1 p(x 0 ) 2 P; p(x 0 ) c 2 2 F; c 1 c 2 is satis able in Dg If P contains also constrained facts p(x) c, then these are always contained in SP (F ) The SP operator has been introduced to study the non ground semantics of CLP programs in [GDL95] where also its connection to the ground semantics is investigated: the set of ground instances of a xpoint of the SP operator is the corresponding xpoint of the TP operator, formally lfp(T P ) lfp(S P ) D and gfp(TP ) gfp(SP ) D (see Appendix A) Thus, Theorem 2 leads to the ....

M. Gabbrielli, M. G. Dore, and G. Levi. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation, 2(5):133-171, 1995.

....evaluation can be used to automatically check properties (speci ed in temporal logic like CTL) of the original program. Finally, a formal de nition of the bottom up semantics can be useful for studying equivalence, compositionality and abstract interpretation (as for traditional logic programs [BGLM94, GDL95]) The reason we selected LO in this preliminary work is that we were looking for a relatively simple linear logic language with a uniform proof presentation, state based computations and aspects of concurrency. Operationally, LO programs behave like a set of multiset rewriting rules. In ....

....consider a formulation of LO with . and . Following the semantic framework of (constraint) logic programming [GDL95, JM94], we formulate the bottom up evaluation procedure in two steps. We rst de ne what one could call a ground semantics via a xpoint operator T P de ned over an extended notion of Herbrand interpretation consisting of multisets of atomic formulas. This way, we capture the uniformity of LO ....

M. Gabbrielli, M. G. Dore and G. Levi. Observable semantics for Constraint Logic Programs. Journal of Logic and Computation, 5(2): 133-171, 1995.

.... F ) SP (F ) fp(x) c 1 c 2 j p(x) c 1 p(x 0 ) 2 P; p(x 0 ) c 2 2 F; c 1 c 2 is satisfiable in Dg If P contains also constrained facts p(x) c, then these are always contained in SP (F ) The SP operator has been introduced to study the non ground semantics of CLP programs in [GDL95] where also its connection to the ground semantics is investigated: the set of ground instances of a fixpoint of the SP operator is the corresponding fixpoint of the TP operator, formally lfp(TP ) lfp(S P ) D and gfp(TP ) gfp(SP ) D . Thus, Theorem 2 leads to the characterization of CTL ....

M. Gabbrielli, M. G. Dore, and G. Levi. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation, 2(5):133--171, 1995.

....evaluation can be used to automatically check properties (specified in temporal logic like CTL) of the original program. Finally, a formal definition of the bottom up semantics can be useful for studying equivalence, compositionality and abstract interpretation (as for traditional logic programs [BGLM94, GDL95]) The reason we selected LO in this preliminary work is that we were looking for a relatively simple linear logic language with a uniform proof presentation, state based computations and aspects of concurrency. Operationally, LO programs behave like a set of multiset rewriting rules. In ....

....a formulation of LO with . Gammaffi; and . Following the semantic framework of (constraint) logic programming [GDL95, JM94], we formulate the bottom up evaluation procedure in two steps. We first define what one could call a ground semantics via a fixpoint operator T P defined over an extended notion of Herbrand interpretation consisting of multisets of atomic formulas. This way, we capture the uniformity of ....

M. Gabbrielli, M. G. Dore and G. Levi. Observable semantics for Constraint Logic Programs. Journal of Logic and Computation, 5(2): 133-171, 1995.

....the arithmetic solver employed in the CLP(R) system [7] deals only with linear constraints, whereas non linear ones are delayed. The languages truly implemented by these real systems are not captured by standard semantics constructions such as the original one of [5] and its refinements [4]. These semantics, indeed, refer to the ideal domain of computation, thus pretending that the system is endowed with a complete solver. The idea of [6] consists in parameterizing the semantics construction with respect to the constraint system and two functions modeling the inferential power of ....

....expressing the fact that X = Y Z (being non linear) is delayed until it becomes linear, namely, until either Y or Z are constrained to take a single value. If we are interested in complete constraint solvers only, then the semantics defined in [6] boils down to the CLP version of the s semantics [4]. In this case the AND compositionality problem has been deeply studied, even with respect to a much more concrete semantics, yielding stronger results [3] ....

M. Gabbrielli, G. M. Dore, and G. Levi. Observable semantics for constraint logic programs. J. Logic Computation, 5(2):133--171, 1995.

....; chain) We have chosen this simple example for brevity. Nevertheless, it makes the important point that we can prove steadfastness in CLP. Moreover, steadfastness is the natural characterisation of correctness of CLP modules. In this respect, it is related to the compositional semantics of [2, 7]. Thus we have shown that our approach can be used to synthesise reusable CLP modules. More interestingly, parameters of an open constraint specification framework can themselves be instantiated by constraint domains. Our approach therefore provides a basis for object oriented CLP software ....

....T with isoinitial model D T that is a sub structure of D and is elementarily equivalent to D. Whichever way we choose to axiomatise (D; L) we need suitable conditions for the validity of our halting criterion. We are concerned with a good semantics for synthesis, not just for computation (e.g. [7]) which enables us to reason precisely and soundly about synthesis. To this end, our constraint specification frameworks are endowed with isoinitial semantics. This provides an abstract semantics for defining the meaning of specifications and program correctness. This semantics is thus a major ....

M. Gabbrielli, G.M. Dore and G. Levi. Observable semantics for constraint logic programs. To appear in J. Logic and Computation.

....a suspension free cc program P can be safely translated into a clp program P 0 by removing all synchronizations (i.e. ask guards) The abstract interpretation of P 0 , computed within any framework for clp, yields a correct approximation of the semantics of the original program P . Following [14], we will introduce a semantics for the translated programs which is compositional wrt the union of programs. This semantics is then approximated by applying the theory of abstract interpretation. Obviously, the outlined methodology provides results in an implicative form: if a computation of ....

....step approach described in the previous section. Assuming that the whole program is suspension free, we can safely approximate its semantics by using the translation NoSynch. As the result of the transformation is a clp program, we can easily perform its analysis in a compositional way. Following [14], we now introduce a semantics for clp programs which is compositional wrt the union of programs. This semantics has been obtained by taking the Omega Gammae 276 tics defined for pure logic programs in [2] and lifting it to the clp case. 5.1 The semantics of Omega programs Definition 5.1 ....

[Article contains additional citation context not shown here]

M. Gabbrielli, M.G. Dore, and G. Levi. Observable semantics for Constraint Logic Programs. Journal of Logic and Computation, 5(2):133--171, 1995.

.... definite logic programs were the ones of Lassez and Maher [LM84, Mah88] further work has been done by Mancarella and Pedreschi [MP88] and Brogi et al. BLM92] In [GS89] Gaifman and Shapiro proposed a compositional semantics, which was further extended in [BGLM94] and for CLP programs in [GDL95] Compositionality vs. non monotonicity. However, in the development of semantics for normal logic programs, logic programs which employ the negation operator) compositionality has been widely disregarded. Notable exception to this are the papers by Maher [Mah93] and Ferrand and Lallouet [FL95] ....

M. Gabbrielli, G.M. Dore, and G. Levi. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation, 5(2):133--171, 1995.

.... but this does not make a difference, since the argument works also with the proper clause 8z: z = t oe flR(z) 7 Embedding CLP proofs into LLP Various operational semantics for general CLP schemes have been proposed which use a notion of resolution, or derivation rules (see for example [AV96, GDL95, And92] or the survey [JM94] This section illustrates one such (top down) procedural semantics, as proposed in [DG94] or [AV96] 15 , closely related to our l , and proves that the answer constraints generated by them are equivalent to the constraints extracted from an abstract proof of the ....

M. Gabrielli, G. M. Dore, and G. Levi. Observable semantics for constraint logic programs. Journal of Logic and Computation, 5(2):133--171, 1995.

....T P is continuos on the set of D interpretations ordered by . Interpretations can be represented by using set of facts, i.e. clauses a c. The fixpoint semantics can be rephrased in terms of collections of facts, in the following denoted by interpretations according to the S semantics of CLP [1]. Let B the base be the set fp( X) true j p 2 Pig. Let I be a interpretation, i.e. a subset of the B . The non ground immediate consequences operator is defined as follows. S P (I) fp( X) c j 9 a renamed clause p( x) Gammac 0 ; b 1 ; b n in P; a i c i ) 2 I ....

M. Gabbrielli, M.G. Dore, and G. Levi. Observable semantics for Constraint Logic Programs. Journal of Logic and Computation, 2(5):133--171, 1995.

....6 Conclusion and related work The idea to use constraints instead of substitutions is used in many areas of automated deduction. It is presented in the unification algorithm of Martelli and Montanari [14] it is used in the semantics of logic programs, for example in Gabbrielli, Dore and Levi [6], and Maher [13] It is implemented in the theorem proving system based on the sequent calculus for classical logic (Anufriev and Aselderov [1] Degtyarev, Zhezherun and Lyaletski [3] see also the survey by Degtyarev and Voronkov [2] The use of constraints in automated deduction is investigated ....

M. Gabbrielli, G.M. Dore, and G. Levi. Observable semantics for constraint logic programs. Journal of Logic and Computation, 5(2):133--171, 1995.

.... of definite logic programs were the ones of Lassez and Maher [18, 20] further work has been done by Mancarella and Pedreschi [22] and Brogi et al. 4] In [12] Gaifman and Shapiro proposed a compositional semantics, which was further extended in [3] and for CLP programs in [11]. Compositionality vs. non monotonicity. However, in the development of semantics for normal logic programs, which employ the negation operator) compositionality has been widely disregarded. Notable exception to this are the papers by Maher [21] and Ferrand and Lallouet [9] comparison between ....

M. Gabbrielli, G.M. Dore, and G. Levi. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation, 5(2):133--171, 1995.

....t n )g) fB 1 ; Bm g; 4. E 0 is E [ fs 1 = t 1 ; s n = t n g. By an constraint SLD derivation we mean a sequence of pairs (G 1 ; E 1 ) G N ; EN ) 1) such that (G i ; E i ) G i 1 ; E i 1 ) holds for all 1 i N Gamma 1. By the standard technique (see for example [19, 12]) one can prove that a goal G succeeds with respect P if and only if there is a constraint SLD derivation (1) such that (1) G 1 = G; 2) E 1 = 3) GN = 2; and (4) EN is solvable in E. It is easy to define a nondeterministic Turing machine M that decides the query answering problem by ....

M. Gabbrielli, G.M. Dore, and G. Levi. Observable semantics for constraint logic programs. Journal of Logic and Computation, 5(2):133-- 171, 1995.

....substitution that makes the answer constraint true is a correct answer substitution. This semantics is not oriented to a particular collection of built in relations R i , but rather to a general constraint logic programming scheme. The definitions and notations used here are similar to those in [GaDoLe 95] The difference comes out in bounded quantification. Definition 4.1 (Constraint) A constraint is a formula of the form 9x 1 : x n (A 1 : Am ) where n 0; m 0 and A i , for i 2 f1; mg, are atoms B i (t 1 ; t k i ) When n; m = 0 the constraint is denoted by true. A ....

....a support for arrays, and in particular for their parallel processing. General constraints over finite sets are considered in [Kozen 95] The semantics of constraint logic programs were described in many papers (see the recent survey [JafMah 94] Our presentation is very similar to that of [GaDoLe 95] except for the handling of bounded quantifiers and a slightly different treatment of bound variables in constraints. The operational semantics for finite domains that is similar to our semantics is introduced in the form of cardinality operators of [VaSiDi 92, VaSaDe 95] Our paper is different ....

M. Gabbrielli, G.M. Dore, and G. Levi. Observable semantics for constraint logic programs. Journal of Logic and Computation, 5(2):133--171, 1995.

....of programs. Over the last few years we have developed a general approach to the semantics [21, 7] whose 2 aim was modeling the observable behaviors (possibly in a compositional way) for a variety of logic languages, ranging from positive logic programs [17, 9, 8] to constraint logic programs [20, 19] and general logic programs [35] Our approach is based on the idea of choosing (equivalence classes of) sets of clauses as semantic domains. Denotations (that we call interpretations) are not interpretations in the conventional mathematical logic sense and can be computed both by a top down ....

M. Gabbrielli, G.M. Dore and G. Levi. Observable Semantics for Constraint Logic Programs. To appear in the Journal of Logic and Computation.

....for CLP programs (modules) is correct iff it maps a program (a module) into an ( c ) equivalent one. 3. 1 A compositional semantics for CLP modules The correctness proofs for our transformation system will be carried out by showing that the system preserves a semantics (borrowed from [13]) which models answer constraints and is compositional w.r.t. Phi. This implies that it is also correct w.r.t. c , in the sense that if two modules have the same semantics then they are c equivalent. From this property it follows the desired correctness result. Basically, the semantics we ....

....r(X,Y) X=a. r(X,Y) Y=b. and Q 2 : q(X,Y) true 2 r(X,Y) r(X,Y) X=a. r(X,Y) Y=b. The query q(X,Y) has the computed answer constraint X = a Y = b in Q 1 and not in Q 2 . The following Lemma shows that the equivalence relation is correct wrt the congruence relation c . Lemma 3. 8 [13] Let M = hP; i and M 0 = hP 0 ; i be two modules with the same set of open predicates. If P P 0 then M c M 0 . 2 We are now able to define the semantic domain. For the sake of simplicity, we will denote the equivalence class of a clause c by c itself. Definition 3.9 ....

[Article contains additional citation context not shown here]

M. Gabbrielli, G.M. Dore, and G. Levi. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation, 1994.

....congruent ones. We now give a result which provides a condition sufficient to guarantee the operational congruence of two modules. Here, and in the sequel, given a set of predicate symbols we call a derivation any derivation c 2 C ; b 2 B such that P red( B) Theorem 4. 3 [16] Let M 1 = hP 1 ; i and M 2 = hP 2 ; i be two modules. If ffl for each derivation in M 1 there exists a similar derivation in M 2 then, for every module M such that M 1 Phi M and M 2 Phi M are defined, we have that for any refutation in M 1 Phi M there exists a similar refutation in M 2 ....

....there exists a derivation true 2 C P ; d 0 2 D of length n such that 5 The condition on clauses used in the derivation is needed to avoid variable name clashes. i) d j c d 0 (ii) the variables that d 0 2 D and c have in common are a subset of the variables of C. 2 Claim 2 [16] Let P be a program, and c 1 c 2 2 C 1 ; C 2 be a query. Then, there exists a derivation c 1 c 2 2 C 1 ; C 2 P ; d 2 D of length n iff there exist two derivations 1 : c 1 2 C 1 P ; d 1 2 D 1 and 2 : c 2 2 C 2 P ; d 2 2 D 2 such that (i) D j D 1 ; D 2 , ....

M. Gabbrielli, G.M. Dore, and G. Levi. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation, 5(2):133--171, 1995.

....to a semantics which maps atoms to sets of constraints. This indicates that our closure based semantics is the analogue for (constraint) logic languages with dynamic scheduling of the S semantics [Falaschi et al. 1989] for logic programming with fixed atom scheduling and of its extension to CLP [Gabbrielli et al. 1995]. 5.1 About full abstraction We recall that a denotational semantics is fully abstract if, whenever it distinguishes two syntactic objects, then there is a context in which the observational behavior of these two objects is different. In our case, the syntactic objects are essentially the ....

Gabbrielli, M., Dore, G., and Levi, G. 1995. Observable Semantics for Constraint Logic Programs. Journal of Logic and Computation 5, 2, 133--171.

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M. Gabbrielli, N. Dore, and G. Levi. Observable semantics for Constraints Logic Programs. To appear in Journal of Logic and Computation, 1994. 31

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