F. Fages, "Constructive negation by pruning", LIENS technical report 94-14, revised 95-24. To appear in the Journal of Logic Programming, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....semantic definition of normal programs could be considered solved. On one hand, Clark Kunen s completion [9] and well founded semantics [15] provide two alternative approaches to define the (model based) meaning of programs. On the other hand, the works of Kunen, Fages and, Lucio, Orejas and Pino [5, 11] provide fixpoint operators to compute its logical consequences (i.e. the set of correct answers) Finally, constructive negation, as introduced by Chan [1] Stuckey [14] and Drabent [3] is accepted as a complete and sound mechanisms to describe the operational semantics of the whole class of ....

....as introduced by Chan [1] Stuckey [14] and Drabent [3] is accepted as a complete and sound mechanisms to describe the operational semantics of the whole class of normal programs. However, from an implementation viewpoint, the proposed operational semantics for constructive negation (see [3, 5, 14] for a representative group of them) are difficult to implement in a practical way. In general, the core of the difficulties for developing practical implementations are the need of (complex) subsidiary computations each time a negative goal is selected during the computation and or the need of ....

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F. Fages. Constructive negation by pruning. Journal of Logic Programming, 32:85-- 118, 1997.

....semantic definition of normal programs could be considered solved. On one hand, Clark Kunen s completion [9] and well founded semantics [15] provide two alternative approaches to define the (model based) meaning of programs. On the other hand, the works of Kunen, Fages and, Lucio, Orejas and Pino [5, 11] provide fixpoint operators to compute its logical consequences (i.e. the set of correct answers) Finally, constructive negation, as introduced by Chan [1] Stuckey [14] and Drabent [3] is accepted as a complete and sound mechanisms to describe the operational semantics of the whole class of ....

....as introduced by Chan [1] Stuckey [14] and Drabent [3] is accepted as a complete and sound mechanisms to describe the operational semantics of the whole class of normal programs. However, from an implementation viewpoint, the proposed operational semantics for constructive negation (see [3, 5, 14] for a representative group of them) are difficult to implement in a practical way. In general, the core of the difficulties for developing practical implementations are the need of (complex) subsidiary computations each time a negative goal is selected during the computation and or the need of ....

[Article contains additional citation context not shown here]

F. Fages. Constructive negation by pruning. Journal of Logic Programming, 32:85-- 118, 1997.

....of the underlying constraint structure. We prove that the admissible closure condition is also necessary to guarantee the existence of an e ective implementation of Constructive Negation. Keywords: Constraint Logic Programming, Constructive Negation. 1 Introduction Constructive Negation (CN) [3, 2, 6, 8, 15] is a technique for handling negation in (Constraint) Logic Programming. The technique relies on the explicit construction of answers to a goal which may possibly involve negation. This is commonly achieved by explicitly computing the negation of the solutions to the corresponding positive goal. ....

....context of Logic Programming. The semantic properties of CN in logic programming have been studied by various authors (e.g. 6, 13] CN has been extended to the more general context of Constraint Logic Programming (CLP) by Stuckey [15] who gave the rst completeness result for this method. In [8] Fages introduced a pruning technique to implement Constructive Negation in CLP and Concurrent Constraint languages. In [15] Stuckey develops a general framework for handling CN in CLP, and proves its soundness and completeness (in 3 valued logic) The development of an actual implementation of ....

F. Fages. Constructive Negation by Pruning. Journal of Logic Programming, 32(2):85-118, 1997.

....j n n 0 which are SLDFA computed answers for A of rank k Gamma 1 and FET Sigma j= j 1 : j n oe 1 : oe m (iv) There is no a node of the form . 20 Drabent [11] proved that this operational semantics, SLDFA resolution, is sound and complete with respect to Comp(P ) Fages [13], provided a fixpoint characterization for other version of constructive negation based on three valued models. Later we present other fixpoint characterization for constructive negation proposed by Lucio, Orejas and Pino [ which is based on intuitionistic structures. This structures are ....

F. Fages. Constructive negation by pruning. Journal of Logic Programming, 32:85--118, 1997.

....which the corresponding positive literal is evaluated. In [5, 29, 32] a negative literal is solved using Clark s completed definitions at run time, possibly with partial evaluation. Quantified complex formulas have to be transformed into a disjunctive normal form and be dealt with explicitly. In [11, 12, 17], substitutions called fail answers are generated for variables in a negative literal A based upon a frontier of the positive literal A. This is a powerful technique since A does not have to be completely evaluated before an answer for A is derived. Since a subgoal can have many different ....

....in a subgoal. Delaying non ground negative literals directly means that constructive negation can be applied even if the constraints and bindings of variables in a negative literal are not completely determined. This achieves the same effect that is realized by the notion of frontiers in [11, 12] and the TU forests in [10] The open problem is how to control the delaying of non ground negative literals and the iterated propagation of constraints so that repetition of computation is avoided. 27 Several minor aspects of SLGCN resolution can be refined. In all anti subsumption constraints, ....

F. Fages. Constructive negation by pruning. Journal of Logic Programming, to appear.

....to [7] is quite more direct, in the sense that the construction of our least models is closely related to ranked resolution as defined there. There is also a certain relation between the construction of our least models and Fitting s fixpoint semantics [9] or rather with the variation defined in [13], although not as close as it may seem: one may notice that in each layer of our least models we add not just the immediate consequences of the previous layer, but all logical consequences. We are convinced that, not only with respect to compositionality issues, our semantics is just the ....

F. Fages, Constructive negation by pruning, Journal of Logic Programming, to appear.

.... the other hand, M(Q P; M(FQ (P ) M(Comp(P ) is straightforward because Comp(P ) j= 3 Q P; The other inclusion can be proved showing, by induction, FET [ FQ (P ) j= 3 (T P i) 8 for every i 2 N, since M(Comp(P ) M(T P ) being T P the Fages immediate consequence operator [8]. 4 Compositionality and Full Abstraction In this section we study the compositionality and full abstraction of the two semantics de ned above. The rst result shows that the FQ semantics is weakly compositional, in the sense that, for any two open programs P1 and P 2, the FQ semantics ....

Fages, F., Constructive negation by pruning, The Journal of Logic Programming 32:85-11 (1997).

....rst occurrence of p(x)2x = s i (0) The operator A P given in [32] is a non ground version of P relative to a structure A where the constraints are interpreted. It ranges over (non ranked) partial constrained interpretations and is neither continuous. The continuous operator de ned in [18], to obtain a fully abstract xpoint semantics characterizing the operational semantics with respect to answer constraints, is in some sense closer to 1 In order to make the reading easier we underline the negative parts. 26 our TP . However, there are two di erences that may be remarked. ....

....to [13] is quite more direct, in the sense that the construction of our least model is closely related to ranked resolution as de ned there. There is also a certain relation between the construction of our least model and Fitting s x point semantics [20] or rather with the version de ned in [18], 35 although not as close as it may seem: notice that in each layer of our least model we add not just the immediate consequences of the previous layer, but all logical consequences. Acknowledgments The authors would like to thank the referees for their detailed comments that have greatly ....

Fages, F., Constructive negation by pruning, The Journal of Logic Programming 32:85-11 (1997).

....to [14] is quite more direct, in the sense that the construction of our least model is closely related to ranked resolution as defined there. There is also a certain relation between the construction of our least model and Fitting s fix point semantics [21] or rather with the version defined in [19], although not as close as it may seem: notice that in each layer of our least model we add not just the immediate consequences of the previous layer, but all logical consequences. Acknowledgments This work has been partially supported by the Spanish CICYT project COSMOS (ref. TIC951016 ....

Fages, F., Constructive negation by pruning, The Journal of Logic Programming 32:85-11 (1997).

....Fages et al. [8] allow the programmer to program the ordering among solutions but the semantics of optimization is provided via negation. Recently, Fages shows how constructive negation by pruning provides a form of branch and bound technique for various optimization goals in CLP languages [7]. Whereas related work has mostly focus on optimization predicates that can be expressed as maximizing or minimizing some objective function, a key distinguishing of our approach is that it allows the programmer to program the preference criteria to suit the application at hand. This capability to ....

F. Fages. Constructive Negation by Pruning. J. of Logic Programming, 32(2):85--118, 1997.

....which is sound and complete with respect to the three valued consequences of the completion of the program can be thought of as a generalisation of Chan s. Fages proposes a simple concurrent pruning mechanism over standard SLD derivation trees for constructive negation in constraint logic programs [6]. Two derivation trees are concurrently constructed. The computed answers from one of the trees are used to prune the nodes of the other. Fages method admits an efficient implementation as it is not necessary to deal with complex goals with explicit quantifiers outside the constraint part. Cleary ....

F. Fages. Constructive Negation by Pruning. Journal of Logic Programming, 32(2):85-- 118, 1997.

.... Programming with a negation operator is a major issue and gives rise to a lot of works ( 2] the first job consists in allowing LC to deal with such an operator and then in executing logic programs with negation (the so called normal programs) Using the ideas of constructive negation (see [12] [7]) LC has been extended in [1] in order to provide a complete negation framework. At the same time, introduction of Constraints in Logic Programming (CLP ) supplies not only more operational efficiency than usual resolution, but also a greater expressive power. So, it was essential to integrate ....

....the intended set of answers. 4 SOUNDNESS AND COMPLETENESS RESULTS FOR CLC In this section, we introduce the standard declarative semantics L(P ) for a CLP program P and we compare it with our operational definition. Concerning the declarative meaning, we follow the point of view developed in [7]. 4.1 Semantics A constrained atom is a couple (c; A) where c is a satisfiable constraint such that V ar(c) V ar(A) The set of constrained atoms is denoted B. Note that (true; A) 2 B for each A 2 Atom. Sometimes, it is convenient to introduce an extended domain which will play for CLP the ....

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F. Fages, `Constructive Negation by Pruning', Technical Report 94-14, LIENS CNRS, Ecole Normale Superieure, 45, Rue d'Ulm, 75230 Paris Cedex 05, (September 1994).

....context, keeping loop avoiding and synthesis abilities. Two extensions have been proposed. Since extension of Logic Programming with a negation operator is a major issue, a lot of works have been developed in this way (see [2] for a survey) Inspired by the ideas of constructive negation (see [10, 11, 16, 34] for instance) LC has been extended in [1] in order to provide a complete negation framework. On another hand, introduction of constraints in Logic Programming (CLP ) see [21] for a survey) not only allows more operational efficiency than usual resolution, but also gives a greater expressive ....

....exhibit their relationship. 4 Soundness and Completeness results for CLC In this section, we introduce the standard declarative semantics L(P ) for a CLP program P and we compare it with our operational definition. Concerning the declarative meaning, we follow the point of view developed in [16]. Since what we observe about a goal G is a set of constraints produced by its computations with regard to a program P , namely the computed answers constraints, these constraints are a more natural choice of observables than the success set as defined in [28] Following theses ideas, we emphasize ....

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F. Fages. Constructive Negation by Pruning. Technical Report 94-14, LIENS CNRS, Ecole Normale Sup'erieure, 45, Rue d'Ulm, 75230 Paris Cedex 05, September 1994.

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F. Fages, "Constructive negation by pruning", LIENS technical report 94-14, revised 95-24. To appear in the Journal of Logic Programming, 1996.

.... of the structure th(X ) This is achieved in pure CLP for various observable properties of the execution: the existence of a successful derivation to a query [13] the set of computed answer constraints [21, 8] finite failures [13] the set of computed constraints with constructive negation [37, 6], etc. For example, computed answer constraints (i.e. final states of computations) can be observed logically: any computed constraint entails the initial goal (modulo the logical translation of the program P and the constraint system C) conversely any constraint c entailing a goal G is ....

F. Fages. Constructive negation by pruning. J. of Logic Programming, 32(2), 1997.

.... theory of the structure th(X ) This is achieved in pure CLP for various observable properties of the execution: the existence of a successful derivation to a query [13] the set of computed answer constraints [21, 8] nite failures [13] the set of computed constraints with constructive negation [37, 6], etc. For example, computed answer constraints (i.e. nal states of computations) can be observed logically: any computed constraint entails the initial goal (modulo the logical translation of the program P and the constraint system C) conversely any constraint c entailing a goal G is ....

F. Fages. Constructive negation by pruning. J. of Logic Programming, 32(2), 1997.

....of constructive negation, which provides normal logic programs with a sound and complete [21] operational semantics w.r.t. Kunen s logical semantics [14] We propose an analysis method for normal logic programs interpreted with constructive negation, based on the generalized S semantics given in [9] and on the hierarchy described in [10] We present here an analysis based on the depth(k) domain which approximates the computed answers obtained by constructive negation and therefore the three valued consequences of the program completion and CET (Clark s equational theory) Other well known ....

....[21] operational semantics w.r.t. Kunen s logical semantics [14] In logic programming, Kunen s semantics is simply the set of three valued consequences of the program s completion and the theory CET . The S semantics of de nite logic programs [2] has been generalized to normal logic programs in [9] for a version of constructive negation, called constructive negation by pruning. The idea of the xpoint operator, which captures the set of computed answer constraints, is to consider a non ground nitary (hence continuous) version of Fitting s operator. Here we give a de nition of the operator ....

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F. Fages. Constructive negation by pruning. Journal of Logic Programming, 32(2):85-118, 1997.

....constructive negation by pruning rule introduced by Chan [4] provides normal CLP programs with a complete operational semantics w.r.t. Kunen s semantics [20] However that scheme has not received yet a fully abstract xpoint semantics characterizing the computed answer constraints for instance. In [8] such a xpoint semantics has been given for a variant of the constructive negation rule, called constructive negation by pruning. That semantics generalizes the S semantics of de nite logic programs [1] to normal CLP programs. In this paper we show that the di erent properties characterizing the ....

.... characterizing the operational behavior of a normal CLP program can be organized in a hierarchy of xpoint semantics related by Galois insertions, corresponding to a hierarchy of observable properties and program equivalences (see gure 1) The S semantics and the xpoint operator T S P de ned in [8] are at the center of the hierarchy developed in this paper. In section 3 we de ne the most abstract Kunen s semantics, as a xpoint semantics and de ne a collecting semantics as being a pair (C; T ) where C is a semantic domain, and T a monotone operator on C. The other semantics of the hierarchy ....

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F. Fages, \Constructive negation by pruning", LIENS technical report 94-14, revised 95-24. To appear in the Journal of Logic Programming, 1996.

....and for which we give a complete top down evaluation procedure. More general local optimization predicates with set of protected variables [4] 19] have the power of general CLP programs with negation, we derive a complete top down procedure from the scheme of constructive negation by pruning [5] in this context, and propose an alternative bottom up evaluation procedure based on a finitary version of Fitting s operator [5] We discuss several variants of the top down procedure and relate it to other proposals [19] 4] Our claim is not that the complete execution models we describe can be ....

.... variables [4] 19] have the power of general CLP programs with negation, we derive a complete top down procedure from the scheme of constructive negation by pruning [5] in this context, and propose an alternative bottom up evaluation procedure based on a finitary version of Fitting s operator [5]. We discuss several variants of the top down procedure and relate it to other proposals [19] 4] Our claim is not that the complete execution models we describe can be used directly to solve efficiently complex optimization problems but that they can serve as a basis for designing more efficient ....

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F. Fages, "Constructive negation by pruning", submitted LIENS tech report 94-14, revised 95-24, 1995.

....quite complicated, and the generalization of the S semantics, which captures only the set of computed answer constraints, to normal logic programs has been open for a while [1] In this paper we present an analysis method for normal CLP programs. It is based on the generalized S semantics given in [9] and on the hierarchy described in [10] One important contribution of the paper is the definition of a normal form for first order constraints on the Herbrand Universe, which is suitable for analysis. In fact the normal form allows us to define an abstraction function which is a congruence wrt ....

....w.r.t. Kunen s logical semantics [14] In the case of a normal CLP(X ) program, Kunen s semantics is simply the set of three valued consequences of the program s completion and the theory th(X ) The S semantics of definite logic programs [1] has been generalized to normal CLP(X ) programs in [9] for a version of constructive negation, called constructive negation by pruning. The idea of the fixpoint operator, which captures the set of computed answer constraints, is to consider a non ground finitary (hence continuous) version of Fitting s operator. Here we give a definition of the ....

[Article contains additional citation context not shown here]

F. Fages, "Constructive negation by pruning", Journal of Logic Programming, 32(2):85-118, 1997.

....In the previous example, taking the third truth value u for p provides a model of P as u :u. Kunen proved a completeness result [18] for the negation as failure rule w.r.t. the three valued completion of the program, followed by stronger completeness results for the constructive negation rule [19,7]. In this paper we study extended logic programs as introduced by Gelfond and Lifschitz [13,14] see also [21,1] to deal with two kinds of negation: explicit negation allowed in clause heads and bodies and negation by failure allowed in clause bodies only. These two negations allow two different ....

....4 valued A, A : j= 2 P : implies A : j= 2 OE : By Proposition 7 (v) this is equivalent to P 4 OE. 2 Operational Semantics. The reduction to normal programs allows to consider that literals A and :A have a separate life . Hence SLDNF resolution [18] resp. constructive negation [19,7]) provide extended programs with correct (resp. complete) operational semantics, in the following way: the answer to a given goal G in an extended logic program is obtained by combining the answers to G and G : in the corresponding normal program) each answer sets the value of one component v ....

F. Fages, Constructive Negation by Pruning, to appear in J. of Logic Programming (1996).

....negation by pruning rule introduced by Chan [4] provides normal CLP programs with a complete operational semantics w.r.t. Kunen s semantics [20] However that scheme has not received yet a fully abstract fixpoint semantics characterizing the computed answer constraints for instance. In [8] such a fixpoint semantics has been given for a variant of the constructive negation rule, called constructive negation by pruning. That semantics generalizes the S semantics of definite logic programs [1] to normal CLP programs. In this paper we show that the different properties characterizing ....

.... the operational behavior of a normal CLP program can be organized in a hierarchy of fixpoint semantics related by Galois insertions, corresponding to a hierarchy of observable properties and program equivalences (see figure 1) The S semantics and the fixpoint operator T S P defined in [8] are at the center of the hierarchy developed in this paper. In section 3 we define the most abstract Kunen s semantics, as a fixpoint semantics and define a collecting semantics as being a pair (C; T ) where C is a semantic domain, and T a monotone operator on C. The other semantics of the ....

[Article contains additional citation context not shown here]

F. Fages, "Constructive negation by pruning", LIENS technical report 94-14, revised 95-24. To appear in the Journal of Logic Programming, 1996.

....of constructive negation, which provides normal logic programs with a sound and complete [21] operational semantics w.r.t. Kunen s logical semantics [14] We propose an analysis method for normal logic programs interpreted with constructive negation, based on the generalized S semantics given in [9] and on the hierarchy described in [10] We present here an analysis based on the depth(k) domain which approximates the computed answers obtained by constructive negation and therefore the three valued consequences of the program completion and CET (Clark s equational theory) Other well known ....

....[21] operational semantics w.r.t. Kunen s logical semantics [14] In logic programming, Kunen s semantics is simply the set of three valued consequences of the program s completion and the theory CET . The S semantics of definite logic programs [2] has been generalized to normal logic programs in [9] for a version of constructive negation, called constructive negation by pruning. The idea of the fixpoint operator, which captures the set of computed answer constraints, is to consider a non ground finitary (hence continuous) version of Fitting s operator. Here we give a definition of the ....

[Article contains additional citation context not shown here]

F. Fages. Constructive negation by pruning. Journal of Logic Programming, 32(2):85--118, 1997.

.... first introduced for the Horn clause fragment of classical logic, has been smoothly applied to constraint logic programming, where the logical nature of the constraint system extends to the goals and program declarations, and states strong connections between operational semantics and entailment [11, 16, 32, 6]. For instance, success constraints (i.e. final states of computations) can be observed logically : Current affiliation : McGill University. Post doctoral fellowship from the INRIA. any success entails the initial state (modulo the logical translation of the program P and the constraint ....

F. Fages. Constructive negation by pruning. J. of Logic Programming, 32(2), 1997.

....is complete on success for 2 level programs, and complete both on success and failure for hierarchical programs. Note that a concurrent pruning procedure complete both on success and failures can be derived for OCLP programs from the scheme for constructive negation based on pruning described in [6]. 3.4 Operational semantics for relational optimization The main difficulty with the previous scheme for handling preference constraints is that all preferences need to be explicitly encoded by an objective function. We show here how this difficulty can be overcomed by expressing the preference ....

....a simple form p(X) q(X) X 3. In general however the process of collecting a complete frontier in the derivation tree of the selected subgoal must be performed at each resolution step with a complex subgoal. Finally note that the scheme for constructive negation based on pruning described in [6], eliminates the need to consider complex subgoals with explicit quantifiers, and should be of practical value for handling program defined preferences in a very general setting. 4 Comparison We have presented two different approaches for integrating preferences inside the CLP framework. We now ....

[Article contains additional citation context not shown here]

F Fages. Constructive negation by pruning. In Ecole de Printemps d'Informatique Th'eorique, Chatillon sur Seine, France, May 1994. Working paper.

.... has been smoothly applied to constraint logic programming (with, or without negation by failure, or constructive negation) where the logical nature of the constraint system extends to the goals and program declarations, and states strong connections between operational semantics and entailment [11, 18, 32, 6]. For instance, success constraints (i.e. final states of computations) can be observed logically: any success entails the initial state (modulo the logical translation of the program P and the constraint system C) conversely any constraint c entailing a goal G is covered (again modulo P ....

F. Fages. Constructive negation by pruning. To appear in J. of Logic Programming, 1996.

....and complete w.r.t. the threevalued logical consequences of the program s completion. We emphasize a full abstraction result which permits to go beyond the theorem proving point of view and to completely characterize the operational behavior of normal CLP programs. These results are based on [8]. Then we show how constructive negation by pruning allows to define optimization higher order predicates for CLP programs. We show that in this context the operational semantics specializes into an efficient branch and bound like procedure proved correct and complete in a full first order ....

....by Bruscoli et al. 5] named intensional negation, performs all disjunctive normal form transformations once and for all at compile time, but still all quantifiers need be explicit at run time and derivation rules need be defined for complex goals. Another way undertaken independently in [7] and [8] is to use the principle of constructive negation as a concurrent pruning mechanism over standard CSLD 3 It is worth noting that a complete notion of three valued logic programming could also take into account the set L u 3 (P ) fcjp(X) 2 B : P ; th(A) j=3 c (p(X) u)g of constrained ....

[Article contains additional citation context not shown here]

F. Fages, "Constructive negation by pruning", Technical report 94-14, Ecole Normale Sup'erieure, Paris. Sept. 1994. Submitted for publication.

.... of the structure th(X ) This is achieved in pure CLP for various observable properties of the execution: the existence of a successful derivation to a query [13] the set of computed answer constraints [21, 8] finite failures [13] the set of computed constraints with constructive negation [33, 6], etc. For example, computed answer constraints (i.e. final states of computations) can be observed logically: any computed constraints entails the initial goal (modulo the logical translation of the program P and the constraint system C) conversely any constraint c entailing a goal G is ....

F. Fages. Constructive negation by pruning. J. of Logic Programming, 32(2), 1997.

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