Une nouvelle sémantique pour la programmation logique capturant la sémantique des modèles stables : la sémantique des extensions

National audienceAnswer set programming is a well studied framework in logic programming. Many research works had been done in order to de ne a semantics for logic programs. Most of these semantics are iterated xed point semantics. The main idea is the canonical model approach which is a declarative semantics for logic programs that can be de ned by selecting for each program one of its canonical models. The notion of canonical models of a logic program is what it is called the stable models. The stable models of a logic program are in a certain sense the minimal Herbrand models of its "reduct" programs. Here we introduce a new semantics for logic programs that is di erent from the known xed point semantics. In our approach, logic programs are expressed as CNF formulas (sets of clauses) of a propositional logic for which we de ne a notion of extension. We prove in this semantics, that each consistent CNF formula admits at least an extension and for each given stable model of a logic program there exists an extension of its corresponding CNF formula which logically entails it. On the other hand, we show that some of the extensions do not entail any stable model, in this case, we de ne a simple descrimination condition which allows to recognize such extensions. These extensions could be very important, but are not captured by the stable models semantics. Our approach, extends the stable model semantics in this sense. Following the new semantics, we give a full characterization of the stable models of a logic program by means of the extensions of its CNF encoding verifying a simple condition, and provide a procedure which can be used to compute such extensions from which we deduce the stable models of the given logic program.La programmation par ensembles r éponses (Answer Set Programming) est un cadre bien étudi é en programmation logique. Plusieurs travaux ont été faits pour d éfinir une s émantique pour les programmes logiques. La plupart de ces s émantiques sont en fait des s émantiques de point fi xe. L'id ée principale est le calcul de mod èles canoniques du programme logique consid ér é, appel és mod èles stables. Les mod èles stables sont dans un certain sens des mod èles minimaux des programmes r éduits. Nous introduisons une nouvelle s emantique pour les programmes logiques, à partir d'une notion d'extension d'une formule propositionnelle classique. Ces extensions peuvent être calcul és de mani ère it érative. Un programme logique est alors cod é par un ensemble de clauses de la logique propositionnelle. On prouve que chaque formule consistante admet au moins une extension et que, pour chaque mod èle stable d'un programme logique, il existe une extension de son codage qui l'implique logiquement. Certaines des extensions ne correspondent pas à un mod èle stable mais sont int eréssantes. Nous donnons une condition discriminante simple qui permet de reconnaitre de telles extensions. En fin, nous d écrivons un algorithme qui calcule les extensions de la formule CNF codant le programme logique. De cet ensemble d'extension on peut extraire les mod èles stables du programme logique initial

Linear-Time Temporal Answer Set Programming

[Abstract]: In this survey, we present an overview on (Modal) Temporal Logic Programming in view of its application to Knowledge Representation and Declarative Problem Solving. The syntax of this extension of logic programs is the result of combining usual rules with temporal modal operators, as in Linear-time Temporal Logic (LTL). In the paper, we focus on the main recent results of the non-monotonic formalism called Temporal Equilibrium Logic (TEL) that is defined for the full syntax of LTL but involves a model selection criterion based on Equilibrium Logic, a well known logical characterization of Answer Set Programming (ASP). As a result, we obtain a proper extension of the stable models semantics for the general case of temporal formulas in the syntax of LTL. We recall the basic definitions for TEL and its monotonic basis, the temporal logic of Here-and-There (THT), and study the differences between finite and infinite trace length. We also provide further useful results, such as the translation into other formalisms like Quantified Equilibrium Logic and Second-order LTL, and some techniques for computing temporal stable models based on automata constructions. In the remainder of the paper, we focus on practical aspects, defining a syntactic fragment called (modal) temporal logic programs closer to ASP, and explaining how this has been exploited in the construction of the solver telingo, a temporal extension of the well-known ASP solver clingo that uses its incremental solving capabilities.Xunta de Galicia; ED431B 2019/03We are thankful to the anonymous reviewers for their thorough work and their useful suggestions that have helped to improve the paper. A special thanks goes to Mirosaw Truszczy´nski for his support in improving the quality of our paper. We are especially grateful to David Pearce, whose help and collaboration on Equilibrium Logic was the seed for a great part of the current paper. This work was partially supported by MICINN, Spain, grant PID2020-116201GB-I00, Xunta de Galicia, Spain (GPC ED431B 2019/03), R´egion Pays de la Loire, France, (projects EL4HC and etoiles montantes CTASP), European Union COST action CA-17124, and DFG grants SCHA 550/11 and 15, Germany

Extremal problems in logic programming and stable model computation

We study the following problem: given a class of logic programs C, determine the maximum number of stable models of a program from C. We establish the maximum for the class of all logic programs with at most n clauses, and for the class of all logic programs of size at most n. We also characterize the programs for which the maxima are attained. We obtain similar results for the class of all disjunctive logic programs with at most n clauses, each of length at most m, and for the class of all disjunctive logic programs of size at most n. Our results on logic programs have direct implication for the design of algorithms to compute stable models. Several such algorithms, similar in spirit to the Davis-Putnam procedure, are described in the paper. Our results imply that there is an algorithm that finds all stable models of a program with n clauses after considering the search space of size O(3^{n/3}) in the worst case. Our results also provide some insights into the question of representability of families of sets as families of stable models of logic programs

On the Equivalence between Logic Programming Semantics and Argumentation Semantics

Publisher PD

Epistemic Foundation of Stable Model Semantics

Stable model semantics has become a very popular approach for the management of negation in logic programming. This approach relies mainly on the closed world assumption to complete the available knowledge and its formulation has its basis in the so-called Gelfond-Lifschitz transformation. The primary goal of this work is to present an alternative and epistemic-based characterization of stable model semantics, to the Gelfond-Lifschitz transformation. In particular, we show that stable model semantics can be defined entirely as an extension of the Kripke-Kleene semantics. Indeed, we show that the closed world assumption can be seen as an additional source of `falsehood' to be added cumulatively to the Kripke-Kleene semantics. Our approach is purely algebraic and can abstract from the particular formalism of choice as it is based on monotone operators (under the knowledge order) over bilattices only.Comment: 41 pages. To appear in Theory and Practice of Logic Programming (TPLP