A New Logical Characterisation of Stable Models and Answer Sets
05 Sep 1996-pp 57-70
TL;DR: This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics with equilibrium logic, and shows that on logic programs equilibrium logic coincides with the inference operation associated with the stable model and answer set semantics of Gelfond and Lifschitz.
Abstract: This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics. We define a nonmonotonic logic, called equilibrium logic, based on the least constructive extension, N2, of the intermediate logic of “here-and-there”. We show that on logic programs equilibrium logic coincides with the inference operation associated with the stable model and answer set semantics of Gelfond and Lifschitz. We thereby obtain a very simple characterisation of answer set semantics as a form of minimal model reasoning in N2, while equilibrium logic itself provides a natural generalisation of this semantics to arbitrary theories. We discuss briefly some consequences and applications of this result.
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TL;DR: The main theorem shows that the verification of strong equivalence can be accomplished by cheching the equivalence of formulas in a monotonic logic, called the logic of here-and-there, which is intermediate between classical logic and intuitionistic logic.
Abstract: A logic program Π1 is said to be equivalent to a logic program Π2 in the sense of the answer set semantics if Π1 and Π2 have the same answer sets. We are interested in the following stronger condition: for every logic program, Π, Π1, ∪ Π has the same answer sets as Π2 ∪ Π. The study of strong equivalence is important, because we learn from it how one can simplify a part of a logic program without looking at the rest of it. The main theorem shows that the verification of strong equivalence can be accomplished by cheching the equivalence of formulas in a monotonic logic, called the logic of here-and-there, which is intermediate between classical logic and intuitionistic logic.
597 citations
Book•
19 Dec 2012TL;DR: This book presents a practical introduction to ASP, aiming at using ASP languages and systems for solving application problems, and introduces ASP's solving technology, modeling language and methodology.
Abstract: Answer Set Programming (ASP) is a declarative problem solving approach, initially tailored to modeling problems in the area of Knowledge Representation and Reasoning (KRR). More recently, its attractive combination of a rich yet simple modeling language with high-performance solving capacities has sparked interest in many other areas even beyond KRR. This book presents a practical introduction to ASP, aiming at using ASP languages and systems for solving application problems. Starting from the essential formal foundations, it introduces ASP's solving technology, modeling language and methodology, while illustrating the overall solving process by practical examples. Table of Contents: List of Figures / List of Tables / Motivation / Introduction / Basic modeling / Grounding / Characterizations / Solving / Systems / Advanced modeling / Conclusions
503 citations
TL;DR: An approach to computing answer sets of logic programs, based on concepts successfully applied in Satisfiability (SAT) checking, to view inferences in Answer Set Programming (ASP) as unit propagation on nogoods, and presents the first full-fledged algorithmic framework for native conflict-driven ASP solving.
376 citations
Cites background from "A New Logical Characterisation of S..."
...programs admit language extensions like classical negation and disjunctions [68], nested expressions [89], propositional formulas [106,41], cardinality and weight constraints [119], or aggregates [39], to name some of them....
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TL;DR: The study of equivalent transformations of programs with nested expressions shows that any such program is equivalent to a set of disjunctive rules, possibly with negation as failure in the heads.
Abstract: We extend the answer set semantics to a class of logic programs with nested expressions permitted in the bodies and heads of rules. These expressions are formed from literals using negation as failure, conjunction (,) and disjunction (s) that can be nested arbitrarily. Conditional expressions are introduced as abbreviations. The study of equivalent transformations of programs with nested expressions shows that any such program is equivalent to a set of disjunctive rules, possibly with negation as failure in the heads. The generalized answer set semantics is related to the Lloyd–Topor generalization of Clark’s completion and to the logic of minimal belief and negation as failure.
361 citations
Cites background from "A New Logical Characterisation of S..."
...We are grateful to Esra Erdem, Michael Gelfond, Katsumi Inoue, David Pearce, David S. Warren and Phoebe Weidmann, and to anonymous referees, for useful comments on the subject of this paper....
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...Several authors, including Lin and Shoham [16], Lifschitz [12], Herre and Wagner [8] and Pearce [20], described ways to embed logic programs under the answer set semantics into languages that are closed under the formation rules of propositional logic....
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Proceedings Article•
13 Jul 2008TL;DR: Answer set programming (ASP) is a form of declarative programming oriented towards difficult search problems, particularly useful in knowledge-intensive applications.
Abstract: Answer set programming (ASP) is a form of declarative programming oriented towards difficult search problems. As an outgrowth of research on the use of nonmonotonic reasoning in knowledge representation, it is particularly useful in knowledge-intensive applications. ASP programs consist of rules that look like Prolog rules, but the computational mechanisms used in ASP are different: they are based on the ideas that have led to the creation of fast satisfiability solvers for propositional logic.
357 citations
References
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Proceedings Article•
01 Jan 1988
TL;DR: This paper introduces a succinct abstract representation of constraint atoms in which a constraint atom is represented compactly and shows that this representation provides a means to characterize dependencies of atoms in a program with constraint atoms, so that some standard characterizations and properties relying on these dependencies in the past for logic programs with ordinary atoms can be extended.
3,539 citations
TL;DR: It is shown that some facts of commonsense knowledge can be represented by logic programs and disjunctive databases more easily when classical negation is available.
Abstract: An important limitation of traditional logic programming as a knowledge representation tool, in comparison with classical logic, is that logic programming does not allow us to deal directly with incomplete information. In order to overcome this limitation, we extend the class of general logic programs by including classical negation, in addition to negation-as-failure. The semantics of such extended programs is based on the method of stable models. The concept of a disjunctive database can be extended in a similar way. We show that some facts of commonsense knowledge can be represented by logic programs and disjunctive databases more easily when classical negation is available. Computationally, classical negation can be eliminated from extended programs by a simple preprocessor. Extended programs are identical to a special case of default theories in the sense of Reiter.
2,451 citations
Book•
01 Jan 1930
454 citations
"A New Logical Characterisation of S..." refers background in this paper
...I was rst presented by Heyting in his [13] and later appeared in a paper by G odel [11]....
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01 Jan 1988
TL;DR: A non-woven fabric adapted for use in footwear is impregnated with a polymer prepared by polymerizing acrylic acid or methacrylic acid, mixtures thereof, or mixtures with other vinyl monomers in the presence of polyethylene oxide to improve its water absorption characteristics.
Abstract: A non-woven fabric adapted for use in footwear is impregnated with a polymer prepared by polymerizing acrylic acid or methacrylic acid, mixtures thereof, or mixtures with other vinyl monomers in the presence of polyethylene oxide to improve its water absorption characteristics.
409 citations