Ultimate Well-Founded and Stable Semantics for Logic Programs with Aggregates

Denecker, Marc; Pelov, Nikolay; Bruynooghe, Maurice

doi:10.1007/3-540-45635-x_22

Cited by 44 publications

(54 citation statements)

References 13 publications

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“…For non-disjunctive programs, the semantics defined this way coincides with the one based on conditional satisfaction (Son et al 2006;, and with the ones equivalent to it (Denecker et al 2001). …”

Section: Introductionmentioning

confidence: 87%

“…In recent years, researchers have paid particular attention to extensions of ASP with means to model aggregate constraints in particular, and constraints on sets in general (Calimeri et al 2005;Denecker et al 2001;Elkabani et al 2004;Elkabani et al 2005;Faber et al 2004;Ferraris 2005;Liu et al 2007;Liu and Truszczynski 2005;Liu and Truszczynski 2006;Marek et al 2008; element (set inclusive) in 2 D \ {∅} is a singleton in 2 D . In this case, any minimal element B in 2 D and D form a pair with B being the bottom element and D being the top.…”

Section: Introductionmentioning

confidence: 99%

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Characterizations of stable model semantics for logic programs with arbitrary constraint atoms

Shen¹,

You²,

Yuan³

2009

Theory and Practice of Logic Programming

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This paper studies the stable model semantics of logic programs with (abstract) constraint atoms and their properties. We introduce a succinct abstract representation of these constraint atoms in which a constraint atom is represented compactly. We show two applications. First, under this representation of constraint atoms, we generalize the Gelfond-Lifschitz transformation and apply it to define stable models (also called answer sets) for logic programs with arbitrary constraint atoms. The resulting semantics turns out to coincide with the one defined by Son et al., which is based on a fixpoint approach. One advantage of our approach is that it can be applied, in a natural way, to define stable models for disjunctive logic programs with constraint atoms, which may appear in the disjunctive head as well as in the body of a rule. As a result, our approach to the stable model semantics for logic programs with constraint atoms generalizes a number of previous approaches. Second, we show that our abstract representation of constraint atoms 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 to logic programs with constraint atoms.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Characterizations of stable model semantics for logic programs with arbitrary constraint atoms

Shen¹,

You²,

Yuan³

2009

Theory and Practice of Logic Programming

View full text Add to dashboard Cite

show abstract

“…Aggregate functions currently supported by the DLV system are: #count (number of terms), #sum (sum of non-negative integers), #times (product of positive integers), #min (minimum term), #max (maximum term) 2 .…”

Section: Examplementioning

confidence: 99%

“…For the interpretation I = {p(2, 2)}, each extending total interpretation contains either p(2, 1) or not p(2, 1). Therefore, either I(S) = [2] or I(S) = [1,2] and the application of #sum yields either 2 > 1 or 3 > 1, hence A is true w.r.t. I.…”

Section: Examplementioning

confidence: 99%

“…The introduction of aggregate atoms [1][2][3][4][5][6][7][8] is one of the major linguistic extensions to Answer Set Programming of the recent years. While both semantic and computational properties of standard (aggregate-free) logic programs have been deeply investigated, relatively few works have focused on logic programs with aggregates; some of their semantic properties and their computational features are still far from being fully clarified.…”

Section: Introductionmentioning

confidence: 99%

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Unfounded Sets for Disjunctive Logic Programs with Arbitrary Aggregates

Faber

2005

Logic Programming and Nonmonotonic Reasoning

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Abstract. Aggregates in answer set programming (ASP) have recently been studied quite intensively. The main focus of previous work has been on defining suitable semantics for programs with arbitrary, potentially recursive aggregates. By now, these efforts appear to have converged. On another line of research, the relation between unfounded sets and (aggregate-free) answer sets has lately been rediscovered. It turned out that most of the currently available answer set solvers rely on this or closely related results (e.g., loop formulas). In this paper, we unite these lines and give a new definition of unfounded sets for disjunctive logic programs with arbitrary, possibly recursive aggregates. While being syntactically somewhat different, we can show that this definition properly generalizes all main notions of unfounded sets that have previously been defined for fragments of the language. We demonstrate that, as for restricted languages, answer sets can be crisply characterized by unfounded sets: They are precisely the unfounded-free models. This result can be seen as a confirmation of the robustness of the definition of answer sets for arbitrary aggregates. We also provide a comprehensive complexity analysis for unfounded sets, and study its impact on answer set computation.

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Answer Set Programming

Faber

Leone

Ricca

2008

Wiley Encyclopedia of Computer Science and Engineering

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Answer set programming (ASP) evolved from logic programming, deductive databases, knowledge representation, and nonmonotonic reasoning, and it serves as a flexible language for solving problems in a declarative way. The user does not need to provide an algorithm for solving the problem; rather, it is sufficient that (s)he specifies the properties of the desired solution for its computation. ASP is highly expressive and allows us to solve problems even beyond the complexity class NP. The main representation feature of ASP are rules, which are interpreted according to common sense principles. In ASP, one writes a program, that is, a collection of rules, which represent a problem to be solved. This program, together with some input, usually expressed using factual rules, possesses a collection of models (possibly also no model), called answer sets, which correspond to the solutions of the modeled problem. Efficient systems for computing answer sets are available and have stimulated the development of applications relying on ASP, in both academia and, business. The field of ASP is growing, and several extensions of the basic language have been proposed and used in applications.

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Ultimate Well-Founded and Stable Semantics for Logic Programs with Aggregates

Cited by 44 publications

References 13 publications

Characterizations of stable model semantics for logic programs with arbitrary constraint atoms

Characterizations of stable model semantics for logic programs with arbitrary constraint atoms

Unfounded Sets for Disjunctive Logic Programs with Arbitrary Aggregates

Answer Set Programming

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