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Ultimate Wellfounded and Stable Semantics for Logic Programs With Aggregates (Extended Abstract)
 In Proceedings of ICLP01, LNCS 2237
, 2001
"... is relatively straightforward. Another advantage of the ultimate approximation is that in cases where TP is monotone the ultimate wellfounded model will be 2valued and will coincide with the least fixpoint of TP . This is not the case for the standard wellfounded semantics. For example in the sta ..."
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is relatively straightforward. Another advantage of the ultimate approximation is that in cases where TP is monotone the ultimate wellfounded model will be 2valued and will coincide with the least fixpoint of TP . This is not the case for the standard wellfounded semantics. For example in the standard wellfounded model of the program: # p. p. p is undefined while the associated TP operator is monotone and p is true in the ultimate wellfounded model. One disadvantage of using the ultimate semantics is that it has a higher computational cost even for programs without aggregates. The complexity goes one level higher in the polynomial hierarchy to # 2 for the wellfounded model and to 2 for a stable model which is also complete for this class [2]. Fortunately, by adding aggregates the complexity does not increase further. To give an example of a logic program with aggregates we consider the problem of computing the length of the shortest path between two nodes in a direc
Integrating answer set programming and constraint logic programming
 Annals of Mathematics and Artificial Intelligence
, 2008
"... We introduce a knowledge representation language AC(C) extending the syntax and semantics of ASP and CRProlog, give some examples of its use, and present an algorithm, ACsolver, for computing answer sets of AC(C) programs. The algorithm does not require full grounding of a program and combines “cla ..."
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We introduce a knowledge representation language AC(C) extending the syntax and semantics of ASP and CRProlog, give some examples of its use, and present an algorithm, ACsolver, for computing answer sets of AC(C) programs. The algorithm does not require full grounding of a program and combines “classical” ASP solving methods with constraint logic programming techniques and CRProlog based abduction. The AC(C) based approach often allows to solve problems which are impossible to solve by more traditional ASP solving techniques. We belief that further investigation of the language and development of more efficient and reliable solvers for its programs can help to substantially expand the domain of applicability of the answer set programming paradigm. 1
Answer sets for logic programs with arbitrary abstract constraint atoms
 J. ARTIFICIAL INTELLIGENCE RESEARCH
, 2007
"... In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (catoms). These approaches generalize the fixpointbased and the level mapping based answer set semantics of normal logic programs to the case of logic p ..."
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Cited by 21 (2 self)
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In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (catoms). These approaches generalize the fixpointbased and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of catoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpointbased semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negationasfailure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of catoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly wellsupported models, are generalizations of the notion of wellsupported models of normal logic programs to the case of programs with catoms. As for the case of fixpointbased semantics, the difference between these two definitions is rooted in the treatment of naf atoms. We prove that answer sets by reduct (resp. by complement) are equivalent to weakly (resp. strongly) wellsupported models of a program, thus generalizing the theorem on the correspondence between stable models and wellsupported models of a normal logic program to the class of programs with catoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone catoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with catoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature.
Properties of programs with monotone and convex constraints
 In Proceedings of the 20th National Conference on Artificial Intelligence (AAAI05
, 2005
"... We study properties of programs with monotone and convex constraints. We extend to these formalisms concepts and results from normal logic programming. They include tight programs and Fages Lemma, program completion and loop formulas, and the notions of strong and uniform equivalence with their char ..."
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We study properties of programs with monotone and convex constraints. We extend to these formalisms concepts and results from normal logic programming. They include tight programs and Fages Lemma, program completion and loop formulas, and the notions of strong and uniform equivalence with their characterizations. Our results form an abstract account of properties of some recent extensions of logic programming with aggregates, especially the formalism of smodels.
Logic programs with abstract constraint atoms: the role of computations
 Proceedings of the 23rd International Conference on Logic Programming (ICLP 2007), LNCS, Springer, 2007 (this
, 2005
"... Abstract. We provide new perspectives on the semantics of logic programs with constraints. To this end we introduce several notions of computation and propose to use the results of computations as answer sets of programs with constraints. We discuss the rationale behind different classes of computat ..."
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Abstract. We provide new perspectives on the semantics of logic programs with constraints. To this end we introduce several notions of computation and propose to use the results of computations as answer sets of programs with constraints. We discuss the rationale behind different classes of computations and study the relationships among them and among the corresponding concepts of answer sets. The proposed semantics generalize the answer set semantics for programs with monotone, convex and/or arbitrary constraints described in the literature. 1
Properties and applications of programs with monotone and convex constraints
 J. ARTIFICIAL INTELLIGENCE RESEARCH
, 2006
"... We study properties of programs with monotone and convex constraints. We extend to these formalisms concepts and results from normal logic programming. They include the notions of strong and uniform equivalence with their characterizations, tight programs and Fages Lemma, program completion and loop ..."
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We study properties of programs with monotone and convex constraints. We extend to these formalisms concepts and results from normal logic programming. They include the notions of strong and uniform equivalence with their characterizations, tight programs and Fages Lemma, program completion and loop formulas. Our results provide an abstract account of properties of some recent extensions of logic programming with aggregates, especially the formalism of lparse programs. They imply a method to compute stable models of lparse programs by means of offtheshelf solvers of pseudoboolean constraints, which is often much faster than the smodels system.
Logic programs with monotone abstract constraint atoms
 UNDER CONSIDERATION FOR PUBLICATION IN THEORY AND PRACTICE OF LOGIC PROGRAMMING
, 2006
"... We introduce and study logic programs whose clauses are built out of monotone constraint atoms. We show that the operational concept of the onestep provability operator generalizes to programs with monotone constraint atoms, but the generalization involves nondeterminism. Our main results demonstra ..."
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Cited by 11 (6 self)
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We introduce and study logic programs whose clauses are built out of monotone constraint atoms. We show that the operational concept of the onestep provability operator generalizes to programs with monotone constraint atoms, but the generalization involves nondeterminism. Our main results demonstrate that our formalism is a common generalization of (1) normal logic programming with its semantics of models, supported models and stable models, (2) logic programming with weight atoms (lparse programs) with the semantics of stable models, as defined by Niemelä, Simons and Soininen, and (3) of disjunctive logic programming with the possiblemodel semantics of Sakama and Inoue.
Logic programs with monotone cardinality atoms
 UNDER CONSIDERATION FOR PUBLICATION IN THEORY AND PRACTICE OF LOGIC PROGRAMMING
"... We investigate mcaprograms, that is, logic programs with clauses built of monotone cardinality atoms of the form kX, where k is a nonnegative integer and X is a finite set of propositional atoms. We develop a theory of mcaprograms. We demonstrate that the operational concept of the onestep prova ..."
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Cited by 7 (0 self)
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We investigate mcaprograms, that is, logic programs with clauses built of monotone cardinality atoms of the form kX, where k is a nonnegative integer and X is a finite set of propositional atoms. We develop a theory of mcaprograms. We demonstrate that the operational concept of the onestep provability operator generalizes to mcaprograms, but the generalization involves nondeterminism. Our main results show that the formalism of mcaprograms is a common generalization of (1) normal logic programming with its semantics of models, supported models and stable models, (2) logic programming with cardinality atoms and with the semantics of stable models, as defined by Niemelä, Simons and Soininen, and (3) of disjunctive logic programming with the possiblemodel semantics of Sakama and Inoue.
A Default Approach to Semantics of Logic Programs with Constraint Atoms
"... Abstract. We define the semantics of logic programs with (abstract) constraint atoms in a way closely tied to default logic. Like default logic, formulas in rules are evaluated using the classical entailment relation, so a constraint atom can be represented by an equivalent propositional formula. Th ..."
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Abstract. We define the semantics of logic programs with (abstract) constraint atoms in a way closely tied to default logic. Like default logic, formulas in rules are evaluated using the classical entailment relation, so a constraint atom can be represented by an equivalent propositional formula. Therefore, answer sets are defined in a way closely related to default extensions. The semantics defined this way enjoys two properties generally considered desirable for answer set programming − minimality and derivability. The derivability property is very important because it guarantees free of selfsupporting loops in answer sets. We show that when restricted to basic logic programs, this semantics agrees with the conditionalsatisfaction based semantics. Furthermore, answer sets by the minimalmodel based semantics can be recast in our approach. Consequently, the default approach gives a unifying account of the major existing semantics for logic programs with constraint atoms. This also makes it possible to characterize, in terms of the minimality and derivability properties, the precise relationship between them and contrast with others. 1
A Generalized GelfondLifschitz Transformation for Logic Programs with Abstract Constraints
"... We present a generalized GelfondLifschitz transformation in order to define stable models for a logic program with arbitrary abstract constraints on sets (catoms). The generalization is based on a formal semantics and a novel abstract representation of catoms, as opposed to the commonly used powe ..."
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We present a generalized GelfondLifschitz transformation in order to define stable models for a logic program with arbitrary abstract constraints on sets (catoms). The generalization is based on a formal semantics and a novel abstract representation of catoms, as opposed to the commonly used power set form representation. In many cases, the abstract representation of a catom results in a substantial reduction of size from its power set form representation. We show that any catom A =(Ad,Ac) in the body of a clause can be characterized using its satisfiable sets, so that given an interpretation I the catom can be handled simply by introducing a special atom θA together with a new clause θA ← A1,..., An for each satisfiable set {A1,..., An} of A. We also prove that the latest fixpoint approach presented by Son et al. and our approach using the generalized GelfondLifschitz transformation are semantically equivalent in the sense that they define the same set of stable models.