Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms

In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based 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 negation-as-failure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly well-supported models, are generalizations of the notion of well-supported models of normal logic programs to the case of programs with c-atoms. As for the case of fixpoint-based 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) well-supported models of a program, thus generalizing the theorem on the correspondence between stable models and well-supported models of a normal logic program to the class of programs with c-atoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone c-atoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with c-atoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature

Rewriting recursive aggregates in answer set programming: back to monotonicity

Aggregation functions are widely used in answer set programming for representing and reasoning on knowledge involving sets of objects collectively. Current implementations simplify the structure of programs in order to optimize the overall performance. In particular, aggregates are rewritten into simpler forms known as monotone aggregates. Since the evaluation of normal programs with monotone aggregates is in general on a lower complexity level than the evaluation of normal programs with arbitrary aggregates, any faithful translation function must introduce disjunction in rule heads in some cases. However, no function of this kind is known. The paper closes this gap by introducing a polynomial, faithful, and modular translation for rewriting common aggregation functions into the simpler form accepted by current solvers. A prototype system allows for experimenting with arbitrary recursive aggregates, which are also supported in the recent version 4.5 of the grounder gringo, using the methods presented in this paper

Towards a unified theory of logic programming semantics: Level mapping characterizations of selector generated models

Currently, the variety of expressive extensions and different semantics created for logic programs with negation is diverse and heterogeneous, and there is a lack of comprehensive comparative studies which map out the multitude of perspectives in a uniform way. Most recently, however, new methodologies have been proposed which allow one to derive uniform characterizations of different declarative semantics for logic programs with negation. In this paper, we study the relationship between two of these approaches, namely the level mapping characterizations due to [Hitzler and Wendt 2005], and the selector generated models due to [Schwarz 2004]. We will show that the latter can be captured by means of the former, thereby supporting the claim that level mappings provide a very flexible framework which is applicable to very diversely defined semantics.Comment: 17 page

Ultimate approximations in nonmonotonic knowledge representation systems

We study fixpoints of operators on lattices. To this end we introduce the notion of an approximation of an operator. We order approximations by means of a precision ordering. We show that each lattice operator O has a unique most precise or ultimate approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We apply our theory to logic programming and introduce the ultimate Kripke-Kleene, well-founded and stable semantics. We show that the ultimate Kripke-Kleene and well-founded semantics are more precise then their standard counterparts We argue that ultimate semantics for logic programming have attractive epistemological properties and that, while in general they are computationally more complex than the standard semantics, for many classes of theories, their complexity is no worse.Comment: This paper was published in Principles of Knowledge Representation and Reasoning, Proceedings of the Eighth International Conference (KR2002

Stable Models of Formulas with Generalized Quantifiers (Preliminary Report)

Applications of answer set programming motivated various extensions of the stable model semantics, for instance, to allow aggregates or to facilitate interface with external ontology descriptions. We present a uniform, reductive view on these extensions by viewing them as special cases of formulas with generalized quantifiers. This is done by extending the first-order stable model semantics by Ferraris, Lee and Lifschitz to account for generalized quantifiers and then by reducing the individual extensions to this formalism

Stable Models of Formulas with Generalized Quantifiers (Preliminary Report)

Applications of answer set programming motivated various extensions of the stable model semantics, for instance, to allow aggregates or to facilitate interface with external ontology descriptions. We present a uniform, reductive view on these extensions by viewing them as special cases of formulas with generalized quantifiers. This is done by extending the first-order stable model semantics by Ferraris, Lee and Lifschitz to account for generalized quantifiers and then by reducing the individual extensions to this formalism