Abstract. The addition of aggregates has been one of the most relevant enhancements to the language of answer set programming (ASP). They strengthen the modeling power of ASP, in terms of concise problem representations. While many important problems can be encoded using nonrecursive aggregates, some relevant examples lend themselves for the use of recursive aggregates. Previous semantic definitions typically agree in the nonrecursive case, but the picture is less clear for recursion. Some proposals explicitly avoid recursive aggregates, most others differ, and many of them do not satisfy desirable criteria, such as minimality or coincidence with answer sets in the aggregate-free case. In this paper we define a semantics for disjunctive programs with arbitrary aggregates (including monotone, antimonotone, and nonmonotone aggregates). This semantics is a fully declarative, genuine generalization of the answer set semantics for disjunctive logic programming (DLP). It is defined by a natural variant of the Gelfond-Lifschitz transformation, and treats aggregate and non-aggregate literals in a uniform way. We prove that our semantics guarantees the minimality (and therefore the incomparability) of answer sets, and demonstrate that it coincides with the standard answer set semantics on aggregate-free programs. Finally we analyze the computational complexity of this language, paying particular attention to the impact of syntactical restrictions on programs. 1

We investigate the properties of logic programs with aggregates. We mainly focus on programs with monotone and antimonotone aggregates (LP A m,a programs). We define a new notion of unfounded set for LP A m,a programs, and prove that it is a sound generalization of the standard notion of unfounded set for aggregate-free programs. We show that the answer sets of an LP A m,a program are precisely its unfounded-free models. We define a well-founded operator WP for LP A m,a programs; we prove that its total fixpoints are pre-cisely the answer sets of P, and its least fixpoint Wω P (∅) is contained in the intersection of all answer sets (if P admits an answer set). W ω P (∅) is

We propose and study extensions of logic programming with constraints represented as generalized atoms of the form C(X), where X is a finite set of atoms and C is an abstract constraint (formally, a collection of sets of atoms). Atoms C(X) are satisfied by an interpretation (set of atoms) M , if M C. We focus here on monotone constraints, that is, those collections C that are closed under the superset. They include, in particular, weight (or pseudo-boolean) constraints studied both by the logic programming and SAT communities. We show that key concepts of the theory of normal logic programs such as the one-step provability operator, the semantics of supported and stable models, as well as several of their properties including complexity results, can be lifted to such case.

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.

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. 1

We show that the concepts of strong and uniform equivalence of logic programs can be generalized to an abstract algebraic setting of operators on complete lattices. Our results imply characterizations of strong and uniform equivalence for several nonmonotonic logics including logic programming with aggregates, default logic and a version of autoepistemic logic. 1

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 off-the-shelf solvers of pseudo-boolean constraints, which is often much faster than the smodels system. 1

The paper presents two equivalent definitions of answer sets for logic programs with aggregates. These definitions build on the notion of unfolding of aggregates, and they are aimed at creating methodologies to translate logic programs with aggregates to normal logic programs or positive programs, whose answer set semantics can be used to defined the semantics of the original programs. The first definition provides an alternative view of the semantics for logic programming with aggregates described in [32,34]. In particular, the unfolding employed by the first definition in this paper coincides with the translation of programs with aggregates into normal logic programs described in [33]. This indicates that the approach proposed in this paper captures the same meaning as the semantics discussed in [32,34]. The second definition is similar to the traditional answer set definition for normal logic programs, in that, given a logic program with aggregates and an interpretation, the unfolding process produces a positive program. The paper shows how this definition can be extended to consider aggregates in the head of the rules.

In this paper, we present a translational semantics for normal logic programs with aggregates. We propose two different translations of logic programs with aggregates into normal logic programs, whose answer set semantics is used to defined the semantics of the original programs. Differently from many of the earlier proposals in this area, our semantics does not impose any syntactic restrictions on the aggregates and the programs. The semantics naturally extends the traditional answer set semantics for normal logic programs, and it subsumes many of the previous proposals in this area, yet it overcomes several drawbacks of those proposals, e.g., by disallowing non-minimal answer sets. We also discuss how the proposed approach can be extended to logic programs with aggregates in the head, a class of programs that has rarely been considered. The new semantics is natural and intuitive, and it can be directly implemented using available answer set solvers. We describe a system, called ASP A, that is capable of computing answer sets of programs with arbitrary (possibly recursively defined) aggregates. We also present a preliminary comparison of ASP A with another system for computing answer sets of programs with aggregates, DLV A. 1

We show that the concept of strong equivalence of logic programs can be generalized to an abstract algebraic setting of operators on complete lattices. Our results imply characterizations of strong equivalence for several nonmonotonic logics including logic programming with aggregates, default logic and a version of autoepistemic logic.