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 by 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. The proposed views of logic programming with aggregates are simple and coincide with the ultimate stable model semantics, and with other semantic characterizations for large classes of program (e.g., programs with monotone aggregates and programs that are aggregate-stratified). Moreover, it can be directly employed to support an implementation using available answer set solvers. The paper describes a system, called ASP^A, that is capable of computing answer sets of programs with arbitrary (e.g., recursively defined) aggregates.
@cite_38: Equilibrium logic, introduced by David Pearce, extends the concept of an answer set from logic programs to arbitrary sets of formulas. Logic programs correspond to the special case in which every formula is a “rule” — an implication that has no implications in the antecedent (body) and consequent (head). The semantics of equilibrium logic looks very different from the usual definitions of an answer set in logic programming, as it is based on Kripke models. In this paper we propose a new definition of equilibrium logic which uses the concept of a reduct, as in the standard definition of an answer set. Second, we apply the generalized concept of an answer set to the problem of defining the semantics of aggregates in answer set programming. We propose, in particular, a semantics for weight constraints that covers the problematic case of negative weights. Our semantics of aggregates is an extension of the approach due to Faber, Leone, and Pfeifer to a language with choice rules and, more generally, arbitrary rules with nested expressions.
@cite_18: Dept. of Computer Science, K.U.LeuvenCelestijnenlaan 200A, B-3001 Heverlee, BelgiumE-mail: pelov,marcd,maurice @cs.kuleuven.ac.beAbstract. We deﬁne a translation of aggregate programs to normal lo-gic programs which preserves the set of partial stable models. We thendeﬁne the classes of deﬁnite and stratiﬁed aggregate programs and showthat the translation of such programs are, respectively, deﬁnite and strat-iﬁed logic programs. Consequently these two classes of programs havea single partial stable model which is two-valued and is also the well-founded model. Our deﬁnition of stratiﬁcation is more general than theexisting one and covers a strictly larger class of programs.
@cite_22: We introduce a family of partial stable model semantics for logic programs with arbitrary aggregate relations. The semantics are parametrized by the interpretation of aggregate relations in three-valued logic. Any semantics in this family satisfies two important properties: (i) it extends the partial stable semantics for normal logic programs and (ii) total stable models are always minimal. We also give a specific instance of the semantics and show that it has several attractive features.
@cite_41: 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.
@cite_36: The Smodels system is one of the state-of-the-art implementations of stable model computation for normal logic programs. In order to enable more realistic applications, the basic modeling language of normal programs has been extended with new constructs including cardinality and weight constraints and corresponding implementation techniques have been developed. This paper summarizes the extensions that have been included in the system, demonstrates their use, provides basic application methodology, illustrates the current level of performance of the system, and compares it to state-of-the-art satisfiability checkers.
@cite_21: 
@cite_15: 
@cite_34: 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 ∩ X ∈ 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.
@cite_17: 
In this section, we relate our definition of -answer sets to several formulations of aggregates proposed in the literature. We begin with a comparison of our unfolding approach with the two most recently proposed semantics for LP with aggregates, i.e., the @cite_22 @cite_18 @cite_17 , the @cite_41 @cite_38 , and the semantics for abstract constraint atoms @cite_34 . We then relate our work to earlier proposals, such as perfect models of aggregate-stratified programs (e.g., @cite_21 ), the fixpoint answer set semantics of aggregate-monotonic programs @cite_15 , and the semantics of programs with weight constraints @cite_36 . Finally, we briefly discuss the relation of -answer sets to other proposals.