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_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.
The above example shows that our characterization of programs with aggregates differs from the proposal in @cite_41 . Apart from the lack of support for aggregates in the heads of rules, the semantics of @cite_41 might accept answer sets that are not -answer sets. Observe that the two semantical characterizations coincide for large classes of programs (e.g., for programs that have only monotone aggregates).