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_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.
This lemma allows us to prove the following relationship between @math and @math . Proof. The result is a trivial consequence of the fact that @math and @math has the same set of partial stable models as @math @cite_18 . @math