Unfounded Sets for Disjunctive Logic Programs with Arbitrary Aggregates

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.