The Complexity Boundary of Answer Set Programming with Generalized Atoms under the FLP Semantics

In recent years, Answer Set Programming (ASP), logic programming under the stable model or answer set semantics, has seen several extensions by generalizing the notion of an atom in these programs: be it aggregate atoms, HEX atoms, generalized quantifiers, or abstract constraints, the idea is to have more complicated satisfaction patterns in the lattice of Herbrand interpretations than traditional, simple atoms. In this paper we refer to any of these constructs as generalized atoms. It is known that programs with generalized atoms that have monotonic or antimonotonic satisfaction patterns do not increase complexity with respect to programs with simple atoms (if satisfaction of the generalized atoms can be decided in polynomial time) under most semantics. It is also known that generalized atoms that are nonmonotonic (being neither monotonic nor antimonotonic) can, but need not, increase the complexity by one level in the polynomial hierarchy if non-disjunctive programs under the FLP semantics are considered. In this paper we provide the precise boundary of this complexity gap: programs with convex generalized atom never increase complexity, while allowing a single non-convex generalized atom (under reasonable conditions) always does. We also discuss several implications of this result in practice.