On the Complexity of Enumerating the Extensions of Abstract Argumentation Frameworks

Several computational problems of abstract argumentation frameworks (AFs) such as skeptical and credulous reasoning, existence of a non-empty extension, verification, etc. have been thoroughly analyzed for various semantics. In contrast, the enumeration problem of AFs (i.e., the problem of computing all extensions according to some semantics) has been left unexplored so far. The goal of this paper is to fill this gap. We thus investigate the enumeration complexity of AFs for a large collection of semantics and, in addition, consider the most common structural restrictions on AFs.

Trichotomy and dichotomy results on the complexity of reasoning with disjunctive logic programs

We present trichotomy results characterizing the complexity of reasoning with disjunctive logic programs. To this end, we introduce a certain definition schema for classes of programs based on a set of allowed arities of rules. We show that each such class of programs has a finite representation, and for each of the classes definable in the schema, we characterize the complexity of the existence of an answer set problem. Next, we derive similar characterizations of the complexity of skeptical and credulous reasoning with disjunctive logic programs. Such results are of potential interest. On the one hand, they reveal some reasons responsible for the hardness of computing answer sets. On the other hand, they identify classes of problem instances, for which the problem is “easy” (in P) or “easier than in general” (in NP). We obtain similar results for the complexity of reasoning with disjunctive programs under the supported-model semantics.