Complexity of Nonempty Existence Problems in Incomplete Argumentation Frameworks

The Abstract argumentation frameworks (AFs) are a prevailing model for the formal representation of argumentations in AI research. The generalized model of incomplete AFs extends the basic model by allowing for the representation of unquantified uncertainty about the existence of elements in an argumentation. For this extended model of AFs, we formally define a natural generalization of the nonempty existence problem, which asks whether there exists a nonempty set of arguments satisfying the conditions specified by a given semantics. Focusing on the fundamental semantics for incomplete AFs and considering a possible and a necessary variant of the nonempty existence problem, this yields a family of related problems, and we provide a full analysis of their computational complexity.