Recursion in Abstract Argumentation is Hard - On the Complexity of Semantics Based on Weak Admissibility

We study the computational complexity of abstract argumentation semantics based on  weak admissibility, a recently introduced concept to deal with arguments of self-defeating  nature. Our results reveal that semantics based on weak admissibility are of much higher  complexity (under typical assumptions) compared to all argumentation semantics which  have been analysed in terms of complexity so far. In fact, we show PSPACE-completeness  of all non-trivial standard decision problems for weak-admissible based semantics. We then  investigate potential tractable fragments and show that restricting the frameworks under  consideration to certain graph-classes significantly reduces the complexity. We also show  that weak-admissibility based extensions can be computed by dividing the given graph into  its strongly connected components (SCCs). This technique ensures that the bottleneck  when computing extensions is the size of the largest SCC instead of the size of the graph  itself and therefore contributes to the search for fixed-parameter tractable implementations  for reasoning with weak admissibility.