Finding Admissible and Preferred Arguments Can be Very Hard

Bondarenko et al. have recently proposed an extension of the argumentation-theoretic semantics of admissible and preferred arguments, originally proposed for logic programming only, to a number of other nonmonotonic reasoning formalisms. In this paper we analyse the computational complexity of credulous and sceptical reasoning under the semantics of admissible and preferred arguments for (the propositional variant of) some well-known frameworks for nonmonotonic reasoning, i.e. Theorist, Circumscription and Autoepistemic Logic. While the new semantics have been assumed to mitigate the computational problems of nonmonotonic reasoning under the standard semantics of stable extensions, we show that in many cases reasoning under the new semantics is computationally harder than under the standard semantics. In particular, for Autoepistemic Logic, the sceptical reasoning problem under the semantics of preferred arguments is located at the fourth level of the polynomial hierarchy, two levels above the same problem under the standard semantics. In some cases, however, reasoning under the new semantics becomes easier – reducing to reasoning in the monotonic logics underlying the nonmonotonic frameworks.