On Undisputed Sets in Abstract Argumentation

We introduce the notion of an undisputed set for abstract argumentation frameworks, which is a conflict-free set of arguments, such that its reduct contains no non-empty admissible sets. We show that undisputed sets, and the stronger notion of strongly undisputed sets, provide a meaningful approach to weaken admissibility and deal with the problem of attacks from self-attacking arguments, in a similar manner as the recently introduced notion of weak admissibility. We investigate the properties of our new semantical notions and show certain relationships to classical semantics, in particular that undisputed sets are a generalisation of preferred extensions and strongly undisputed sets are a generalisation of stable extensions. We also investigate the computational complexity of standard reasoning tasks with these new notions and show that they lie on the second and third level of the polynomial hierarchy, respectively.