On the Existence of Semi-Stable Extensions

In this paper, we describe an open problem in abstract argumentation theory: the precise conditions under which semi-stable extensions exist. Although each finite argumentation framework can be shown to have at least one semi-stable extension, this is no longer the case when infinite argumentation frameworks are considered. This puts semi-stable semantics between stable and preferred semantics. Where stable semantics does not warrant the existence of extensions (even for finite argumentation frameworks) and preferred semantics always warrants the existence of extensions (even for infinite argumentation frameworks), semistable semantics warrants the existence of extensions only for finite argumentation frameworks, but not for infinite argumentation frameworks. We illustrate this with a counter-example of the latter. The question is then studied if, even for infinite argumentation frameworks, one can identify specific conditions under which semi-stable extensions do exist.