Extension-Based Semantics for Incomplete Argumentation Frameworks

Incomplete Argumentation Frameworks (IAFs) have been defined to incorporate some qualitative uncertainty in abstract argumentation: information such as "I am not sure whether this argument exists" or "I am not sure whether this argument attacks that one" can be expressed. Reasoning with IAFs is classically based on a set of completions, i.e. standard argumentation frameworks that represent the possible worlds encoded in the IAF. The number of these completions may be exponential with respect to the number of arguments in the IAF. This leads, in some cases, to an increase of the complexity of reasoning, compared to the complexity of standard AFs. In this paper, we follow an approach that was initiated for Partial AFs (a subclass of IAFs), which consists in defining new forms of conflict-freeness and defense, the properties that underly the definition of Dung's semantics for AFs. We generalize these semantics from PAFs to IAFs. We show that, among three possible types of admissibility, only two of them satisfy some desirable properties. We use them to define two new families of extension-based semantics. We study the properties of these semantics, and in particular we show that their complexity remains the same as in the case of Dung's AFs. Finally, we propose a logical encoding of these semantics, that paves the way to the development of SAT-based solvers for reasoning with our new semantics for IAFs.