Extension-based semantics for incomplete argumentation frameworks: properties, complexity and algorithms

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 (AFs) 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 Argumentation Frameworks (PAFs) (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, and we show experimentally that this encoding can be used efficiently to reason with IAFs, thanks to the power of modern SAT solvers.