On the Maximal Number of Complete Extensions in Abstract Argumentation Frameworks

Abstract argumentation frameworks are by now a major research area in knowledge representation and reasoning.

Various aspects of AFs have been extensively studied over the last 25 years.

Contributing to understanding the expressive power of AFs, researchers found lower and upper bounds for the maximal number of extensions, that is, acceptable points of view, in AFs.

One of the classical and most important concepts in AFs are so-called complete extensions.

Surprisingly, the exact bound for the maximal number of complete extensions in an AF has not yet been formally established, although there is a reasonable conjecture tracing back at least to 2015.

Recently the notion of modularization was introduced and it was shown that this concept plays a key role for the understanding of relations between semantics as well as intrinsic properties.

In this paper, we will use this property to give a formal proof of the conjecture regarding complete semantics.