On the Characteristic Functions in Listing Stable Arguments

An abstract argumentation framework (af) is viewed as a directed graph such that graph vertices represent abstract arguments while graph edges denote attacks between these arguments. We say that a set, S, of arguments, are conflict-free if and only if for every \((x,y) \in S \times S\) , x does not attack y. A set, S, of arguments of a given af, is called a stable extension in af if S is conflict-free and such that every argument outside S is attacked by an argument inside S. To the best of our knowledge, a thorough mathematical analysis of the truth of what so-called characteristic functions (which are an essential component in generating all stable extensions of a given af) is not previously addressed in the literature. We fill this gap; we rigorously analyze the verity of characteristic functions employed in listing all stable extensions in a given af.