Stratified Labelings for Abstract Argumentation

argumentation frameworks [Dun95] take a very simple view on argumentation as they do not presuppose any internal structure of an argument. Abstract argumentation frameworks only consider the interactions of arguments by means of an attack relation between arguments. Definition 1 (Abstract Argumentation Framework). An abstract argumentation framework AF is a tuple AF = (Arg,→) where Arg is a set of arguments and → is a relation →⊆ Arg × Arg. For two arguments A,B ∈ Arg the relation A → B means that argument A attacks argument B. Abstract argumentation frameworks can be concisely represented by directed graphs, where arguments are represented as nodes and edges model the attack relation. Example 1. Consider the abstract argumentation framework AF = (Arg,→) depicted in Figure 1. Here it is Arg = {A1,A2,A3,A4,A5} and →= {(A1,A2), (A2,A1), (A2,A3), (A3,A4), (A4,A5), (A5,A4), (A5,A3)}. Semantics are usually given to abstract argumentation frameworks by means of extensions [Dun95] or labelings [WC10]. An extension E of an argumentation framework AF = (Arg,→) is a set of arguments E ⊆ Arg that gives some coherent view on the argumentation underlying