On graded semantics of abstract argumentation: Extension-based case

Based on Grossi and Modgil's recent work [1], this paper considers some issues on extension-based semantics for abstract argumentation framework (AAF, for short). First, an alternative fundamental lemma is given, which generalizes the corresponding result obtained in [1]. This lemma plays a central role in constructing some special extensions in terms of iterations of the defense function. Applying this lemma, some flaws in [1] are corrected and a number of structural properties of various extension-based semantics are given. Second, the operator so-called reduced meet modulo an ultrafilter is presented. A number of fundamental semantics for AAF, including conflict-free, admissible, complete and stable semantics, are shown to be closed under this operator. Based on this fact, we provide a concise and uniform proof method to establish the universal definability of a family of range related semantics. Thirdly, using model-theoretical tools, we characterize the class of extension-based semantics that is closed under reduced meet modulo any ultrafilter, which brings us a metatheorem concerning the universal definability of range related semantics. Finally, in addition to range related semantics, some graded variants of traditional semantics of AAF are also considered in this paper, e.g., ideal semantics, eager semantics, etc.