More on extension-based semantics of argumentation

TL;DR

This paper advances the theory of extension-based semantics in argumentation by generalizing key lemmas, correcting previous flaws, and introducing new operators to analyze infinite argumentation frameworks.

Contribution

It provides a generalized fundamental lemma, corrects prior work, and introduces the reduced meet operator for infinite frameworks, enhancing the structural understanding of argumentation semantics.

Findings

Generalized the fundamental lemma for graded semantics.

Corrected flaws in Grossi and Modgil's previous work.

Introduced the reduced meet operator and proved its properties.

Abstract

After a few decades of development, computational argumentation has become one of the active realms in AI. This paper considers extension-based concrete and abstract semantics of argumentation. For concrete ones, based on Grossi and Modgil's recent work, this paper considers some issues on graded extension-based semantics of abstract argumentation framework (AAF, for short). First, an alternative fundamental lemma is given, which generalizes the corresponding result due to Grossi and Modgil by relaxing the constraint on parameters. This lemma provides a new sufficient condition for preserving conflict-freeness and brings a Galois adjunction between admissible sets and complete extensions, which is of vital importance in constructing some special extensions in terms of iterations of the defense function. Applying such a lemma, some flaws in Grossi and Modgil's work are corrected, and the structural property and universal definability of various extension-based semantics are given. Second, an operator so-called reduced meet modulo an ultrafilter is presented, which is a simple but powerful tool in exploring infinite AAFs. The neutrality function and the defense function, which play central roles in Dung's abstract argumentation theory, are shown to be distributive over reduced meets modulo any ultrafilter. A variety of fundamental semantics of AAFs, including conflict-free, admissible, complete and stable semantics, etc, are shown to be closed under this operator. Based on this fact, a number of applications of such operators are considered. In particular, we provide a simple and uniform method to prove the universal definability of a family of range related semantics. Since all graded concrete semantics considered in this paper are generalizations of corresponding non-graded ones, all results about them obtained in this paper also hold in the traditional situation.