A general approach to extension-based semantics in abstract argumentation

Abstract

After a few decades of development, computational argumentation has become one of the active areas in AI. This paper proposes a general approach to deal with semantic related problems of abstract argumentation frameworks (AAFs, for short) and further develops the graded semantics presented by Grossi and Modgil. We pay attention not only to issues of various extension-based semantics at the object level but also the metatheoretical level. 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, universal definability of various extension-based semantics and relationships among semantics are considered. Second, the operator ⋂D 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 for AAFs (including conflict-free, admissible, complete and stable semantics, etc) and some derived semantics are shown to be closed under this operator. Based on these facts, a number of applications of the operator ⋂D are considered, which show this operator can deal with various aspects of argumentation theory. In particular, we provide a simple and uniform method to prove the universal definability of a family of range related semantics. In addition to these results at the object level, this paper also explores ones at the metatheoretical level. To illustrate the universal applicability of the operator ⋂D as a tool for handling semantic related problems, we introduce the first order language FS(Ω) and establish a connection between ultraproducts of models in model theory and reduced meets of extensions modulo ultrafilters. On the basis of this, we characterize the extension-based semantics that is closed under the operator ⋂D in terms of the FS(Ω)-definability, which provides an example of utilizing model theory to explore the theoretical problems of AAFs and brings a series of metatheorems about extension-based semantics. In a word, our work shows that the operator ⋂D can deal with various problems related to various semantics, which suggests that it provides a general method to study AAFs.