More on extension-based semantics of argumentation
  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.
