Abstract The acceptability of attacks plays a key role in defining semantics of higher-order argumentation frameworks (HO-AF). Several approaches have been put forward to describe the recursive acceptability of attacks. The diversity, on the one hand, leads to the variety of HO-AF semantics. However, on the other hand, it also brings difficulties in the formal examination of the relationship among these distinct semantics and the exploration of new semantics. This work aims at exploring the underlying commons in existing HO-AF semantics and examining the role of such commons in the study of HO-AF semantics. To this end, we introduce novel notion of renovation sets to represent the acceptability of attacks, and identify several types of renovation sets for different kinds of acceptability. We then use renovation sets to represent and compare HO-AF semantics built in four different approaches. The proposal of renovation sets provides a general tool for the study of HO-AFs. The representation demonstrates a method of using renovation sets to establish HO-AF semantics. The comparison shows the feasibility of investigating the relationship between different HO-AF semantics with the help of renovation sets.

Abstract As a useful reasoning model, higher-order argumentation frameworks (HO-AF) have been studied in many works. This paper contributes to the theory of HO-AFs in three aspects. First, the acceptability of attacks depending on a set of argument is explored, and credulously accepted attacks w.r.t. a set of arguments are formally characterised by introducing regular renovation sets. Second, a new semantics, named regular semantics (r-semantics) is established for HO-AFs. Every complete (or conflict-free, admissible) set in Modgil’s semantics is complete (or conflict-free, admissible) in r-semantics. Some maximality-based r-semantics are just the argument part of the corresponding semantics of Baroni et al. Third, an efficient method is put forward to calculate these maximality-based Baroni’s semantics through r-semantics.

In this paper we combine fuzzy set theory and argumentation to facilitate the use of fuzzy arguments and attacks. Unlike many existing approaches, our work does not require the use of any parameters, bringing it closer to Dung's work in spirit. We begin by introducing Fuzzy Argumentation Frameworks, and specialise them using the Gödel t-norm. We then examine this framework's properties and show that the standard Dung extensions are obtained, though the stable semantics coincide with the preferred. Finally, we examine the relationship between our framework and Dung's original system, as well as the existing fuzzy frameworks, describing where they overlap and differ.