Revisiting initial sets in abstract argumentation

Abstract

We revisit the notion of initial sets by Xu and Cayrol (In Proceedings of the 1st Chinese Conference on Logic and Argumentation (CLAR’16) 2016), i. e., non-empty minimal admissible sets in abstract argumentation frameworks. Initial sets are a simple concept for analysing conflicts in an abstract argumentation framework and to explain why certain arguments can be accepted. We contribute with new insights on the structure of initial sets and devise a simple non-deterministic construction principle for any admissible set, based on iterative selection of initial sets of the original framework and its induced reducts. In particular, we characterise many existing admissibility-based semantics via this construction principle, thus providing a constructive explanation on the structure of extensions. We also investigate certain problems related to initial sets with respect to their computational complexity.

Serialisable Semantics for Abstract Argumentation

“…Initial sets are not supposed to be used to solve the whole argumentation represented in an argumentation framework, but rather a single atomic conflict within the framework. We can also differentiate between three types of initial sets [6].…”

“…In [6] the notion of serialisability has been introduced as a new approach for constructing admissible sets (and extensions of a variety of semantics) iteratively using initial sets. This approach relies also on the notion of the reduct [8].…”

“…In [6] it has been shown that many of the mainstream extension-based semantics can be serialised, meaning that there is non-deterministic construction principle that allows to iteratively construct extensions by selecting minimal non-empty admissible setscalled initial sets [7]-and moving to the reduct [8]. Individual semantics can be distinguished by the way they select initial sets (via a so-called selection function) and how they terminate the construction (via a so-called termination function).…”

“…Individual semantics can be distinguished by the way they select initial sets (via a so-called selection function) and how they terminate the construction (via a so-called termination function). For example, preferred semantics can be serialised by selecting initial sets arbitrarily until no further initial sets can be found [6]. Satisfaction of this principle of serialisability by a semantics allows a deeper inspection of the reasons why certain arguments are contained in an extension and therefore facilitates the explanatory power of an argumentation semantics [9,10].…”

“…This parametrisation of a semantics allows the easy development of further semantics simply by defining those two components. In [6], a specific candidate for such a semantics has already been suggested, which we coin here the unchallenged semantics. This semantics is defined by exhaustively adding unattacked and unchallenged initial sets (the formal definitions of these terms will be introduced in Section 3) and it has some interesting connections to preferred and ideal semantics.…”