On the preferred extensions of argumentation frameworks: bijections with naive sets

TL;DR

This paper explores bijections between preferred extensions and naive sets in argumentation frameworks, providing new methods for their identification and listing, especially under certain structural conditions.

Contribution

It introduces novel bijections linking preferred extensions to naive sets, and offers tractable algorithms for specific classes of frameworks with lattice properties.

Findings

Naive-bijective frameworks have equal naive sets and preferred extensions.

Recognizing naive-bijective frameworks is tractable with bounded in-degree.

Preferred extensions can be listed efficiently for frameworks with lattice properties.

Abstract

This paper deals with the problem of finding the preferred extensions of an argumentation framework by means of a bijection with the naive sets of another framework. First, we consider the case where an argumentation framework is naive-bijective: its naive sets and preferred extensions are equal. Recognizing naive-bijective argumentation frameworks is hard, but we show that it is tractable for frameworks with bounded in-degree. Next, we give a bijection between the preferred extensions of an argumentation framework being admissible-closed (the intersection of two admissible sets is admissible) and the naive sets of another framework on the same set of arguments. On the other hand, we prove that identifying admissible-closed argumentation frameworks is coNP-complete. At last, we introduce the notion of irreducible self-defending sets as those that are not the union of others. It turns out there exists a bijection between the preferred extensions of an argumentation framework and the naive sets of a framework on its irreducible self-defending sets. Consequently, the preferred extensions of argumentation frameworks with some lattice properties can be listed with polynomial delay and polynomial space.