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

Mohammed Elaroussi (1) , Lhouari Nourine (2, 3) , Mohammed Said Radjef (1) , Simon Vilmin (4, 5)

1 Unité de recherche LaMOS = Research unit LaMOS - Unité de recherche Laboratoire de Modélisation et Optimisation des Systèmes [Université de Béjaïa]
2 LIMOS - Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes
3 UCA - Université Clermont Auvergne
4 LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
5 BD - Base de Données

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.

Mots clés: Argumentation system; preferred extension

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Mohammed Elaroussi, Lhouari Nourine, Mohammed Said Radjef, Simon Vilmin. On the preferred extensions of argumentation frameworks: Bijections with naive sets. Information Processing Letters, 2023, 181, pp.106354. ⟨10.1016/j.ipl.2022.106354⟩. ⟨hal-03565645⟩