Lattice point of view for argumentation framework

The main purpose of this article is to develop a lattice point of view for the study of argumentation framework extensions. We first characterize self-defending sets of an argumentation framework by the closed sets of an implicational system that can be computed in polynomial time from the argumentation framework. On the other hand, for any implicational system \(\Sigma \) over the set of arguments, we associate an argumentation framework whose admissible sets are in bijection with closed sets of \(\Sigma \). Second, we propose conflict-closed sets reduction rules, based on implicational system, to find out minimal subsets of vertex cover closed while maintaining all potential admissible extensions as well as preferred extensions. This leads to a polynomial delay and space algorithm to enumerate admissible sets of argumentation frameworks without even cycles. Finally, based on the implicational system, a new decomposition of the argumentation framework is defined and leads to a polynomial delay and space algorithm to enumerate admissible sets for a bipartite argumentation framework. The proposed algorithm improves the exponential space complexity of previous algorithms.