On the preferred extensions of argumentation frameworks: bijections with naive sets
This work addresses computational challenges in formal argumentation theory, offering incremental improvements in algorithm design for specific classes of frameworks.
The paper tackles the problem of finding preferred extensions in argumentation frameworks by establishing bijections with naive sets, showing that recognizing naive-bijective frameworks is hard but tractable for bounded in-degree cases, and proving that identifying admissible-closed frameworks is coNP-complete, while enabling polynomial-time listing for frameworks with certain lattice properties.
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.