Maxi-Consistent Operators in Argumentation

This paper studies an instantiation of Dung-style argumentation system with classical propositional logic. Our goal is to explore the link between the result obtained by using argumentation to deal with an inconsistent knowledge base and the result obtained by using maximal consistent subsets of the same knowledge base. Namely, for a given attack relation and semantics, we study the question: does every extension of the argumentation system correspond to exactly one maximal consistent subset of the knowledge base? We study the class of attack relations which satisfy that condition. We show that such a relation must be conflict-dependent, must not be valid, must not be conflict-complete, must not be symmetric etc. Then, we show that some attack relations serve as lower or upper bounds with respect to the condition we study (e.g. we show that if an attack relation contains “canonical undercut” then it does not satisfy this condition). By using our results, we show for each attack relation and each semantics whether or not they satisfy the aforementioned condition. Finally, we interpret our results and discuss more general questions, like does (and when) this link is a desirable property. This work will help us obtain our long-term goal, which is to better understand the role of argumentation and, more particularly, the expressivity of logic-based instantiations of Dung-style argumentation frameworks.