Bipolar Argumentation Frameworks with a dual relation between defeat and defence

Abstract

Abstract Bipolar Argumentation Frameworks ($\textit{BAF}$s) extend Dung’s Abstract Argumentation Frameworks ($\textit{AAF}$s) by incorporating an explicit notion of support between arguments. However, there is a price to pay: the semantics for $\textit{BAF}$s often involve more intricate definitions and computational procedures than those for $\textit{AAF}$s. In this paper, we establish a dual relation between defeat and defence. Taking profit from this dual perspective, we define conflict-free sets, acceptability, extension-based and labelling-based semantics as in $\textit{AAF}$s. We also show that our definitions collapse into the corresponding concepts proposed for $\textit{AAF}$s when the support relation is ignored. In particular, we prove the semantics $\beta $-admissible, $\beta $-complete, $\beta $-grounded, $\beta $-preferred, $\beta $-stable and $\beta $-semi-stable defined here for $\textit{BAF}$s are generalisations of the corresponding semantics for $\textit{AAF}$s. Besides generalising $\textit{AAF}$s semantics to $\textit{BAF}$s, our approach also preserves some of their most remarkable results, including Dung’s Fundamental Lemma.