Ordering Argumentation Frameworks

Abstract

This paper introduces two orderings over abstract argumentation frameworks to compare justification status under argumentation semantics. Given two argumentation frameworks \(AF_1\) and \(AF_2\) and an argumentation semantics \(\sigma \), \(AF_2\) is more \(\sharp \)-general than (or equal to) \(AF_1\) (written \(AF_1\sqsubseteq _{\sigma }^{\sharp } AF_2\)) if for any \(\sigma \)-extension F of \(AF_2\) there is a \(\sigma \)-extension E of \(AF_1\) such that \(E\subseteq F\). In contrast, \(AF_2\) is more \(\flat \)-general than (or equal to) \(AF_1\) (written \(AF_1\sqsubseteq _{\sigma }^{\flat } AF_2\)) if for any \(\sigma \)-extension E of \(AF_1\) there is a \(\sigma \)-extension F of \(AF_2\) such that \(E\subseteq F\). We show that if \(AF_1\sqsubseteq _{\sigma }^{\sharp } AF_2\) then \(AF_2\) skeptically accepts arguments more than \(AF_1\) (under the \(\sigma \)-semantics) while if \(AF_1\sqsubseteq _{\sigma }^{\flat } AF_2\) then \(AF_2\) credulously accepts arguments more than \(AF_1\). Mathematically, these orders constitute pre-order sets over the set of all argumentation frameworks. Next we consider comparing two AFs under dynamic environments by observing the effect of incorporating new information into given AFs. We introduce two orderings in such dynamic environments and show its connection to strong equivalence between argumentation frameworks.