Encoding Argumentation Frameworks to Propositional Logic Systems

Abstract

This paper generalizes the encoding of argumentation frameworks beyond the classical 2-valued propositional logic system ($P{L}_2$) to 3-valued propositional logic systems ($P{L}_3$s) and fuzzy propositional logic systems ($P{L}_{[0,1]}s$), employing two key encodings: normal encoding ($e{c}_1$) and regular encoding ($e{c}_2$). Specifically, via $e{c}_1$ and $e{c}_2$, we establish model relationships between Dung's classical semantics (stable and complete semantics) and the encoded semantics associated with Kleene's $P{L}_3$ and Łukasiewicz's $P{L}_3$. Through $e{c}_1$, we also explore connections between Gabbay's real equational semantics and the encoded semantics of $P{L}_{[0,1]}s$, including showing that Gabbay's $E{q}_{\text{max}}^R$ and $E{q}_{\text{inverse}}^R$ correspond to the fuzzy encoded semantics of $P{L}_{[0,1]}^G$ and $P{L}_{[0,1]}^P$ respectively. Additionally, we propose a new fuzzy encoded semantics ($E{q}^L$) associated with Łukasiewicz's $P{L}_{[0,1]}$ and investigate interactions between complete semantics and fuzzy encoded semantics. This work strengthens the links between argumentation frameworks and propositional logic systems, providing a framework for constructing new argumentation semantics.

37 pages