Encoding higher-order argumentation frameworks with supports to propositional logic systems

TL;DR

This paper introduces a higher-order argumentation framework with supports (HAFS) that explicitly models complex attack-support interactions and provides logical encodings into propositional logic systems, enhancing formalization and computational integration.

Contribution

It proposes a novel higher-order argumentation framework with supports (HAFS), defines multiple semantics, and develops a normal encoding methodology into propositional logic systems, establishing model equivalence.

Findings

HAFS allows attacks and supports to act as both sources and targets.

Semantic encodings into propositional logic systems are proven model-equivalent.

Transformations between fuzzy and complete semantics models are established.

Abstract

Argumentation frameworks ($AF$s) have been extensively developed, but existing higher-order bipolar $AF$s suffer from critical limitations: attackers and supporters are restricted to arguments, multi-valued and fuzzy semantics lack unified generalization, and encodings often rely on complex logics with poor interoperability. To address these gaps, this paper proposes a higher-order argumentation framework with supports ($HAFS$), which explicitly allows attacks and supports to act as both targets and sources of interactions. We define a suite of semantics for $HAFS$s, including extension-based semantics, adjacent complete labelling semantics (a 3-valued semantics), and numerical equational semantics ([0,1]-valued semantics). Furthermore, we develop a normal encoding methodology to translate $HAFS$s into propositional logic systems ($\mathcal{PLS}$s): $HAFS$s under complete labelling semantics are encoded into {\L}ukasiewicz's three-valued propositional logic ($\mathcal{PL}_3^L$), and those under equational semantics are encoded into fuzzy $\mathcal{PLS}$s ($\mathcal{PL}_{[0,1]}$) such as G\"odel and Product fuzzy logics. We prove model equivalence between $HAFS$s and their encoded logical formulas, establishing the logical foundation of $HAFS$ semantics. Additionally, we investigate the relationships between 3-valued complete semantics and fuzzy equational semantics, showing that models of fuzzy encoded semantics can be transformed into complete semantics models via ternarization, and vice versa for specific t-norms. This work advances the formalization and logical encoding of higher-order bipolar argumentation, enabling seamless integration with lightweight computational solvers and uniform handling of uncertainty.