Encoding argumentation frameworks with set attackers to propositional logic systems
Shuai Tang, Jiachao Wu, Ning Zhou
TL;DR
This paper tackles the lack of a unified encoding for advanced argumentation frameworks by introducing AFSA, a general framework that supports higher-order attacks and sets of attackers. It builds HLAF, BHAF, SETAF, and the unified HSAF into AFSA, provides complete semantics and three numerical equational semantics, and develops encodings into $PL_3^L$ and $PL_{[0,1]}$ to establish model-equivalence results. A formal equational approach and Brouwer fixed-point arguments guarantee solution existence for the numerical semantics, and semantic transformations show how to translate between AFSA families while preserving semantics. The work lays a solid logical foundation for automated reasoning tools in AI, decision support, and multi-agent systems, enabling robust handling of complex attack structures in uncertain or graded settings.
Abstract
Argumentation frameworks ($AF$s) have been a useful tool for approximate reasoning. The encoding method is an important approach to formally model $AF$s under related semantics. The aim of this paper is to develop the encoding method from classical Dung's $AF$s ($DAF$s) to $AF$s with set attackers ($AFSA$s) including higher-level argumentation frames ($HLAF$s), Barringer's higher-order $AF$s ($BHAF$s), frameworks with sets of attacking arguments ($SETAF$s) and higher-order set $AF$s ($HSAF$s). Regarding syntactic structures, we propose the $HSAF$s where the target of an attack is either an argument or an attack and the sources are sets of arguments and attacks. Regarding semantics, we translate $HLAF$s and $SETAF$s under respective complete semantics to Łukasiewicz's 3-valued propositional logic system ($PL_3^L$). Furthermore, we propose complete semantics of $BHAF$s and $HSAF$s by respectively generalizing from $HLAF$s and $SETAF$s, and then translate to the $PL_3^L$. Moreover, for numerical semantics of $AFSA$s, we propose the equational semantics and translate to fuzzy propositional logic systems ($PL_{[0,1]}$s). This paper establishes relationships of model equivalence between an $AFSA$ under a given semantics and the encoded formula in a related propositional logic system ($PLS$). By connections of $AFSA$s and $PLS$s, this paper provides the logical foundations for $AFSA$s associated with complete semantics and equational semantics. The results advance the argumentation theory by unifying $HOAF$s and $SETAF$s under logical formalisms, paving the way for automated reasoning tools in AI, decision support, and multi-agent systems.