Argumentation frameworks (AFs) have been a useful tool for approximate reasoning. The encoding method is an important approach to formally model AFs under related semantics. The aim of this paper is to develop the encoding method from classical Dung's AFs (DAFs) to AFs with set attackers (AFSAs) including higher-level argumentation frames (HLAFs), Barringer's higher-order AFs (BHAFs), frameworks with sets of attacking arguments (SETAFs) and higher-order set AFs (HSAFs). Regarding syntactic structures, we propose the HSAFs 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 HLAFs and SETAFs under respective complete semantics to Łukasiewicz's 3-valued propositional logic system (PL₃L). Furthermore, we propose complete semantics of BHAFs and HSAFs by respectively generalizing from HLAFs and SETAFs, and then translate to the PL₃L. Moreover, for numerical semantics of AFSAs, we propose the equational semantics and translate to fuzzy propositional logic systems (PL₀, ₁s). 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 AFSAs and PLSs, this paper provides the logical foundations for AFSAs associated with complete semantics and equational semantics. The results advance the argumentation theory by unifying HOAFs and SETAFs under logical formalisms, paving the way for automated reasoning tools in AI, decision support, and multi-agent systems.
Tang et al. (Fri,) studied this question.
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