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Abstract

Abstract argumentation concerns the construction and evaluation of arguments according to their interactions. In Dung’s abstract argumentation frameworks (AAFs), arguments interact negatively via an attack relation. Since then, a plethora of extensions have been proposed with the aim of expressing other common argument relationships. Two such proposals are bipolar argumentation frameworks (BAFs), in which both attack (negative) and support (positive) relations coexist; and frameworks with sets of attacking arguments (SETAFs), where attacks can be collective, originating from a set of arguments. In this paper, we show equivalences between a specific notion of support ( $β$ -semantics) and of joint attacks in argumentation, by providing direct translations from BAFs and SETAFs (and vice versa) in a one-to-one correspondence between (BAF) $β$ -complete and (SETAF) complete labellings, $β$ -grounded and grounded labellings, $β$ -preferred and preferred labellings, $β$ -stable and stable labellings, and $β$ -semi-stable and semi-stable labellings. Besides semantic equivalences, we show structural (or syntactic) equivalences between BAFs and SETAFs by finding subsets of them for which the proposed translations are each other’s inverse up to isomorphism.

Keywords

bipolar argumentation framework with sets of attacking arguments (SETAF)intertranslatability semantic equivalence syntactic equivalence

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On the equivalence between bipolar argumentation frameworks and frameworks with sets of attacking arguments (SETAFs)

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