Abstract Argumentation Frameworks (AAFs) and Normal Logic Programs (NLPs) are closely related formalisms for which many equivalences have already been elicited. In this paper, we extend this line of research by considering Bipolar Argumentation Frameworks (BAFs), in which arguments have an explicit support relation, independent of the attack relation. We provide direct translations from BAFs to NLPs (and vice versa) in a one-to-one correspondence between several argumentation and 3-valued logic programming semantics. This includes the equivalence involving L-stable semantics. Besides, we deepen the connection between NLPs and BAFs by finding subsets of them for which the proposed translations are each other’s inverse up to isomorphism.

Abstract Bipolar Argumentation Frameworks ($\textit{BAF}$s) extend Dung’s Abstract Argumentation Frameworks ($\textit{AAF}$s) by incorporating an explicit notion of support between arguments. However, there is a price to pay: the semantics for $\textit{BAF}$s often involve more intricate definitions and computational procedures than those for $\textit{AAF}$s. In this paper, we establish a dual relation between defeat and defence. Taking profit from this dual perspective, we define conflict-free sets, acceptability, extension-based and labelling-based semantics as in $\textit{AAF}$s. We also show that our definitions collapse into the corresponding concepts proposed for $\textit{AAF}$s when the support relation is ignored. In particular, we prove the semantics $\beta $-admissible, $\beta $-complete, $\beta $-grounded, $\beta $-preferred, $\beta $-stable and $\beta $-semi-stable defined here for $\textit{BAF}$s are generalisations of the corresponding semantics for $\textit{AAF}$s. Besides generalising $\textit{AAF}$s semantics to $\textit{BAF}$s, our approach also preserves some of their most remarkable results, including Dung’s Fundamental Lemma.

Abstract A framework with sets of attacking arguments ( $\textit{SETAF}$ ) is an extension of the well-known Dung’s Abstract Argumentation Frameworks ( $\mathit{AAF}$ s) that allows joint attacks on arguments. In this paper, we provide a translation from Normal Logic Programs ( $\textit{NLP}$ s) to $\textit{SETAF}$ s and vice versa, from $\textit{SETAF}$ s to $\textit{NLP}$ s. We show that there is pairwise equivalence between their semantics, including the equivalence between $L$ -stable and semi-stable semantics. Furthermore, for a class of $\textit{NLP}$ s called Redundancy-Free Atomic Logic Programs ( $\textit{RFALP}$ s), there is also a structural equivalence as these back-and-forth translations are each other’s inverse. Then, we show that $\textit{RFALP}$ s are as expressive as $\textit{NLP}$ s by transforming any $\textit{NLP}$ into an equivalent $\textit{RFALP}$ through a series of program transformations already known in the literature. We also show that these program transformations are confluent, meaning that every $\textit{NLP}$ will be transformed into a unique $\textit{RFALP}$ . The results presented in this paper enhance our understanding that $\textit{NLP}$ s and $\textit{SETAF}$ s are essentially the same formalism.

Abstract Abstract Dialectical Frameworks (ADFs) are argumentation frameworks where each node is associated with an acceptance condition. This allows us to model different types of dependencies as supports and attacks. Previous studies provided a translation from Normal Logic Programs (NLPs) toADFs and proved the stable models semantics for a normal logic program has an equivalent semantics to that of the correspondingADF. However, these studies failed in identifying a semantics forADFs equivalent to a three-valued semantics (as partial stable models and well-founded models) forNLPs. In this work, we focus on a fragment ofADFs, called Attacking Dialectical Frameworks (ADF+s), and provide a translation fromNLPs toADF+s robust enough to guarantee the equivalence between partial stable models, well-founded models, regular models, stable models semantics forNLPs and respectively complete models, grounded models, preferred models, stable models forADFs. In addition, we define a new semantics forADF+s, calledL-stable, and show it is equivalent to theL-stable semantics forNLPs.