Revisiting the notions of extension and acceptance over incomplete abstract argumentation frameworks

We revisit the completion-based reasoning over incomplete Abstract Argumentation Frameworks (iAAFs), whose core is the notion of i-extension under the possible or necessary perspective, i.e. a set of arguments that, in some collective way, prove coherent and justified in at least one or in every completion, respectively. In particular, we show that, under the possible perspective, i-extensions exhibit a counterintuitive behavior, as they do not ensure conflict-freeness. Thus, we introduce the alternative notion of i*-extension, that not only fixes this behavior, but has also a positive computational impact: under various semantics of extensions, moving from i- to i*- extensions makes the complexity of the verification problem under the possible perspective move from NP-complete to P. Starting from this, we revisit the notion of accepted argument and define it as an argument belonging to at least one or every i*-extension, depending on whether the credulous or skeptical perspective is adopted. We show that this new definition does not in general coincide with the one in the literature (that is not based on the notion of i- or i*- extension), and can provide the analyst with a different perspective for reasoning over the justification of arguments in the presence of incompleteness. In this regard, we have thoroughly investigated the differences of the revisited acceptance problem with the acceptance problem in the literature, also in terms of computational complexity.