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Abstract: The postulate of relevance provides a suitable and general notion of minimal change for belief contraction. Relevance is tightly connected to smooth kernel contractions when an agent’s epistemic state is represented as a logically closed set of formulae. This connection, however, breaks down when an agent’s epistemic state is represented as a set of formulae not necessarily logically closed. We investigate the cause behind this schism, and we reconnect relevance with smooth kernel contractions by constraining the behaviour of their choice mechanisms and epistemic preference relations. Our first representation theorem connects smooth kernel contractions with a novel class of epistemic preference relations. For our second representation theorem, we introduce the principle of symmetry of removal that relates relevance to epistemic choices. For the last theorem, we devise a novel class of smooth kernel contractions, that satisfy relevance, which are based on epistemic preference relations that capture the principle of symmetry of removal.

Abstract: We propose a general framework to investigate semantics of Dung-style argumentation frameworks (AFs) by means of generic defeat operators. After establishing the technical foundations, we propose natural generic versions of Dung’s classical semantics. We demonstrate how classical as well as recent proposals can be captured by our approach when utilizing suitable notions of defeat. We perform an investigation of basic properties which semantics inherit from the underlying defeat operator. In particular, we show under which conditions a counterpart to Dung’s fundamental lemma can be inferred and how it ensures the existence of the generalized version of complete extensions. We contribute to a principle-based study of AF semantics by discussing properties tailored to compare different defeat operators. Finally, we report computational complexity results for basic reasoning tasks which hold in our general framework.