1. Argumentation, by considering arguments and their interactions, is a way of reasoning that has proven successful in many contexts, for instance, in multi-agent applications (Carrera & Iglesias,...

We revisit the notion of initial sets by Xu and Cayrol 2016), i. e., non-empty minimal admissible sets in abstract argumentation frameworks. Initial sets are a simple concept for analysing conflicts in an abstract argumentation framework and to explain why certain arguments can be accepted. We contribute with new insights on the structure of initial sets and devise a simple non-deterministic construction principle for any admissible set, based on iterative selection of initial sets of the original framework and its induced (...) reducts. In particular, we characterise many existing admissibility-based semantics via this construction principle, thus providing a constructive explanation on the structure of extensions. We also investigate certain problems related to initial sets with respect to their computational complexity. (shrink)