In the recently proposed SCC-recursive approach to argumentation semantics, the strongly connected components of an argumentation framework are used as the basic elements for the incremental construction of extensions. In this paper we argue that a finer decomposition, considering some suitably defined internal substructures of strongly connected components, called autonomous fragments, may be appropriate and support, in some cases, more intuitive results than the original approach. We cast this proposal within the SCC-recursive framework, show that it satisfies some fundamental requirements and provide some examples of its potential advantages.
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