Recently, stage and cf2 semantics for abstract argumentation attracted specific attention. By distancing from the notion of defence, they are capable to select arguments out of odd-length cycles. In case of cf2 semantics, the SCC-recursive schema guarantees that important evaluation criteria for argumentation semantics, like directionality, weak- and CF-reinstatement, are fulfilled. Beside several desirable properties, both stage and cf2 semantics still have some drawbacks. The stage semantics does not satisfy the above mentioned evaluation criteria, whereas cf2 semantics produces some questionable results on frameworks with cycles of length ≥ 6. Therefore, we suggest to combine stage semantics with the SCC-recursive schema of cf2 semantics. The resulting stage2 semantics overcomes the problems regarding cf2 and stage semantics. We study properties of stage2 semantics and its relations to existing semantics, show that it fulfills the mentioned evaluation criteria, study strong equivalence for stage2 semantics and provide a comprehensive complexity analysis of the associated reasoning problems. Besides the analysis of stage2 semantics, we also complement existing complexity results for cf2 by an analysis of tractable fragments and fixed parameter tractability. Furthermore, we provide answer-set programming (ASP) encodings for stage2 semantics and labelling-based algorithms for cf2 and stage2 semantics.

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