Technical Note : Complexity of Stage Semantics in Argumentation Frameworks

In this work, we answer two questions about the complexity of semi-stable semantics for abstract argumentation frameworks raised by Du nne and Caminada (2008): we show Π2 -completeness for the problem of deciding whether an argume nt is skeptically accepted, and respectively, Σ2 -completeness for the problem of deciding whether an argument is credulously accepted under the semi-stable seman tics. Furthermore, we extend these complexity bounds to the according decision problems for tage semantics as introduced by Verheij (1997). We also discuss two approaches towa rds tractability: first, we prove that the problems under consideration are fixed-param eter tractable with respect to tree-width; second, we show that the problems remain intrac table when considering frameworks of bounded cycle-rank. 1Institute for Information Systems 184/2, Technische Unive rsität Wien, Favoritenstrasse 9-11, 1040 Vienna, Austria. E-mail: dvorak@dbai.tuwien.ac.at 2Institute for Information Systems 184/2, Technische Unive rsität Wien, Favoritenstrasse 9-11, 1040 Vienna, Austria. E-mail: woltran@dbai.tuwien.ac.at Copyright c © 2009 by the authors