The Complexity of Reasoning for Fragments of Default Logic

Authors:
Olaf Beyersdorff;Arne Meier;Michael Thomas;Heribert Vollmer
Affiliations:
Institut für Theoretische Informatik, Gottfried Wilhelm Leibniz Universität, Hannover, Germany 30167;Institut für Theoretische Informatik, Gottfried Wilhelm Leibniz Universität, Hannover, Germany 30167;Institut für Theoretische Informatik, Gottfried Wilhelm Leibniz Universität, Hannover, Germany 30167;Institut für Theoretische Informatik, Gottfried Wilhelm Leibniz Universität, Hannover, Germany 30167
Venue:
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
Year:
2009

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Abstract

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as $\Sigma^{\rm P}_2$-complete, and the complexity of the credulous and skeptical reasoning problem as $\Sigma^{\rm P}_2$-complete, resp. $\Pi^{\rm P}_2$-complete. Additionally, he investigated restrictions on the default rules, i. e., semi-normal default rules. Selman made in 1992 a similar approach with disjunction-free and unary default rules. In this paper we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a trichotomy ($\Sigma^{\rm P}_2$-, NP-complete, trivial) for the extension existence problem, whereas for the credulous and sceptical reasoning problem we get a finer classification down to NL-complete cases.