and G. Schwarz 1 M. Truszczynski 2 Robotics Laboratory, Computer Science Department, Stanford University,and Stanford, CA 943-5-4110, USA 2Computer Science Department, University of Kentucky, Lexington, KY 40506-0046, USA
We establish the complexity of decision problems associated with the nonmonotonic modal logic S4. We prove that the problem of existence of an S4-expansion for a given set
Aof premisses is [Sigma]P 2-complete. Similarly, we show that for a given formula [phi] and a set Aof premisses, it is [Sigma]P 2-complete to decide whether [phi] belongs to at least one S4-expansion for A, and it is IIP P 2-compete to decide whether [phi] belongs to all S4-expansions for A. This refutes a conjecture of Gottlob that these problems are PSPACE-complete. An interesting aspect of these results is that reasoning (testing satisfiability and provability) in the monotonic modal logic S4 is PSPACE-complete. To the best of our knowledge, the nonmonotonic logic S4 is the first example of a nonmonotonic formalism which is computationally easier than the monotonic logic that underlies it (assuming PSPACE does not collapse to [Sigma]P 2). Nonmonotonic logics, expansions, S, S4 F, complexity.
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