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PSPACE Bounds for Rank 1 Modal Logics

Lutz Schröder; Dirk Pattinson
In: Rajeev Alur (Hrsg.). Twenty-First Annual IEEE Symposium on Logic in Computer Science (LICS 2006). IEEE Symposium on Logic in Computer Science (LICS-06), August 12-15, Seattle, Washington, USA, Pages 231-240, IEEE, 2006.


For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all mboxrank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatization, in PSPACE. This leads not only to a unified derivation of (known) tight PSPACE-bounds for a number of logics including K, coalition logic, and graded modal logic (and to a new algorithm in the latter case), but also to a previously unknown tight PSPACE-bound for probabilistic modal logic, with rational probabilities coded in binary. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

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