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Products of ‘transitive” modal logics

Published online by Cambridge University Press:  12 March 2014

D. Gabelaia ,
A. Kurucz ,
F. Wolter  and
M. Zakharyaschev
D. Gabelaia
Affiliation:
Department of Computer Science, King's College London, Strand, London WC2R 2LS, UKE-mail:, [email protected]
A. Kurucz
Affiliation:
Department of Computer Science, King's College London, Strand, London WC2R 2LS, UKE-mail:, [email protected]
F. Wolter
Affiliation:
Department of Computer Science, King's College London, Strand, London WC2R 2LS, UKE-mail:, [email protected]
M. Zakharyaschev
Affiliation:
Department of Computer Science, University of Liverpool, Liverpool L69 7ZF, UKE-mail:, [email protected]
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Abstract

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4, K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if l1 and l2 are classes of transitive frames such that their depth cannot be bounded by any fixed n < ω, then the logic of the class {5ℑ1 × ℑ2 ∣ ℑ1 ∈ l1, ℑ2, ∈ l2} is undecidable. (On the contrary, the product of, say, K4 and the logic of all transitive Kripke frames of depth ≤ n, for some fixed n < ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π11-complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005 , pp. 993 - 1021
Copyright
Copyright © Association for Symbolic Logic 2005

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