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On modal logics between K × K × K and S5 × S5 × S5

Published online by Cambridge University Press:  12 March 2014

R. Hirsch ,
I. Hodkinson  and
A. Kurucz
R. Hirsch
Affiliation:
Department of Computer Science, University College, Gower Street, London WC1E 6BT, UK, E-mail: [email protected]
I. Hodkinson
Affiliation:
Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK, E-mail: [email protected]
A. Kurucz
Affiliation:
Department of Computer Science, King's College, Strand, London WC2R 2LS, UK, E-mail: [email protected]
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Abstract

We prove that every n-modal logic between Kn and S5n is undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite relation algebras.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 221 - 234
Copyright
Copyright © Association for Symbolic Logic 2002

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