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Journal of Logic and Computation, Volume 11, Issue 6, pp. 909-931: Abstract.

On the Products of Linear Modal Logics

Mark Reynolds1, and Michael Zakharyaschev2

1School of Information Technology, Murdoch University, Perth, Western Australia. E-mail: m.reynolds@murdoch.edu.au
2Department of Computer Science, King's College London, Strand, London WC2R 2LS, UK. E-mail: mz@dcs.kcl.ac.uk

We study two-dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4.3, S4.3, GL.3, Grz.3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems posed by Gabbay and Shehtman. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatization for the square K4.3 × K4.3 of the minimal liner logic using non-structural Gabbay-type inference rules.

Keywords: Modal logic, Kripke frame, Cartesian product, decidability

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