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    Products of 'transitive' modal logics

    Gabelaia, D. and Kurucz, A. and Wolter, F. and Zakharyaschev, Michael (2005) Products of 'transitive' modal logics. The journal of Symbolic Logic 70 (3), pp. 993-1021. ISSN 0022-4812.

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    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.


    Item Type: Article
    School: Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences
    Depositing User: Sarah Hall
    Date Deposited: 25 Oct 2021 17:59
    Last Modified: 09 Aug 2023 12:52


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