14 found
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  1.  37
    Topological completeness of the provability logic GLP.Lev Beklemishev & David Gabelaia - 2013 - Annals of Pure and Applied Logic 164 (12):1201-1223.
    Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.
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  2.  65
    Some Results on Modal Axiomatization and Definability for Topological Spaces.Guram Bezhanishvili, Leo Esakia & David Gabelaia - 2005 - Studia Logica 81 (3):325-355.
    We consider two topological interpretations of the modal diamond—as the closure operator (C-semantics) and as the derived set operator (d-semantics). We call the logics arising from these interpretations C-logics and d-logics, respectively. We axiomatize a number of subclasses of the class of nodec spaces with respect to both semantics, and characterize exactly which of these classes are modally definable. It is demonstrated that the d-semantics is more expressive than the C-semantics. In particular, we show that the d-logics of the six (...)
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  3.  34
    Modal logics of metric spaces.Guram Bezhanishvili, David Gabelaia & Joel Lucero-Bryan - 2015 - Review of Symbolic Logic 8 (1):178-191.
  4.  14
    Modal languages for topology: Expressivity and definability.Balder ten Cate, David Gabelaia & Dmitry Sustretov - 2009 - Annals of Pure and Applied Logic 159 (1-2):146-170.
    In this paper we study the expressive power and definability for modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt–Thomason definability theorem in terms of the well-established first-order topological language.
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  5.  34
    Products of 'transitive' modal logics.David Gabelaia, Agi Kurucz, Frank Wolter & Michael Zakharyaschev - 2005 - Journal of Symbolic Logic 70 (3):993-1021.
    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.
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  6.  12
    Non-primitive recursive decidability of products of modal logics with expanding domains.David Gabelaia, Agi Kurucz, Frank Wolter & Michael Zakharyaschev - 2006 - Annals of Pure and Applied Logic 142 (1):245-268.
    We show that—unlike products of ‘transitive’ modal logics which are usually undecidable—their ‘expanding domain’ relativisations can be decidable, though not in primitive recursive time. In particular, we prove the decidability and the finite expanding product model property of bimodal logics interpreted in two-dimensional structures where one component—call it the ‘flow of time’—is • a finite linear order or a finite transitive tree and the other is composed of structures like • transitive trees/partial orders/quasi-orders/linear orders or only finite such structures expanding (...)
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  7.  15
    A negative solution of Kuznetsov’s problem for varieties of bi-Heyting algebras.Guram Bezhanishvili, David Gabelaia & Mamuka Jibladze - 2022 - Journal of Mathematical Logic 22 (3).
    Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. In this paper, we show that there exist (continuum many) varieties of bi-Heyting algebras that are not generated by their complete members. It follows that there exist (continuum many) extensions of the Heyting–Brouwer logic [math] that are topologically incomplete. This result provides further insight into the long-standing open problem of Kuznetsov by yielding a negative solution of the reformulation of the problem from extensions of [math] to extensions of [math].
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  8. Polyhedral Completeness of Intermediate Logics: The Nerve Criterion.Sam Adam-day, Nick Bezhanishvili, David Gabelaia & Vincenzo Marra - forthcoming - Journal of Symbolic Logic:1-41.
    We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted inn-dimensional polyhedra. An intermediate logic ispolyhedrally completeif it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial characterisation of polyhedrally (...)
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  9.  13
    Products of ‘transitive” modal logics.David Gabelaia, Agi Kurucz, Frank Wolter & Michael Zakharyaschev - 2005 - Journal of Symbolic Logic 70 (3):993-1021.
    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 asK4,S4,S4.1,K4.3,GL, orGrzare 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, ifl1andl2are classes of transitive frames such that their depth cannot be (...)
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  10.  25
    Connected modal logics.Guram Bezhanishvili & David Gabelaia - 2011 - Archive for Mathematical Logic 50 (3-4):287-317.
    We introduce the concept of a connected logic (over S4) and show that each connected logic with the finite model property is the logic of a subalgebra of the closure algebra of all subsets of the real line R, thus generalizing the McKinsey-Tarski theorem. As a consequence, we obtain that each intermediate logic with the finite model property is the logic of a subalgebra of the Heyting algebra of all open subsets of R.
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  11.  20
    Topological completeness of logics above s4.Guram Bezhanishvili, David Gabelaia & Joel Lucero-Bryan - 2015 - Journal of Symbolic Logic 80 (2):520-566.
  12.  11
    Temporal logic of surjective bounded morphisms between finite linear processes.David Gabelaia, Evgeny Kuznetsov, Radu Casian Mihailescu, Konstantine Razmadze & Levan Uridia - 2023 - Journal of Applied Non-Classical Logics:1-30.
    In this paper, we study temporal logic for finite linear structures and surjective bounded morphisms between them. We give a characterisation of such structures by modal formulas and show that every pair of linear structures with a bounded morphism between them can be uniquely characterised by a temporal formula up to an isomorphism. As the main result, we prove Kripke completeness of the logic with respect to the class of finite linear structures with bounded morphisms between them.
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  13.  19
    Admissible Bases Via Stable Canonical Rules.Nick Bezhanishvili, David Gabelaia, Silvio Ghilardi & Mamuka Jibladze - 2016 - Studia Logica 104 (2):317-341.
    We establish the dichotomy property for stable canonical multi-conclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics.
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  14. Lecture Notes on Artificial Intelligence 5422, Logic, Language, and Computation 7th International Tbilisi Symposium on Logic, Language, and Computation.Peter Bosch, David Gabelaia & Jérôme Lang (eds.) - 2009 - Springer.