arXiv subjects

Nick Bezhanishvili

Publications and source records attributed to Nick Bezhanishvili.

At least 19 recordsLinked to original sources

arXiv2026

Dynamic Polyhedral Logic

We introduce spatio-temporal polyhedral reachability logics, extending dynamic topological logic with polyhedral semantics and a path-based spatial reachability operator. Formulas are interpreted over polyhedra, with admissible valuations ranging over polyhedral subsets; the spatial modality is interpreted as interior, the binary operator $γ(φ,ψ)$ expresses reachability of a $ψ$-point through a $φ$-region, and the temporal modalities are interpreted by a PL-homeomorphism and its inverse. We define h-dynamic reachability spaces and axiomatize the corresponding h-dynamic extensions of the known reachability logics of topological, finite, Alexandroff, and polyhedral spaces. The main result is soundness and completeness for the intended classes of invertible dynamical systems.

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arXiv2026

The Cardinalities of Intervals of Equational Theories and Logics

We study the cardinality of classes of equational theories (varieties) and logics by applying descriptive set theory. We affirmatively solve open problems raised by Jackson and Lee [Trans. Am. Math. Soc. 370 (2018), pp. 4785-4812] regarding the cardinalities of subvariety lattices, and by Bezhanishvili et al. [J. Math. Log. (2025), in press] regarding the degrees of the finite model property (fmp). By coding equations and formulas by natural numbers, and theories and logics by real numbers, we examine their position in the Borel hierarchy. We prove that every interval of equational theories in a countable language corresponds to a $\boldsymbolΠ^0_1$ set, and every fmp span of a normal modal logic to a $\boldsymbolΠ^0_2$ set. It follows that they have cardinality either $\leq \aleph_0$ or $2^{\aleph_0}$, provably in ZFC. In the same manner, we observe that the set of pretabular extensions of a tense logic is a $\boldsymbolΠ^0_2$ set, so its cardinality is either $\leq \aleph_0$ or $2^{\aleph_0}$. We also point out a negative solution to another open problem raised by Jackson and Lee, op. cit., regarding the existence of independent systems, which relies on Ježek et al. [Bull. Aust. Math. Soc. 42 (1990), pp. 57-70].

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arXiv2026

Modal Measurable Logics via a Modal Loomis-Sikorski Representation Theorem

We investigate a modal extension of the infinitary classical logic with countable meets and joins, formulated with an eye toward measure-theoretic work in dynamical systems and in point-free ergodic theory. We define a modal formalism in this language, which we call modal measurable logics. We also introduce a Kripke-like semantics for these logics in measurable spaces taking a designated modal sigma-ideal into consideration. Using a restriction of Jonsson-Tarski duality and a modal extension of the Loomis-Sikorski theorem, we prove completeness of modal measurable logics with respect to this new semantics.

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arXiv2026

Fischer-Servi logic does not have interpolation

We prove that the Fischer-Servi logic $\mathsf{IK}$ does not have the (Craig) interpolation property. This is obtained by showing that the corresponding class of modal Heyting algebras lacks the amalgamation property. We also generalize this result to some extensions of the Fischer-Servi logic such as $\mathsf{IT}$, $\mathsf{IK4}$, $\mathsf{IS4}$, and $\mathsf{IGL}$.

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arXiv2026

Superamalgamation for modal lattices via non-distributive dualities

We show that the variety of modal lattices has the superamalgamation property. As a consequence, we obtain that the weak positive modal logic has the Craig interpolation property. Our proof employs the recent duality for modal lattices based on modal L-spaces. Moreover, we extend this result to a number of other weak positive modal logics axiomatized by modal axioms corresponding to universal Horn sentences.

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arXiv2026

Modal Fragments

We survey systematic approaches to basis-restricted fragments of propositional logic and modal logics, with an emphasis on how expressive power and computational complexity depend on the allowed operators. The propositional case is well-established and serves as a conceptual template: Post's lattice organizes fragments via Boolean clones and supports complexity classifications for standard reasoning tasks. For modal fragments, we then bring together two historically independent lines of investigation: a general framework where modal fragments are parameterized by a basis of "connectives" defined by arbitrary modal formulas (initially proposed and studied by logicians such as Kuznetsov and Ratsa in the 1970s), and the more tractable class of what we call simple modal fragments parameterized by Boolean functions plus selected modal operators, where Post-lattice methods enable systematic decidability and dichotomy results. Along the way, we collect and extend results on teachability and exact learnability from examples for both propositional fragments and simple modal fragments, and we conclude by identifying several open problems.

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arXiv2026

Weak Simplicial Bisimilarity and Minimisation for Polyhedral Model Checking

The work described in this paper builds on the polyhedral semantics of the Spatial Logic for Closure Spaces (SLCS) and the geometric spatial model checker PolyLogicA. Polyhedral models are central in domains that exploit mesh processing, such as 3D computer graphics. A discrete representation of polyhedral models is given by cell poset models, which are amenable to geometric spatial model checking on polyhedral models using the logical language SLCS$η$, a weaker version of SLCS. In this work we show that the mapping from polyhedral models to cell poset models preserves and reflects SLCS$η$. We also propose weak simplicial bisimilarity on polyhedral models and weak $\pm$-bisimilarity on cell poset models, where by ``weak'' we mean that the relevant equivalence is coarser than the corresponding one for SLCS, leading to a greater reduction of the size of models and thus to more efficient model checking. We show that the proposed bisimilarities enjoy the Hennessy-Milner property, i.e. two points are weakly simplicial bisimilar iff they are logically equivalent for SLCS$η$. Similarly, two cells are weakly $\pm$-bisimilar iff they are logically equivalent in the poset-model interpretation of SLCS$η$. Furthermore we present a model minimisation procedure and prove that it correctly computes the minimal model with respect to weak $\pm$-bisimilarity, i.e. with respect to logical equivalence of SLCS$η$. The procedure works via an encoding into LTSs and then exploits branching bisimilarity on those LTSs, exploiting the minimisation capabilities as included in the mCRL2 toolset. Various examples show the effectiveness of the approach.

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arXiv2025

Esakia order-compactifications and locally Esakia spaces

We introduce Esakia order-compactifications and study how they fit in the general theory of Priestley order-compactifications. We provide an analog of Dwinger's theorem by characterizing Esakia order-compactifications by means of special rings of upsets. These considerations naturally lead to the notion of a locally Esakia space, for which we prove that taking the largest Esakia order-compacification is functorial, thus obtaining an analog of Banaschewski's theorem.

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arXiv2025

Six Proofs of Interpolation for the Modal Logic K

In this chapter, we present six different proofs of Craig interpolation for the modal logic K, each using a different set of techniques (model-theoretic, proof-theoretic, syntactic, automata-theoretic, using quasi-models, and algebraic). We compare the pros and cons of each proof technique.

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arXiv2025

Pre-filtrations, Pre-stable Canonical Rules, and the Kuznetsov-Muravitsky Isomorphism

We introduce pre-filtration and pre-stable canonical rules for the Kuznetsov-Muravitsky system of intuitionistic modal logic and provide a new proof of the Kuznetsov-Muravitsky isomorphism, along with several preservation results. The proofs employ these rules and a duality between modal (Heyting) algebras and their corresponding order-topological spaces.

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arXiv2025

Structural Completeness in bi-IPC

In this note we show that no extension of bi-intuitionistic logic, except for classical logic, is structurally complete; indeed, none of them are passively structurally complete. A direct proof of active structural completeness is given for some simple systems.

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arXiv2025

Blok-Esakia Theorems via Stable Canonical Rules

We present a new uniform method for studying modal companions of superintuitionistic rule systems and related notions, based on the machinery of stable canonical rules. Using this method, we obtain alternative proofs of the Blok-Esakia theorem and of the Dummett-Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalizes smoothly to richer signatures. Using essentially the same argument, we obtain a proof of an analogue of the Blok-Esakia theorem for bi-superintuitionistic and tense rule systems, and of the Kuznetsov-Muravitsky isomorphism between rule systems extending the modal intuitionistic logic $\logic{KM}$ and modal rule systems extending the provability logic $\logic{GL}$. In addition, our proof of the Dummett-Lemmon conjecture also generalizes to the bi-superintuitionistic and tense cases.

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arXiv2025

A calculus for modal compact Hausdorff spaces

The symmetric strict implication calculus $\mathsf{S^2IC}$ is a modal calculus for compact Hausdorff spaces. This is established through de Vries duality, linking compact Hausdorff spaces with de Vries algebras-complete Boolean algebras equipped with a special relation. Modal compact Hausdorff spaces are compact Hausdorff spaces enriched with a continuous relation. These spaces correspond, via modalized de Vries duality, to upper continuous modal de Vries algebras. In this paper we introduce the modal symmetric strict implication calculus $\mathsf{MS^2IC}$, which extends $\mathsf{S^2IC}$. We prove that $\mathsf{MS^2IC}$ is strongly sound and complete with respect to upper continuous modal de Vries algebras, thereby providing a logical calculus for modal compact Hausdorff spaces. We also develop a relational semantics for $\mathsf{MS^2IC}$ that we employ to show admissibility of various $Π_2$-rules in this system.

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arXiv2025

Degrees of the finite model property: the antidichotomy theorem

A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic $\sf K$ is $1$ or $2^{\aleph_0}$. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as $\sf S4$ or $\sf K4$) or for extensions of the intuitionistic propositional calculus $\mathsf{IPC}$. In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem that the degree of fmp of a normal extension of $\sf K$ remains $1$ or $2^{\aleph_0}$. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of $\mathsf{IPC}$: each nonzero cardinal $κ$ such that $κ\leq \aleph_0$ or $κ= 2^{\aleph_0}$ is realized as the degree of fmp of some extension of $\mathsf{IPC}$. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of $\sf S4$ and $\sf K4$.

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arXiv2024

Logics of polyhedral reachability

Polyhedral semantics is a recently introduced branch of spatial modal logic, in which modal formulas are interpreted as piecewise linear subsets of an Euclidean space. Polyhedral semantics for the basic modal language has already been well investigated. However, for many practical applications of polyhedral semantics, it is advantageous to enrich the basic modal language with a reachability modality. Recently, a language with an Until-like spatial modality has been introduced, with demonstrated applicability to the analysis of 3D meshes via model checking. In this paper, we exhibit an axiom system for this logic, and show that it is complete with respect to polyhedral semantics. The proof consists of two major steps: First, we show that this logic, which is built over Grzegorczyk's system $\mathsf{Grz}$, has the finite model property. Subsequently, we show that every formula satisfied in a finite poset is also satisfied in a polyhedral model, thereby establishing polyhedral completeness.

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arXiv2024

A Coalgebraic Semantics for Intuitionistic Modal Logic

We give a new coalgebraic semantics for intuitionistic modal logic with $\Box$. In particular, we provide a colagebraic representation of intuitionistic descriptive modal frames and of intuitonistic modal Kripke frames based on image-finite posets. This gives a solution to a problem in the area of coalgebaic logic for these classes of frames, raised explicitly by Litak (2014) and de Groot and Pattinson (2020). Our key technical tool is a recent generalization of a construction by Ghilardi, in the form of a right adjoint to the inclusion of the category of Esakia spaces in the category of Priestley spaces. As an application of these results, we study bisimulations of intuitionistic modal frames, describe dual spaces of free modal Heyting algebras, and provide a path towards a theory of coalgebraic intuitionistic logics.

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arXiv2024

Weak Simplicial Bisimilarity for Polyhedral Models and SLCS_eta -- Extended Version

In the context of spatial logics and spatial model checking for polyhedral models -- mathematical basis for visualisations in continuous space -- we propose a weakening of simplicial bisimilarity. We additionally propose a corresponding weak notion of $\pm$-bisimilarity on cell-poset models, a discrete representation of polyhedral models. We show that two points are weakly simplicial bisimilar iff their repesentations are weakly $\pm$-bisimilar. The advantage of this weaker notion is that it leads to a stronger reduction of models than its counterpart that was introduced in our previous work. This is important, since real-world polyhedral models, such as those found in domains exploiting mesh processing, typically consist of large numbers of cells. We also propose SLCS_eta, a weaker version of the Spatial Logic for Closure Spaces (SLCS) on polyhedral models, and we show that the proposed bisimilarities enjoy the Hennessy-Milner property: two points are weakly simplicial bisimilar iff they are logically equivalent for SLCS_eta. Similarly, two cells are weakly $\pm$-bisimilar iff they are logically equivalent in the poset-model interpretation of SLCS_eta. This work is performed in the context of the geometric spatial model checker PolyLogicA and the polyhedral semantics of SLCS.

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arXiv2023

On Shehtman's Two Problems

We provide partial solutions to two problems posed by Shehtman concerning the modal logic of the Čech-Stone compactification of an ordinal space. We use the Continuum Hypothesis to give a finite axiomatization of the modal logic of $β(ω^2)$, thus resolving Shehtman's first problem for $n=2$. We also characterize modal logics arising from the Čech-Stone compactification of an ordinal $γ$ provided the Cantor normal form of $γ$ satisfies an additional condition. This gives a partial solution of Shehtman's second problem.

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