arXiv subjects

David Gabelaia

Publications and source records attributed to David Gabelaia.

14 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 $\gamma(\varphi,\psi)$ expresses reachability of a $\psi$-point through a $\varphi$-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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arXiv2024

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$\eta$, a weaker version of SLCS. In this work we show that the mapping from polyhedral models to cell poset models preserves and reflects SLCS$\eta$. 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$\eta$. Similarly, two cells are weakly $\pm$-bisimilar iff they are logically equivalent in the poset-model interpretation of SLCS$\eta$. 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$\eta$. 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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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

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

The Intermediate Logic of Convex Polyhedra

We investigate a recent semantics for intermediate (and modal) logics in terms of polyhedra. The main result is a finite axiomatisation of the intermediate logic of the class of all polytopes -- i.e., compact convex polyhedra -- denoted PL. This logic is defined in terms of the Jankov-Fine formulas of two simple frames. Soundness of this axiomatisation requires extracting the geometric constraints imposed on polyhedra by the two formulas, and then using substantial classical results from polyhedral geometry to show that convex polyhedra satisfy those constraints. To establish completeness of the axiomatisation, we first define the notion of the geometric realisation of a frame into a polyhedron. We then show that any PL frame is a p-morphic image of one which has a special form: it is a 'sawed tree'. Any sawed tree has a geometric realisation into a convex polyhedron, which completes the proof.

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arXiv2022

Polyhedral completeness of intermediate logics: the Nerve Criterion

We investigate a recently-devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if 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-complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally-complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov-Fine formulas of 'starlike trees' all of which are polyhedrally-complete. The polyhedral completeness theorem for these 'starlike logics' is the second main result of this paper.

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arXiv2021

Geometric Model Checking of Continuous Space

Topological Spatial Model Checking is a recent paradigm where model checking techniques are developed for the topological interpretation of Modal Logic. The Spatial Logic of Closure Spaces, SLCS, extends Modal Logic with reachability connectives that, in turn, can be used for expressing interesting spatial properties, such as "being near to" or "being surrounded by". SLCS constitutes the kernel of a solid logical framework for reasoning about discrete space, such as graphs and digital images, interpreted as quasi discrete closure spaces. Following a recently developed geometric semantics of Modal Logic, we propose an interpretation of SLCS in continuous space, admitting a geometric spatial model checking procedure, by resorting to models based on polyhedra. Such representations of space are increasingly relevant in many domains of application, due to recent developments of 3D scanning and visualisation techniques that exploit mesh processing. We introduce PolyLogicA, a geometric spatial model checker for SLCS formulas on polyhedra and demonstrate feasibility of our approach on two 3D polyhedral models of realistic size. Finally, we introduce a geometric definition of bisimilarity, proving that it characterises logical equivalence.

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arXiv2021

A negative solution of Kuznetsov's problem for varieties of bi-Heyting algebras

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 $\mathsf{HB}$ 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 intermediate logics to extensions of $\mathsf{HB}$.

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arXiv2020

Diego's Theorem for nuclear implicative semilattices

We prove that the variety of nuclear implicative semilattices is locally finite, thus generalizing Diego's Theorem. The key ingredients of our proof include the coloring technique and construction of universal models from modal logic. For this we develop duality theory for finite nuclear implicative semilattices, generalizing Köhler duality. We prove that our main result remains true for bounded nuclear implicative semilattices, give an alternative proof of Diego's Theorem, and provide an explicit description of the free cyclic nuclear implicative semilattice.

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arXiv2018

Modal logic of planar polygons

We study the modal logic of the closure algebra $P_2$, generated by the set of all polygons in the Euclidean plane $\mathbb{R}^2$. We show that this logic is finitely axiomatizable, is complete with respect to the class of frames we call "crown" frames, is not first order definable, does not have the Craig interpolation property, and its validity problem is PSPACE-complete.

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arXiv2013

Topological completeness of logics above S4

It is a celebrated result of McKinsey and Tarski [28] that S4 is the logic of the closure algebra X+ over any dense-in-itself separable metrizable space. In particular, S4 is the logic of the closure algebra over the reals R, the rationals Q, or the Cantor space C. By [5], each logic above S4 that has the finite model property is the logic of a subalgebra of Q+, as well as the logic of a subalgebra of C+. This is no longer true for R, and the main result of [5] states that each connected logic above S4 with the finite model property is the logic of a subalgebra of the closure algebra R+. In this paper we extend these results to all logics above S4. Namely, for a normal modal logic L, we prove that the following conditions are equivalent: (i) L is above S4, (ii) L is the logic of a subalgebra of Q+, (iii) L is the logic of a subalgebra of C+. We introduce the concept of a well-connected logic above S4 and prove that the following conditions are equivalent: (i) L is a well-connected logic, (ii) L is the logic of a subalgebra of the closure algebra T_2^+ over the infinite binary tree, (iii) L is the logic of a subalgebra of the closure algebra L_2^+ over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logic L above S4 is connected iff L is the logic of a subalgebra of R+, and transfer our results to the setting of intermediate logics.

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arXiv2012

Topological interpretations of provability logic

Provability logic concerns the study of modality $\Box$ as provability in formal systems such as Peano arithmetic. Natural, albeit quite surprising, topological interpretation of provability logic has been found in the 1970's by Harold Simmons and Leo Esakia. They have observed that the dual $\Diamond$ modality, corresponding to consistency in the context of formal arithmetic, has all the basic properties of the topological derivative operator acting on a scattered space. The topic has become a long-term project for the Georgian school of logic led by Esakia, with occasional contributions from elsewhere. More recently, a new impetus came from the study of polymodal provability logic GLP that was known to be Kripke incomplete and, in general, to have a more complicated behavior than its unimodal counterpart. Topological semantics provided a better alternative to Kripke models in the sense that GLP was shown to be topologically complete. At the same time, new fascinating connections with set theory and large cardinals have emerged. We give a survey of the results on topological semantics of provability logic starting from first contributions by Esakia. However, a special emphasis is put on the recent work on topological models of polymodal provability logic. We also included a few results that have not been published so far, most notably the results of Section 6 (due the second author) and Sections 10, 11 (due to the first author).

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arXiv2011

Topological completeness of the provability logic GLP

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 satisfying all the axioms of GLP are called GLP-spaces. 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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arXiv2006

Modal languages for topology: expressivity and definability

In this paper we study the expressive power and definability for (extended) 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 $L_t$.

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