arXiv subjects

Ilya B. Shapirovsky

Publications and source records attributed to Ilya B. Shapirovsky.

7 recordsLinked to original sources

arXiv2025

Generalizations of The Finite Height Criterion for Local Tabularity

It is well known that for transitive unimodal logics, finite height is both necessary and sufficient for local tabularity. It is also well known that for intermediate logics, finite height is sufficient (but not necessary) for local tabularity. For non-transitive unimodal, and for polymodal logics, finite height is necessary (but not sufficient) for local tabularity. We discuss generalizations of the finite height criterion of local tabularity for families of non-transitive and polymodal logics. Then we discuss the finite modal depth property of modal logics and give a version of the finite height criterion for this property.

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arXiv2025

On pre-local tabularity above $\mathrm{S4}\times \mathrm{S4}$

We investigate pre-local tabularity in normal extensions of the logic $\mathrm{S4}\times \mathrm{S4}$. We show that there are exactly four pre-locally tabular logics in normal extensions of products of finite height, and that every non-locally tabular logic in this family is contained in one of them. We also give an axiomatic criterion of local tabularity above the logic of products with Noetherian skeletons. Finally, we discuss examples of pre-locally tabular extensions of $\mathrm{S4}\times \mathrm{S4}$ outside this class, including logics with the converse and universal modalities.

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arXiv2024

Locally tabular products of modal logics

In the product $L_1\times L_2$ of two Kripke complete consistent logics, local tabularity of $L_1$ and $L_2$ is necessary for local tabularity of $L_1\times L_2$. However, it is not sufficient: the product of two locally tabular logics may not be locally tabular. We provide extra semantic and axiomatic conditions that give criteria of local tabularity of the product of two locally tabular logics, and apply them to identify new families of locally tabular products. We show that the product of two locally tabular logics may lack the product finite model property. We give an axiomatic criterion of local tabularity for all extensions of $S4.1 [ 2 ]\times S5$. Finally, we describe a new prelocally tabular extension of $S{4}\times S{5}$.

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arXiv2022

Sufficient conditions for local tabularity of a polymodal logic

On relational structures and on polymodal logics, we describe operations which preserve local tabularity. This provides new sufficient semantic and axiomatic conditions for local tabularity of a modal logic. The main results are the following. We show that local tabularity does not depend on reflexivity. Namely, given a class $\mathcal{F}$ of frames, consider the class $\mathcal{F}^\mathrm{r}$ of frames, where the reflexive closure operation was applied to each relation in every frame in $\mathcal{F}$. We show that if the logic of $\mathcal{F}^\mathrm{r}$ is locally tabular, then the logic of $\mathcal{F}$ is locally tabular as well. Then we consider the operation of sum on Kripke frames, where a family of frames-summands is indexed by elements of another frame. We show that if both the logic of indices and the logic of summands are locally tabular, then the logic of corresponding sums is also locally tabular. Finally, using the previous theorem, we describe an operation on logics that preserves local tabularity: we provide a set of formulas such that the extension of the fusion of two canonical locally tabular logics with these formulas is locally tabular.

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arXiv2020

Satisfiability problems on sums of Kripke frames

We consider the operation of sum on Kripke frames, where a family of frames-summands is indexed by elements of another frame. In many cases, the modal logic of sums inherits the finite model property and decidability from the modal logic of summands. In this paper we show that, under a general condition, the satisfiability problem on sums is polynomial space Turing reducible to the satisfiability problem on summands. In particular, for many modal logics decidability in PSPACE is an immediate corollary from the semantic characterization of the logic.

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arXiv2018

Glivenko's theorem, finite height, and local tabularity

Glivenko's theorem states that a formula is derivable in classical propositional logic $\mathrm{CL}$ iff under the double negation it is derivable in intuitionistic propositional logic $\mathrm{IL}$: $\mathrm{CL}\vdash\varphi$ iff $\mathrm{IL}\vdash\neg\neg\varphi$. Its analog for the modal logics $\mathrm{S5}$ and $\mathrm{S4}$ states that $\mathrm{S5}\vdash \varphi$ iff $\mathrm{S4} \vdash \neg \Box \neg \Box \varphi$. In Kripke semantics, $\mathrm{IL}$ is the logic of partial orders, and $\mathrm{CL}$ is the logic of partial orders of height 1. Likewise, $\mathrm{S4}$ is the logic of preorders, and $\mathrm{S5}$ is the logic of equivalence relations, which are preorders of height 1. In this paper we generalize Glivenko's translation for logics of arbitrary finite height.

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arXiv2018

On modal logics of model-theoretic relations

Given a class $\mathcal C$ of models, a binary relation ${\mathcal R}$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $\mathcal C$ in $L$ where the modal operator is interpreted via $\mathcal R$. We discuss how modal theories of $\mathcal C$ and ${\mathcal R}$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside $L$. We calculate such theories for the submodel and the quotient relations. We prove a downward L\"owenheim--Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.

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