Ilya B. Shapirovsky

• PhD
• Professor (Assistant) at New Mexico State University

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 fram...

We consider logics derived from Euclidean spaces $\mathbb{R}^n$. Each Euclidean space carries relations consisting of those pairs that are, respectively, distance more than 1 apart, distance less than 1 apart, and distance 1 apart. Each relation gives a uni-modal logic of $\mathbb{R}^n$ called the farness, nearness, and constant distance logics, re...

We consider the bimodal language, where the first modality is interpreted by a binary relation in the standard way, and the second is interpreted by the relation of inequality. It follows from Hughes (1990), that in this language, non-k-colorability of a graph is expressible for every finite k. We show that modal logics of classes of non-k-colorabl...

We consider the bimodal language, where the first modality is interpreted by a binary relation in the standard way, and the second is interpreted by the relation of inequality. It follows from Hughes (1990), that in this language, non-$k$-colorability of a graph is expressible for every finite $k$. We show that modal logics of classes of non-$k$-co...

We describe a family of decidable propositional dynamic logics, where atomic modalities satisfy some extra conditions (for example, given by axioms of the logics K5, S5, or K45 for different atomic modalities). It follows from recent results (Kikot, Shapirovsky, Zolin, 2014; 2020) that if a modal logic $L$ admits a special type of filtration (so-ca...

We study locally tabular polymodal logics. The main results are the following. We show that local tabularity does not depend on reflexivity. Namely, given a class $\mathcal{C}$ of frames, consider the class $\mathcal{C}^\mathrm{r}$ of frames, where the reflexive closure operation was applied to every relation in every frame in $\mathcal{C}$ . We sh...

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 [Babenyshev and Rybakov 2010; Shapirovsky 2018]. In this paper we show that, under a general conditi...

Glivenko’s theorem states that a formula is derivable in classical propositional logic CL iff under the double negation it is derivable in intuitionistic propositional logic IL. In Kripke semantics, IL is the logic of partial orders, and CL is the logic of partial orders of height 1. Likewise, S4 is the logic of preorders, and S5 is the logic of eq...

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 polyno...

We give a sufficient condition for Kripke completeness of modal logics enriched with the transitive closure modality. More precisely, we show that if a logic admits what we call definable filtration (ADF), then such an expansion of the logic is complete; in addition, has the finite model property, and again ADF. This argument can be iterated, and a...

We give a sufficient condition for Kripke completeness of modal logics that have the transitive closure modality. More precisely, we show that if a modal logic admits what we call definable filtration, then its enrichment with the transitive closure modality (and the corresponding axioms) is Kripke complete; in addition, the resulting logic has the...

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 \(\mathcal {R}\) d...

Let \((\omega ^n,\preceq )\) be the n-th direct power of \((\omega ,\le )\), natural numbers with the standard ordering, and let \((\omega ^n,\prec )\) be the n-th direct power of \((\omega ,<)\). We show that for all finite n, the modal algebras of \((\omega ^n,\preceq )\) and of \((\omega ^n,\prec )\) are locally finite. In particular, it follows...

Given a class C of models, a binary relation R on C, and a model-theoretic language L, we consider the modal logic and the modal algebra of the theory of C in L where the modal operator is interpreted via R. We discuss how modal theories of C and R depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality...

Let (ω^n,\preceq) be the direct power of n instances of (ω, ≤), natural numbers with the standard ordering, (ω, \prec) the direct power of n instances of (ω, <). We show that for all finite n, the modal logics of (ω^n,\preceq) and of (ω,\prec) have the finite model property and moreover, their modal algebras are locally finite.

The operation of sum of a family (F_i | i in I) of Kripke frames indexed by elements of another frame I provides a natural way to construct expressive polymodal logics with good semantic and algorithmic properties. This operation has had several important applications over the last decade: it was used by L. Beklemishev in the context of polymodal p...

The well-known Glivenko's theorem states that a formula is derivable in the classical propositional logic ${\mathrm{CL}}$ iff under the double negation it is derivable in the intuitionistic propositional logic $\mathrm{IL}$: $\mathrm{CL}\vdash \varphi$ iff $\mathrm{IL}\vdash\neg\neg\varphi$. Its analog for the modal logics $\mathrm{S5}$ into $\math...

В работе доказана финитная аппроксимируемость и разрешимость одного семейства модальных логик. Бинарное отношение $R$ назовем предтранзитивным, если $R^*=\bigcup_{i\leqslant m} R^i$ для некоторого $m\geqslant 0$, где $R^*$ - транзитивное рефлексивное замыкание $R$. Под высотой шкалы $(W,R)$ будем понимать высоту предпорядка $(W,R^*)$. Построены спе...

According to the classical result by Segerberg and Maksimova, a modal logic containing K4 is locally tabular iff it is of finite height. The notion of finite height can also be defined for logics, in which the master modality is expressible ('pretransitive' logics). We observe that every locally tabular logic is a pretransitive logic of finite heig...

In this paper we prove the finite model property and decidability of a family of modal logics. A binary relation R is said to be pretransitive if R∗ = Ui≤m Rⁱ for some m ≥ 0, where R∗ is the transitive reflexive closure of R. By the height of a frame (W, R) we mean the height of the preorder (W, R∗). We construct special partitions (filtrations) of...

The paper proves finite model property and decidability for a family of modal logics. A binary relation $R$ is called pretransitive, if $R^*=\cup_{i\leq m} R^i$ for some $m\geq 0$, where $R^*$ is the transitive reflexive closure of $R$. By the height of $(W,R)$ we mean the height of the preorder $(W,R^*)$. Special partitionings (filtrations) are de...

Filtration is a standard tool for establishing the finite model property of modal logics. We consider logics and classes of frames that admit filtration, and identify some operations on them that preserve this property. In particular, the operation of adding the inverse or the transitive closure of a relation is shown to be safe in this sense. Thes...

One of natural combinations of Kripke complete modal logics is the product, an operation that has been extensively investigated over the last 15 years. In this paper we consider its analogue for arbitrary modal logics: to this end, we use product-like constructions on general frames and modal algebras. This operation was first introduced by Y. Hasi...

With a set S of words in an alphabet A we associate the frame (S,H), where sHt iff s and t are words of the same length and h(s,t)=1 for the Hamming distance h. We investigate some unimodal logics of these frames. We show that if the length of words n is fixed and finite, the logics are closely related to many-dimensional products S5 n , so in many...

In this paper, we prove undecidability and the lack of finite model property for a certain class of unimodal logics. To do this, we adapt the technique from [7], where products of transitive modal logics were investigated, for the unimodal case. As a particular corollary, we present an undecidable unimodal fragment of Halpern and Shoham's Interval...

In this paper we prove that Japaridze's Polymodal Logic is PSPACE-decidable. To show this, we describe a decision proc edure for satisfiability on hereditarily ordered frames that can be applied to obtain upper complexity bounds for various modal logics.

In this paper we study modal logics of closed domains on the real plane ordered by the chronological future relation. For the modal logic determined by an arbitrary closed convex domain with a smooth bound, we present a finite axiom system and prove the finite modal property.

We study modal logics of regions in a real space ordered by the inclusion and compact inclusion relations. For various systems of regions, we propose complete finite modal axiomatizations; the described logics are finitely approximable and PSPACE-complete.

We introduce a class of propositional modal logics axiomatized by infinite sequences of formulas in special form. Two logics of this type are known from [10] and [13]. Although the axioms are beyond Sahlqvist class and its generalization defined in [8], all the resulting logics are still complete with respect to elementary classes of frames. For tw...

In this paper we identify modal logics of some bimodal Kripke frames corresponding to geometrical structures. Each of these frames is a set of 'geometrical' objects with some natural accessibility relation plus the universal relation. For these logics we present nite axiom systems and prove completeness. We also show that all these logics have the...

The paper studies modal logics of Kripke frames, in which possible worlds are regions in space with natural accessibility relations. These logics are also interpreted as relativistic temporal logics. Together with an overview, we prove some new results on completeness, decidability, complexity, and finite axiomatizability.

This paper makes the next essential step after the past twenty years. It solves one of three problems put by R. Goldblatt in [6] (see also [7]): to axiomatize the modal logic of the frame (R ; )

This paper makes the next essential step after the past twenty years. Itsolves one of three problems put by R. Goldblatt in [6] (see also [7]): toaxiomatize the modal logic of the frame (R; )