arXiv subjects
Publications and source records attributed to Andrey Kudinov.
In this paper, we find the axiomatization for the topological square of S4.1. This is the first known topological square of a modal logic that differs from both the fusion and the Kripke product. We also prove the finite model property and decidability for this logic.
On the product of two neighborhood frames, three natural neighborhood functions can be defined: the horizontal one assigning to a point (x, y) the set of all supersets of the Cartesian product of U and y, where U is a neighborhood of x; the vertical analog; and the product neighborhood function assigning as neighborhoods all supersets of sets Cartesian products of U and V, for neighborhoods U of x and V of y. We define the tri-modal logics Tx+T and Dx+D of classes of full products equipped with all three neighborhood functions of neighborhood frames validating the logic T or D; thereby extending known product results for S4 and D4 to weaker systems. Two interaction principles arise: (sub) = []p -> [1]p & [2]p and (mix) = []p -> [1][2]p & [2][1]p, where the modality [] stands for the product neighborhood function and [1], [2] the horizontal and vertical ones. Namely, we show that Tx+T = T*T*T + (mix) and Dx+D = D*D*D + (mix), where * denotes fusion. Notably, (sub) and (mix) are equivalent over S4*S4*S4 and thus S4*S4*S4 + (mix) axiomatizes the logic of full products of topological spaces.
We study the finite model property of subframe logics with expressible transitive reflexive closure modality. For $m>0$, let $\mathrm{L}_m$ be the logic defined by axiom $\lozenge^{m+1} p\to \lozenge p\vee p$. We construct filtrations for the logics $\mathrm{L}_m$. It follows that these logics and their tense counterparts have the finite model property. Then we show that every canonical subframe logic that contains $\mathrm{L}_m$ have the finite model property.
In this paper we consider the topological products of modal logics of S4.1 and S4. We prove that it is equal to the fusion of logics S4.1 and S4 plus one additional axiom. We also show that this product is decidable. This is an example of a topological product of logics that is greater than the fusion but less than the expanding product of the corresponding logics.
We consider quantified pretransitive Horn modal logic. It is known that such logics are complete with respect to predicate Kripke frames with expanding domains. In this paper we prove that they are also complete with respect to neighbourhood frames with constant domains.
We consider modal logics of products of neighborhood frames. We define n-product of modal logics as the logic of all products of neighborhood frames of corresponding logics and find n-product of any two pretransitive Horn axiomatizable logics. As a corrolary we find the d-logic of products of topological spaces for some classes of topological spaces.
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 described for pretransitive frames of finite height, which implies finite model property and decidability of logics of these frames.
We consider modal logics of products of neighborhood frames and prove that for any pair $L$ and $L'$ of logics from set $\{S4, D4, D, T\}$ modal logic of products of $L$-neighborhood frames and $L'$-neighborhood frames is the fusion of $L$ and $L'$.
In this chapter we study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, $T_1$-spaces, dense-in-themselves spaces, a zero-dimensional dense-in-itself separable metric space, $R^n$ $(n\ge 2)$. We also discuss the correlation between languages with different combinations of the topological, the derivational, the universal and the difference modality in terms of definability.