Guram Bezhanishvili

Modal Logics of Space

1 The interest of Space When thinking about the physical world, modal logicians have taken Time as their main theme, because it fits so well with an interest in the flow of information and computation. Spatial logics have been footnotes to the tradition – even though the axiomatic method was largely geometrical. An exception is Tarskis early work on deviant geometrical primitives, and his decidable first-order axiomatization of elementary geometry. Today Space remains intriguing – both for mathematical reasons, and given the amount of work in CS and AI on visual reasoning and image processing. These two concerns are by no means the same, but both involve logic of spatial structures.

Some results on modal axiomatization and definability for topological spaces

We consider two topological interpretations of the modal diamond—as the closure operator (C-semantics) and as the derived set operator (d-semantics). We call the logics arising from these interpretations C-logics and d-logics, respectively. We axiomatize a number of subclasses of the class of nodec spaces with respect to both semantics, and characterize exactly which of these classes are modally definable. It is demonstrated that the d-semantics is more expressive than the C-semantics. In particular, we show that the d-logics of the six classes of spaces considered in the paper are pairwise distinct, while the C-logics of some of them coincide.

Multimo dal Logics of Products of Topologies

We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 ⊕ S4. We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies.We prove that both of these logics are complete for the product of rational numbers ℚ × ℚ with the appropriate topologies.

Bitopological duality for distributive lattices and heyting algebras

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakias duality.

Locally finite varieties

Abstract. In this paper we present a new and useful criterion for a variety to be locally finite. Many examples are given to justify the effectiveness of the criterion.

Profinite Completions and Canonical Extensions of Heyting Algebras

We show that the profinite completions and canonical extensions of bounded distributive lattices and of Boolean algebras coincide. We characterize dual spaces of canonical extensions of bounded distributive lattices and Heyting algebras in terms of Nachbin order-compactifications. We give the dual description of the profinite completion

An algebraic approach to canonical formulas: Intuitionistic case

\widehat{H}

Euclidean hierarchy in modal logic

of a Heyting algebra H, and characterize the dual space of

Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces

\widehat{H}

An algebraic approach to subframe logics. Intuitionistic case

. We also give a necessary and sufficient condition for the profinite completion of H to coincide with its canonical extension, and provide a new criterion for a variety V of Heyting algebras to be finitely generated by showing that V is finitely generated if and only if the profinite completion of every member of V coincides with its canonical extension. From this we obtain a new proof of a well-known theorem that every finitely generated variety of Heyting algebras is canonical.