Dimiter Vakarelov

The paper is in the field of region-based theory of space and time. This is an extension of the region-based theory of space with time. Its origin goes back to some ideas of Whitehead, De Laguna, and Tarski and is related to the problem of how to build the theory of space without the use of the notion of point. The notion of contact algebra presents an algebraic formulation of RBTS. CA is an extension of Boolean algebra, considered as an algebra of spatial regions with an additional relation of contact. Dynamic contact algebra considered as an algebraic formulation of RBTST is an extension of CA aiming to study regions changing in time. In this paper, we study a version of DCA incorporating an explicit predicate AE of actual existence. We first develop the representation theory of such DCAs by means of the so-called snapshot models. Second, we introduce topological models of DCA and develop the corresponding topological representation and duality theory.

In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called \emph{elementary}. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called \emph{canonical}. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and results, and outline further research perspectives.

In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called elementary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and results, and outline further research perspectives.

Mereotopology is an extension of mereology with some relations of topological nature like contact. An algebraic counterpart of mereotopology is the notion of contact algebra which is a Boolean algebra whose elements are considered to denote spatial regions, extended with a binary relation of contact between regions. Although the language of contact algebra is quite expressive to define many useful mereological relations and mereotopological relations, there are, however, some interesting mereotopological relations which are not definable in it. Such are, for instance, the relation of n-ary contact, internal connectedness and some others. To overcome this disadvantage, we introduce a generalisation of contact algebra, replacing the contact with a binary relation between finite sets of regions and a region, satisfying some formal properties of Tarski consequence relation. The obtained system is called sequent algebra, considered as an algebraic counterpart of a new mereotopology. We develop the topological representation theory for sequent algebras showing in this way certain correspondence between point-free and point-based models of space. As a by-product, we show how one logical relation in nature notion, Tarski consequence relation, may have also certain spatial meaning.

This paper is a continuation of [VAK 01]. The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to Roeper's notion of region-based topology [ROE 97]. The similarity between the local proximity spaces of Leader [LEA 67] and local connection algebras is emphasized. Machinery, analogous to that introduced by Efremovi?c [EFR 51],[EFR 52], Smirnov [SMI 52] and Leader [LEA 67] for proximity and local proximity spaces, is developed. This permits us to give new proximity-type models of local connection algebras, to obtain a representation theorem for such algebras and to give a new shorter proof of the main theorem of Roeper's paper [ROE 97]. Finally, the notion of MVD-algebra is introduced. It is similar to Mormann's notion of enriched Boolean algebra [MOR 98], based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that the connection relation and boundedness can be incorporated into one, mereological in nature relation. In this way a formalization of the Whiteheadian theory of space based on a single mereological relation is obtained.

We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames

We proved in [1] that there exist a continuum consistent monotone intuitionistic modal logics which do not admit the law of the excluded middle p ∨ ¬p. Rieger [2] and Nishimura [3] introduced a sequence of formulas ϕ0, ϕ1, . . . , ϕω of one variable p such that for any intuitionistic formula ϕi containing only the variable p there exists a formula ϕi from this sequence equivalent to ϕ in the intuitionistic propositional logic . In [5] V. Tselkov has proved that for each i ≥ 4 there exist at least countably many consistent monotone intuitionistic modal logics which do not admit the formula ϕi. In this paper we strengthen this result by showing that there exist continuum of such logics. We construct an example of monotone intuitionistic modal logic not admitting all formulas ϕi, where i ≥ 4 and i =6 ω. Using similar constructions we prove that there exist at least countably many maximal monotone intuitionistic modal logics. Let us mention that in the classical case Makinson [6] has shown that there are exactly three such logics