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Computable Contact Algebras

Bazhenov Nikolay

Fundamenta Informaticae

2019

Vol. 167, nr 4

257--269

We investigate computability-theoretic properties of contact algebras. These structures were introduced by Dimov and Vakarelov in [Fundam. Inform. 74 (2006), 209-249] as an axiomatization for the region-based theory of space. We prove that the class of countable contact algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. This means that the class of contact algebras is very rich from the computability-theoretic point of view. As an application of our result, we show that the Π3-theory of contact algebras is hereditarily undecidable. This is a refinement of the result of Koppelberg, Düntsch, and Winter [Algebra Univers., 68 (2012), 353-366].

Modal Logics for Region-based Theories of Space

Balbiani P., Tinchev T., Vakarelov D.

Fundamenta Informaticae

2007

Vol. 81, nr 1-3

29-82

We dedicate thispaper to Professor Andrzej Grzegorczyk. His paper "Axiomatization of geometry without points" is one the first contributions to the region-based theory of space.

Contact Algebras and Region-based Theory of Space: Proximity Approach - II

Dimov G., Vakarelov D.

Fundamenta Informaticae

2006

Vol. 74, nr 2,3

251-282

This paper is the second part of the paper [2]. Both of them are in the field of region-based (or Whitehedian) theory of space, which is an important subfield of Qualitative Spatial Reasoning (QSR). The paper can be considered also as an application of abstract algebra and topology to some problems arising and motivated in Theoretical Computer Science and QSR. In [2], different axiomatizations for region-based theory of space were given. The most general one was introduced under the name ``Contact Algebra". In this paper some categories defined in the language of contact algebras are introduced. It is shown that they are equivalent to the category of all semiregular T0-spaces and their continuous maps and to its full subcategories having as objects all regular (respectively, completely regular; compact; locally compact) Hausdorff spaces. An algorithm for a direct construction of all, up to homeomorphism, finite semiregular T0-spaces of rank n is found. An example of an RCC model which has no regular Hausdorff representation space is presented. The main method of investigation in both parts is a lattice-theoretic generalization of methods and constructions from the theory of proximity spaces. Proximity models for various kinds of contact algebras are given here. In this way, the paper can be regarded as a full realization of the proximity approach to the region-based theory of space.

Contact Algebras and Region-based Theory of Space: A Proximity Approach - I

Dimov G., Vakarelov D.

Fundamenta Informaticae

2006

Vol. 74, nr 2,3

209-249

This work is in the field of region-based (or Whitehedian) theory of space, which is an important subfield of Qualitative Spatial Reasoning (QSR). The paper can be considered also as an application of abstract algebra and topology to some problems arising and motivated in Theoretical Computer Science and QSR Different axiomatizations for region-based (or Whiteheadian) theory of space are given. The most general one is introduced under the name ``Contact Algebra". Adding some extra first- or second-order axioms to those of contact algebras, some new or already known algebraic notions are obtained. Representation theorems and completion theorems for all such algebras are proved. Extension theories of the classes of all semiregular T0-spaces and all N-regular (a notion introduced here) T1-spaces are developed.