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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].

Mereotopologies with Predicates of Actual Existence and Actual Contact

Vakarelov D.

Fundamenta Informaticae

2017

Vol. 156, nr 3/4

413--432

We discuss in this work the importance of some predicates of ontological existence in mereology and in mereotopology especially for systems incorporating time. Tarski showed that mereology can be identified in some sense with complete Boolean algebras with zero 0 deleted. If one prefers to use only first-order language, the first-order theory for Boolean algebras can be used with zero included for simplicity. We extend the language of Boolean algebra with a one-place predicate AE(x), called ”actual existence” and satisfying some natural axioms. We present natural models for Boolean algebras with predicate AE(x) motivating the axioms and prove corresponding representation theorems. Mereotopology is considered as an extension of mereology with some relations of topological nature, like contact. One of the standard mereotopological systems is contact algebra, which is an extension of Boolean algebra with a contact relation C, satisfying some simple and obvious axioms. We consider in this paper a natural generalization of contact algebra as an extension of Boolean algebra with the predicate AE(x) and a contact relation Cα called ”actual contact”, assuming for them natural axioms combining Cα and AE. Relational and topological models are proposed for the resulting system and corresponding representation theorems are proved. I dedicate this paper to my teacher in logic Professor Helena Rasiowa for her 100-th birth anniversary. Professor Rasiowa showed me the importance of algebraic and topological methods in logic and this was her main influence on me.