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1
Content available remote Dynamic Mereotopology: A Point-free Theory of Changing Regions. I. Stable and unstable mereotopological relations
100%
Vakarelov D.
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2010
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tom Vol. 100, nr 1/4
159-180
EN
In this paper we present a point-free theory of Whiteheadean style of space and time. Its algebraic formulation, called dynamic contact algebra (DCA), is a Boolean algebra whose elements symbolized dynamic regions changing in time, with two spatio-temporal mereotopological relations between them: stable and unstable contact. We prove several representation theorems for DCAs, representing them in structures arising from products of contact algebras or from products of topological spaces. We also present a decidable quantifier-free constraint logic for reasoning about stable and unstable mereotopological relations between dynamic regions. We consider the paper as a first step in point-free dynamic mereotopology.
2
Content available remote Arrow Logic with Arbitrary Intersections: Applications to Pawlak's Information Systems
63%
Balbiani P. , Vakarelov D.
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2007
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tom Vol. 75, nr 1-4
1-25
EN
We dedicate this paper to the memory of Zdislaw Pawlak, the founder of rough sets methodology in computer science. A great deal of our scientific work was motivated and influenced by Pawlak's ideas.
3
Content available remote Contact Algebras and Region-based Theory of Space: Proximity Approach - II
63%
Dimov G. , Vakarelov D.
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2006
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tom Vol. 74, nr 2,3
251-282
EN
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.
4
Content available remote Contact Algebras and Region-based Theory of Space: A Proximity Approach - I
63%
Dimov G. , Vakarelov D.
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2006
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tom Vol. 74, nr 2,3
209-249
EN
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.
5
Content available remote Iteration-free PDL with intersection : a complete axiomatization
63%
Balbiani P. , Vakarelov D.
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2001
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tom Vol. 45, nr 3
173-194
EN
This paper is devoted to the completeness issue of PDL0Ç - an iteration-free fragment of Propositional Dynamic Logic with intersection of programs. The trouble with PDL0Ç is that the operation of intersection is not modally definable. Using new techniques connected with rules for intersection and the notions of large and maximal programs, the paper demonstrates that the presented proof theory for PDL0Ç is complete for the standard Kripke semantics of PDL0Ç.
6
Content available remote A Modal Logic for Indiscernibility and Complementarity in Information Systems
63%
Balbiani P. , Vakarelov D.
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2002
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tom Vol. 50, nr 3,4
243-263
EN
In this paper, we study indiscernibility relations and complementarity relations in information systems. The first-order characterization of indiscernibility and complementarity is obtained through a duality result between information systems and certain structures of relational type characterized by first-order conditions. The modal analysis of indiscernibility and complementarity is performed through a modal logic which modalities correspond to indiscernibility relations and complementarity relations in information systems.
7
Content available remote Algorithmic Correspondence and Completeness in Modal Logic. III. Extensions of the Algorithm SQEMA with Substitutions
51%
Conradie W. , Goranko V. , Vakarelov D.
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tom Vol. 92, nr 4
307-343
EN
In earlier papers we have introduced an algorithm, SQEMA, for computing first-order equivalents and proving canonicity of modal formulae. However, SQEMA is not complete with respect to the so called complex Sahlqvist formulae. In this paper we, first, introduce the class of complex inductive formulae, which extends both the class of complex Sahlqvist formulae and the class of polyadic inductive formulae, and second, extend SQEMA to SQEMAsub by allowing suitable substitutions in the process of transformation. We prove the correctness of SQEMAsub with respect to local equivalence of the input and output formulae and d-persistence of formulae on which the algorithm succeeds, and show that SQEMAsub is complete with respect to the class of complex inductive formulae.
8
Content available remote Modal Logics for Region-based Theories of Space
51%
Balbiani P. , Tinchev T. , Vakarelov D.
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2007
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tom Vol. 81, nr 1-3
29-82
EN
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.
9
Content available remote A Modal Logic for Pawlak's Approximation Spaces with Rough Cardinality n
51%
Balbiani P. , Iliev P. , Vakarelov D.
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2008
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tom Vol. 83, nr 4
451-464
EN
The natural modal logic corresponding to Pawlak's approximation spaces is S5, based on the box modality [R]A (and the diamond modality áRn~A=?[R]?A), where R is the corresponding indiscernibility relation of the approximation space S=(W,R). However the expressive power of S5 is too weak and, for instance, we cannot express that the space S has exactly n equivalence classes ( we say that S is roughly-finite and n is the rough cardinality of S). For this reason we extend the modal logic S5 with a new box modality [S]A, where S is the complement of R i.e. the discernibility relation of W. We propose a complete axiomatization, in this new language, of the logic ROUGHn corresponding to the class of approximation spaces with rough cardinality n. We prove that the satisfiability problem for ROUGHn is NP-complete.
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