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1
Content available remote Mereotopologies with Predicates of Actual Existence and Actual Contact
Vakarelov D.
Fundamenta Informaticae
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2017
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Vol. 156, nr 3/4
413--432
EN
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.
2
Content available remote Dynamic Mereotopology: A Point-free Theory of Changing Regions. I. Stable and unstable mereotopological relations
Vakarelov D.
Fundamenta Informaticae
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2010
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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.
3
Content available remote Algorithmic Correspondence and Completeness in Modal Logic. III. Extensions of the Algorithm SQEMA with Substitutions
Conradie W., Goranko V., Vakarelov D.
Fundamenta Informaticae
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2009
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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.
4
Content available remote A Modal Logic for Pawlak's Approximation Spaces with Rough Cardinality n
Balbiani P., Iliev P., Vakarelov D.
Fundamenta Informaticae
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2008
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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.
5
Content available remote Modal Logics for Region-based Theories of Space
Balbiani P., Tinchev T., Vakarelov D.
Fundamenta Informaticae
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2007
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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.
6
Content available remote Arrow Logic with Arbitrary Intersections: Applications to Pawlak's Information Systems
Balbiani P., Vakarelov D.
Fundamenta Informaticae
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2007
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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.
7
Content available remote Contact Algebras and Region-based Theory of Space: Proximity Approach - II
Dimov G., Vakarelov D.
Fundamenta Informaticae
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2006
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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.
8
Content available remote Contact Algebras and Region-based Theory of Space: A Proximity Approach - I
Dimov G., Vakarelov D.
Fundamenta Informaticae
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2006
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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.
9
Content available remote A Modal Logic for Indiscernibility and Complementarity in Information Systems
Balbiani P., Vakarelov D.
Fundamenta Informaticae
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2002
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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.
10
Content available remote Iteration-free PDL with intersection : a complete axiomatization
Balbiani P., Vakarelov D.
Fundamenta Informaticae
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2001
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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.
11
Content available remote On Scott consequence systems
Dimov G., Vakarelov D.
Fundamenta Informaticae
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1998
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Vol. 33, nr 1
43-70
EN
The notion of Scott consequence system (briefly, S-system) was introduced by D.Vakarelov in an analogy to a similar notion given by D. Scott. In part one of the paper we study the category Ssyst of all S-systems and all their morphisms. We show that the category DLat of all distributive lattices and all lattice homomorphisms is isomorphic to a reflective full subcategory of the category Ssyst. Extending the representation theory of D. Vakarelo for S-systems in P-systems, we develop an isomorphism theory for S-systems and for Tarski consequence systems. In part two of the paper we prove that the separation theorem for S-systems is equivalent in ZF to some other separation principles, including the separation theorem for filters and ideals in Boolean algebras and separation theorem for convex sets in convexity spaces.
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