Lattice-based paraconsistent logic (original) (raw)

Contact algebras and region-based theory of space: A proximity approach-I

Fundamenta Informaticae, 2006

This paper is the second part of the paper . 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 T 0 -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 T 0 -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.

Dynamic logics of the region-based theory of discrete spaces

Journal of Applied Non-Classical Logics, 2007

We introduce Boolean proximity algebras as a generalization of Efremovič proximities which are suitable in reasoning about discrete regions. Following Stone's representation theorem for Boolean algebras, it is shown that each such algebra is isomorphic to a substructure of a complete and atomic Boolean proximity algebra.

Algebraisable versions of predicate topological logic

Motivated by questions like: which spatial structures may be characterized by means of modal logic, what is the logic of space, how to encode in modal logic different geometric relations, topological logic provides a framework for studying the confluence of the topological semantics for sfS4\sf S4sfS4 modalities, based on topological spaces rather than Kripke frames. Following research initiated by Sgro, and further pursued algebraically by Georgescu, we prove an interpolation theorem and an omitting types theorem for various extensions of predicate topological logic and Chang's modal logic. Our proof is algebraic addressing expansions of cylindric algebras using interior operators and boxes, respectively. Then we proceed like is done in abstract algebraic logic by studing algebraisable extensions of both logics; obtaining a plethora of results on the amalgamation property for various subclasses of their algebraic counterparts, which are varieties. Notions like atom-canonicity and com...

A mereotopology based on sequent algebras

Journal of Applied Non-Classical Logics, 2017

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 (part-of, overlap, underlap) and mereotopological relations (external contact, tangential part-of, nontangential part-of, self-connectedness), 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 A b 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 pointbased models of space. As a by-product, we show how one logical relation in nature notion, Tarski consequence relation, may have also certain spatial (mereotopological) meaning.

Concrete models of computation for topological algebras

Theoretical Computer Science, 1999

A concrete model of computation for a topological algebra is based on a representation of the algebra made from functions on the natural numbers. The functions computable in a concrete model are computable in the representation in the classical sense of the Chruch-Turing Thesis.

A necessary relation algebra for mereotopology

Studia Logica, 2001

The standard model for mereotopological structures are Boolean subalgebras of the complete Boolean algebra of regular closed subsets of a nonempty connected regular T 0 topological space with an additional "contact relation" C defined by

Complementation in representable theories of region-based space

Abstract Through contact algebras we study theories of mereotopology in a uniform way that clearly separates mereological from topological concepts. We identify and axiomatize an important subclass of closure mereotopologies (CMT) called unique closure mereotopologies (UCMT) whose models always have orthocomplemented contact algebras (OCA) an algebraic counterpart.