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Automated Generation of Logical Constraints on Approximation Spaces Using Quantifier Elimination

Doherty P., Szałas A.

Fundamenta Informaticae

2013

Vol. 127, nr 1-4

135--149

This paper focuses on approximate reasoning based on the use of approximation spaces. Approximation spaces and the approximated relations induced by them are a generalization of the rough set-based approximations of Pawlak. Approximation spaces are used to define neighborhoods around individuals and rough inclusion functions. These in turn are used to define approximate sets and relations. In any of the approaches, one would like to embed such relations in an appropriate logical theory which can be used as a reasoning engine for specific applications with specific constraints. We propose a framework which permits a formal study of the relationship between properties of approximations and properties of approximation spaces. Using ideas from correspondence theory, we develop an analogous framework for approximation spaces. We also show that this framework can be strongly supported by automated techniques for quantifier elimination.

Double Approximation and Complete Lattices

Haruna T., Gunji Y-P.

Fundamenta Informaticae

2011

Vol. 111, nr 1

1-14

We explore lattice theoretic aspects in rough set theory in terms of the duality between algebra and representation. Approximation spaces are dual to complete atomic Boolean algebras in the sense that there is an adjunction between corresponding suitable categories. This is an analogy with the adjunction between the category of topological spaces and the opposite of the category of frames in pointless topology. In this paper we consider a generalization of approximation spaces called double approximation systems. A double approximation system consists of a set and two equivalence relations on it. We construct an adjunction generalizing this concept for approximation spaces. To achieve this goal, we first introduce a natural generalization of complete atomic Boolean algebras called complete prime lattices. Then we select double approximation systems that can be dual to complete prime lattices and prove the adjunction.