Distance Measures Induced by Finite Approximation Spaces and Approximation Operators

Issue title: Concurrency Specification and Programming (CS&P)

Article type: Research Article

Authors: Wolski, Marcin

Affiliations: Department of Logic and Methodology of Science, Maria Curie-Skłodowska University, Poland. [email protected]

Note: [] Address for correspondence: Department of Logic and Methodology of Science, Maria Curie-Skłodowska University, Pl.Marii Curie-Skłodowskiej 4, 20-032 Lublin, Poland

Abstract: In this paper we study metric properties of finite approximation spaces and approximation operators from Rough Set Theory. In the first part of the article we examine finite approximation spaces and finite approximation topological spaces regarded as particular instances of two basic types of information structures from the framework of Information Quanta: information quantum relational systems and property systems, respectively. In the second part of the paper is the Marczewski-Steinhaus metric discussed as a certain distance of sets defined with respect to the approximation operators. We propose two types of á la Marczewski-Steinhaus distance functions: the first type is based on the lower approximation operator; the second one is based on the upper approximation operator. These types can be defined with respect to both finite approximation spaces (information quantum relational systems) and finite approximation topological spaces (property systems), giving us four distancemeasure functions. In order to define a distance of sets which preserves their lower and upper approximations, one can take the sum of two respective functions.

Journal: Fundamenta Informaticae, vol. 85, no. 1-4, pp. 497-512, 2008