A Correspondence Framework between Three-Valued Logics and Similarity-Based Approximate Reasoning

This paper focuses on approximate reasoning based on the use of similarity spaces. Similarity spaces and the approximated relations induced by them are a generalization of the rough set-based approximations of Pawlak [17, 18]. Similarity spaces are used to define neighborhoods around individuals and 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 logic 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 approximate relations, similarity spaces and three-valued logics. Using ideas from correspondence theory for modal logics and constraints on an accessibility relation, we develop an analogous framework for three-valued logics and constraints on similarity relations. In this manner, we can provide a tool which helps in determining the proper three-valued logical reasoning engine to use for different classes of approximate relations generated via specific types of similarity spaces. Additionally, by choosing a three-valued logic first, the framework determines what constraints would be required on a similarity relation and the Address for correspondence: Department of Computer and Information Science, Linköping University, Linköping, Sweden ∗Supported in part by the Wallenberg Foundation. †Supported in part by the MNiI grant 3 T11C 023 29. 2 P. Doherty, A. Szałas / A Correspondence Framework between Three-Valued Logics and Approximate Reasoning approximate relations induced by it. Such information would guide the generation of approximate relations for specific applications.