From Fuzzy Rule-Based Systems to Granular Fuzzy Rule-Based Systems: A Study in Granular Computing

Abstract

In the study, we introduce a concept of granular fuzzy rule-based systems, offer a motivation behind its emergence and elaborate on ensuing algorithmic developments. It is shown that the granularity of the fuzzy rules is directly associated with a reduction (compression) process in which the number of rules becomes reduced in order to enhance the readability (transparency) of the resulting rule base. The retained rules are made more abstract (general) by admitting a granular form of the fuzzy sets forming their antecedents. In other words, while the original rules read as “if A _i then B _i ” their reduced subset comes in the form “if G(A _i ) then B _i ” with G(.) denoting a certain granular extension of the original fuzzy set (which can be realized e.g., in the form of interval-valued fuzzy sets, fuzzy sets of type-2 or rough – fuzzy sets). It is shown that the optimization of the reduced set of rules is realized through an optimal distribution of information granularity among fuzzy sets forming the conditions of the reduced rules. In particular, it is shown that the distribution of information granularity, being regarded as an important design asset, is realized through a minimization of a certain objective function quantifying how well the granular fuzzy set formed by the reduced rule set represents (covers) all rules. In the sequel, we introduce an idea of a granular representation of results of inferences realized in fuzzy rule-based systems.