Approximation capability of SISO Fuzzy Relational Inference systems based on fuzzy implications

In this work, we show that fuzzy inference systems based on Fuzzy Relational Inference (FRI) with implicative interpretation of the rule base are universal approximators under suitable choice of operations for the rest of the components of the fuzzy system. The presented proofs make no assumption on the form or representations of the considered fuzzy implications and hence show that a much larger class of fuzzy implications other than what is typically considered in the literature can be employed meaningfully in FRIs based on implicative models.

I. Introduction

The term approximate reasoning (AR) refers to methods and methodologies that enable reasoning with imprecise inputs to obtain meaningful outputs [6]. Fuzzy Inference Systems (FIS) form one particular type of AR scheme involving fuzzy sets and are one of the best known applications of fuzzy logic in the wider sense. FIS have many degrees of freedom, viz., the underlying fuzzy partition of the input and output spaces, the fuzzy logic operations employed, the fuzzification and defuzzification mechanism used, etc. This freedom gives rise to a variety of FIS with differing capabilities. The best known types of FIS are the Fuzzy Relational Inference (FRI) systems [12], [25], Similarity Based Reasoning (SBR) systems [11], [18], [5] and the Takagi-Sugeno fuzzy systems [15], [16]. In this work, we focus only on the FRI systems.