SISO fuzzy relational inference systems based on fuzzy implications are universal approximators

In this work, we show that single input single output (SISO) 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 other 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. A concept of Weak Coherence is proposed, which plays an important role in enlarging the class of fuzzy implications that can be considered.