The representation problem for additive fuzzy systems

We produce representations algebraically and present a criterion based on partial derivatives. We prove that all bounded scalar functions of one variable have a fuzzy representation and that functions of two or more variables need not. We show that where a representation exists the number of rules in the corresponding fussy system is linear in the number of inputs. Also, use of bipolar membership functions can expand the domain of definition for a representation without increasing computational load. These results help explain the abilities and limitations of fuzzy systems to reproduce exact responses. Where suitable representations exist a substantial reduction in the number of rules is possible