On Fuzzy Implication Relations

For the last decade, fuzzy controllers have been applied with great success to a variety of domains. Nevertheless the underlying theoretical framework still remains unsatisfactory. One of the key questions still open is concerned with approximate reasoning. How should we determine the eeect of applying the fuzzy rule IF x is ~ A THEN y is ~ B] to ~ A 0 with ~ A 0 , ~ A, and ~ B being fuzzy sets? The idea is to use sup{min composition ~ A 0 ~ R ~ A! ~ B] to get the conclusion ~ B 0. But which fuzzy relation or which class of fuzzy relations represents the semantics of the given rule ~ A ! ~ B] best? We claim that the derivation of a proper implication relation has to consider two diierent aspects at the same time: it should fullll our intuitive expectation and obey certain mathematical laws as well. From a fuzzy rule based system we expect a both intuitively plausible and mathematically sound behaviour. Therefore neither a solely intuitive nor a solely mathematical motivated implication relation solves our problem. In this paper we will present a derivation suuciently dealing with both aspects and resulting in a special class of fuzzy implication relations.