Local relation mapping theory of fuzzy reasoning

Many authors discussed the interpolation mechanism of fuzzy logic system applied in control. The input and output of fuzzy control system are always restrained in (fuzzy) numbers-associated with fuzzification and defuzzification procedure. In artificial intelligence area we need use common fuzzy sets to represent vague and fuzzy concept. In this paper, we propose a "local relation mapping" theory, which is a corresponding result of interpolation mechanism of fuzzy logical system without fuzzification and defuzzification. This local relation mapping property holds in Mamdani algorithm (Max-Min fuzzy reasoning model). We find that each CRI fuzzy reasoning method based on the multi-valued logic implication operator has an approximate Mamdani algorithm. The fuzzy relation matrix of CRI fuzzy inference method based on the multivalued logic implication operator has an enveloping surface of Mamdani algorithm. So most CRI fuzzy reasoning methods have local relation mapping property approximately