articles

The effectiveness of operations on fuzzy presented data (fuzzy sets and relations) in the tasks of modeling under uncertainty conditions is largely determined by the complexity of fuzzy computing, which, as a rule, is based on L. Zadeh’s Extension Principle using a crisp function. Operations on membership functions of fuzzy sets and relations, based on the Extension Principle, are equivalent to interval operations on their α-levels, which are much simpler to computing. The proof of these results is based on Theorem H. Nguyen. The implementation of fuzzy computing is generalized to the case of a fuzzy mapping between membership functions of fuzzy sets and relations based on the Generalized Fuzzy Expansion Principle. However, the problem of representing and proving the possibility of implementing the Generalized Fuzzy Expansion Principle in the transition from operations on membership functions of fuzzy sets and relations to alternative operations on their α-levels has not yet been solved. This paper describes the implementation of a fuzzy mapping between fuzzy sets using a fuzzy relation, based on an approach to the fuzzy composition of membership functions of fuzzy sets and fuzzy relations. An alternative approach to the interpretation of the Generalized Fuzzy Expansion Principle is proposed, based on the fuzzy composition of characteristic functions of α-levels of fuzzy sets and relations, as well as the method of fuzzy computing based on this principle. The paper proves the equivalence of the results of fuzzy composition based on the two approaches considered, as well as a comparative assessment of the complexity of computing and the degree of parallelism in their implementation. The use of the proposed Generalized Fuzzy Expansion Principle and the fuzzy computing method based on it makes it possible to significantly simplify the implementation of fuzzy computing by using non-numerical (logical) operations on the values of characteristic functions of α-levels of fuzzy sets and relations instead of computational operations on the real values of the membership functions of these fuzzy sets and relations.