Fuzzy inference based on families of alpha -level sets

Abstract

This paper proposes a fuzzy-inference method in which fuzzy sets are defined by the families of their a-level sets, based on the resolution identity theorem. This method stands in contrast to the conventional fuzzy-inference method, in which fuzzy sets are defined by membership functions. It has the following advantages over cohventional methods: (1) it studies the characteristics of fuzzy inference, in particular the input-output relations of fuzzy inference; (2) it provides fast inference operations and requires less memory capacity for fuzzy sets defined in universes of discourse with a large number of elements; (3) it easily interfaces with two-valued logic; and (4) it effectively matches with systems that include fuzzy-set operations based on the extension principle.This paper first compares fuzzy sets defined by the families of their a-level sets with those defined by membership functions in terms of processing time and required memory capacity in fuzzy logic operations. It then derives the fuzzy-inference method and proves important propositions of fuzzy-inference operations. It also presents some examples of inference operations by the proposed method, and considers fuzzy-inference characteristics and computational efficiency from a-level-set-based fuzzy inference. This paper finally concludes with a brief discussion. (b) Fig. 1. Fuzzy-set definitions (the symbol " " shows an element in the discrete space *Yd). (a) Fuzzy-set definition by a membership function. (b)Fuzzy-set definition by the family of n-level sets.

Hierarchical fuzzy Petri nets and ?-level sets inference

“…The propositions proved in Ref. 18 provide the referred definitions and offer the appropriate theoretical background to the approach developed in the present work. It is assumed that all fuzzy sets as convex fuzzy sets and that the inverted truth value T X is an S function.…”

“…The net introduced there performs the operations for each ␣-level set, as does the algorithm presented in Ref. 18 and reviewed in the preceding section. The complete inference process, though, requires a family of nets like the one described.…”

“…An alternative fuzzy inference method in which fuzzy sets are defined by the families of their ␣-level sets was proposed in Ref. 18. It has a number of advantages over membership-function-based fuzzy inference methods.…”

“…The reader interested in a discussion on these and other advantages is referred to Ref. 18. The method for fuzzy inference adopted here consists of performing fuzzy inferences in the truth spaces using ␣-level sets.…”

“…17, with the modeling of a different approach to inference conducted in the truth space, that represents fuzzy sets by families of ␣-level sets, proposed in Ref. 18. The ␣-level sets inference offers some advantages over the fuzzy sets based inference, including computational efficiency in applications where the number of elements in the universe of discourse is much larger than the number of membership grade levels.…”