On The Fuzzy Balls And The Applications In Fuzzy Logic Systems

Fuzzy logic systems are one of the important research fields in the applications of fuzzy mathematics. Fuzzy logic systems possess not only simple and systematical characteristics in the structures, but also the capability of extracting information from the literal language by using fuzzy logics. In applications, fuzzy logic systems are utilized widely to obtain the solutions of the problems with literal language in engineering. Recently, the theory problems of fuzzy logic systems are distributed mainly over the constructing fuzzy rules, finding fuzzy logics, interpretability and output approximation. How to construct satisfactory fuzzy logic systems is the core of the problems and the goal of the theory research.Generally speaking, fuzzy logic systems are employed to be the approximators, which is one of the important characteristics of fuzzy logic systems. The precision and interpretability are two important aspects reflected from fuzzy logic systems. However, the obtained universal theorems show that it becomes more challenging as producing the fuzzy logic systems with the precision and interpretability. About this case, there may be two causes, one is that increase in the precision results in the increase in the number of fuzzy rules; another is that the increase in the number of fuzzy rules results in decrease in interpretability. Therefore, there may be two ways to be chosen for the solutions of the problems. One is changing the form of fuzzy rules, for instance, the form of rules with one dimension is changed into the vector form of rules; another is to select suitable membership functions for describing the language variables.In this paper, first the concept of fuzzy ball is proposed to describing the language variable "near a point" and similar others. By expanding the concept of fuzzy balls and obtaining their properties, it is proved that the capability of describing the language variable "near a point" with fuzzy balls in geometry. Then the fuzzy logic systems with vector rules are constructed by using fuzzy balls. Since the form of vector rules does not happen in the tensor product with mono-variables, the fuzzy logic systems with vector rules refrain maybe from the dimension disaster. Meanwhile, the results show that the fuzzy logic systems with vector rules possess not only interpretability but also the universal approximation.