Approximate hierarchical fuzzy reasoning based on the law of importation

Abstract

Fuzzy reasoning has been widely used in applied sciences such as fuzzy control, artificial intelligence, image processing, data mining, machine learning, decision-making, prediction, classification and so on. However, one has to face with the rule explosion in the application of fuzzy reasoning. As an efficient tool to deal with the rule explosion of fuzzy system, the law of importation equation I(A(x,y),z)=I(x,I(y,z)) with aggregation functions (LIA) can be utilized to construct the fuzzy hierarchical system. In order to enhance the ability of this fuzzy hierarchical system in the actual environments, this paper aims mainly to construct a hierarchical fuzzy system involved in the fuzzy implications that approximately but not strictly satisfy (LIA) to approximate a fuzzy system with arbitrary accuracy. Concretely, we first define Ulam stability of (LIA). And the Ulam stability of (LIA) is then studied, that is, the fuzzy implications are sought to approximately fulfill (LIA) for some given aggregation functions. Further, a hierarchical fuzzy system based on the Ulam stability of (LIA) is constructed to approximate a fuzzy system with arbitrary accuracy. Finally, an example is presented to substantiate our theoretical discussion.