Approximate fuzzy reasoning based on interpolation in the vague environment of the fuzzy rulebase

In many practical applications of fuzzy logic controllers, fuzzy sets are used to describe a vague value, a value and a kind of density information on the antecedent and consequent universes of the fuzzy rule base. In this case the antecedent and consequent fuzzy partitions (formed by these primary fuzzy sets) can be described by vague environments. Using the concept of vague environment characterized by scaling functions instead of the linguistic term fuzzy sets gives a simple way for fuzzy approximate reasoning. Comparing the description of a universe given by a fuzzy partition to the way of using the concept of vague environment, we can say that the linguistic terms of the fuzzy partition are crisp points in the vague environment, while the shapes of the fuzzy sets (density information) are described by the scaling function. The primary fuzzy sets of the antecedent and the consequent parts of the fuzzy rules can be characterised by crisp points in their vague environments, so the fuzzy rules themselves are points in their vague environment too (in the vague environment of the fuzzy rule base). It means, that the question of approximate fuzzy reasoning can be reduced to the problem of interpolation of the rule points in the vague environment of the fuzzy rule base relation. In other words, using the concept of vague environment, in most cases we can build approximate fuzzy reasoning methods simple enough to be a good alternative to the classical Compositional Rule of Inference (CRI) methods in practical applications. In this paper two methods of approximate fuzzy reasoning based on interpolation in the vague environment of the fuzzy rule base, and a comparison of these methods to the classical CRI are introduced