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

In many practical applications of the fuzzy logic controllers, the linguistic term 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 rulebase. This case the antecedent and consequent fuzzy partitions (formed by these primary fuzzy sets) can be described by vague environments (based on the similarity or indistinguishability of the elements [2]), Using the concept of vague environment characterized by scaling functions [2] instead of the linguistic term fuzz! 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 (the ''density information'') are described by the scaling function, This case 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 rulebase), 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 rulebase relation [4,5]. 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 of the classical Compositional Rule of Inference (CRI) methods in practical applications [6]. In this paper two methods of approximate fuzzy reasoning based on interpolation in the vague environment of the fuzzy rulebase, and comparison of these methods to the classical CRI will be introduced.