Approximate reasoning by linear rule interpolation and general approximation

The problem of sparse fuzzy rule bases is introduced. Because of the high computational complexity of the original compositional rule of inference (CRI) method, it is strongly suggested that the number of rules in a final fuzzy knowledge base is drastically reduced. Various methods of analogical reasoning available in the literature are reviewed. The mapping style interpretation of fuzzy rules leads to the idea of approximating the fuzzy mapping by using classical approximation techniques. Graduality, measurability, and distance in the fuzzy sense are introduced. These notions enable the introduction of the concept of similarity between two fuzzy terms, by their closeness derived from their distance. The fundamental equation of linear rule interpolation is given, its solution gives the final formulas used for interpolating pairs of rules by their α-cuts, using the resolution principle. The method is extended to multiple dimensional variable spaces, by the normalization of all dimensions. Finally, some further methods are shown that generalize the previous idea, where various approximation techniques are used for the α-cuts and so, various approximations of the fuzzy mapping R: X → Y.