Towards a Theory of a Fuzzy Rule Base Interpolation

It is well known that a fuzzy rule base is a characterization of a partially given mapping (fuzzy function) between fuzzy universes. For practical applications, it is desirable to interpolate that function in order to compute its values at points (fuzzy or crisp) other than fuzzy sets (nodes) in antecedents of the rule base. Moreover, interpolation requires that in the case of coincidence between a point and a node, a computation method should produce a fuzzy set that coincides with the corresponding one in the consequence part of the rule base. Moreover, by Lotfi Zadeh, interpolation is a fuzzy logic inference engine. In his early papers [14, 15], he proposed the Generalized Modus Ponens (GMP) in the form of a formal scheme of inference and its possible realization in the form of the Compositional Rule of Inference (CRI). Both schemes were intensively investigated by many authors which resulted in a collection of methods related to a fuzzy rule base interpolation. Let us briefly overview main contributions to the field. In our opinion, all contributions can be classified into the following four groups.