Arithmetic Fuzzy Models

It is well known that a fuzzy rule base can be interpreted in different ways. From a logical point of view, the conjunctive interpretation is preferred, while from a practical point of view, the disjunctive interpretation has been dominantly present. Each of these interpretations results in a specific fuzzy relation that models the fuzzy rule base. Basic interpolation requirements naturally suggest a corresponding inference mechanism: the direct image for the conjunctive interpretation and the subdirect image for the disjunctive interpretation. Interpolation then corresponds to solvability of some system of fuzzy relational equations. In this paper, we show that other types of fuzzy relations, which are closely related to Takagi–Sugeno (T–S) models, are of major interest as well. These fuzzy relations are based on addition and multiplication only, from which we get the name arithmetic fuzzy models. Under some mild requirements, these fuzzy relations turn out to be solutions of the same systems of fuzzy relational equations. The impact of these results is both theoretical and practical: There exist simple solutions to systems of fuzzy relational equations, other than the extremal solutions that have received all the attention so far, which are, moreover, easy to implement.