Closeness in similarity-based reasoning with an interpolation condition

This study considers an approximate reasoning scheme where the knowledge base is a set of fuzzy IF–THEN rules and the inference mechanism is characterized in terms of closeness. We propose the interpretation of this scheme, which uses Ruspini's theory of conditional consistency and implication measures, and its adaptation to fuzzy sets. In the proposed interpretation, we find the necessary and sufficient conditions to ensure that the computation of the conclusion fulfills the interpolation condition. We show that the inference mechanism is equivalent to the compositional rule of inference in the form of inf⁡→inf⁡→ composition. Finally, we construct a particular interpretation where all of the considered requirements are satisfied and the inference operator reduces to a piecewise linear interpolation function.