Universal Quintuple Implicational Algorithm: A Unified Granular Computing Framework

In the field of fuzzy inference, the universal triple I algorithm integrated the CRI (Compositional Rule of Inference) algorithm with the triple I algorithm. Later the triple I algorithm was generalized to the QIP (quintuple implication principle) algorithm. Whether the QIP algorithm and the CRI algorithm can be unified has become an interesting question. Therefore, in this study, a fuzzy inference scheme referred to as the universal quintuple implicational (UQI) algorithm is proposed. First, we establish a unified granular computing framework with the UQI algorithm, which is a generalization of the QIP algorithm, the CRI algorithm as well as the universal triple I algorithm. The optimal UQI solutions derived from the fundamental principle of determining inference results are obtained for the FMP (fuzzy modus ponens) problem, in which some specific solutions are also given. Second, the reversible property of the UQI algorithm is verified for FMP, while aiming at the metric derived from the biresiduum operation, the robustness of the UQI algorithm is validated. Third, under the environment of multiple rules, two general cases of FITA (First-Inference-Then-Aggregate) and FATI (First-Aggregate-Then-Inference) are constructed for the UQI algorithm. The corresponding equivalence relation between continuity and interpolation is analyzed. Fourth, the fuzzy system is established based on the UQI algorithm, the singleton fuzzier as well as the centroid defuzzier. Its response ability is analyzed and it is shown that such fuzzy system is a universal approximator. Lastly, we compare the results of the UQI algorithm with the QIP algorithm by five examples for FMP. It is found that the UQI algorithm is able to acquire more and better forms of the fuzzy inference in contrast with the QIP algorithm.