Differently Implicational Bandler–Kohout Subproduct Method

The Bandler–Kohout subproduct (BKS) method acts as one of the two representative fuzzy relational inference (FRI) strategies. Observing the BKS method using constraint modeling, two fuzzy implications, respectively, produce expression to the factors of inference mechanism and rule base. However, these two factors normally reflect different connotations from the perspectives of artificial intelligence applications and logical meaning. Enlightened by such idea, in this study, we propose and investigate the differently implicational BKS (DBKS) method. Initially, main properties of DBKS are validated. The reversibility and interpolativity of DBKS are proved under certain conditions. The equivalent relationship is verified between interpolativity and continuity for DBKS. The robustness of DBKS is confirmed from both the similarity and the extensional hull. Posteriorly, the computational performance of DBKS is analyzed. In DBKS, the preservation of the indistinguishability holds for input fuzzy sets, and it is proved that the first-aggregate-then-infer (FATI) reasoning strategy of DBKS is equivalent to the first-infer-then-aggregate (FITA) one. To improve the computational efficiency, the hierarchical DBKS method is presented. In addition, the fuzzy system is established on the strength of the DBKS method, the singleton fuzzifier and the centroid defuzzifier. Its response function is analyzed and a universal approximator is built by the fuzzy system via DBKS. At the end, we compare the results of DBKS with BKS by virtue of two examples in affective computing. It is discovered that DBKS can create superior forms of FRI in comparison to those produced by BKS.