Fuzzy inference based on α-cuts and generalized mean: Relations between the methods in its family

This paper clarifies the relations between fuzzy inference methods based on α-cuts and the generalized mean. The basis of conventional fuzzy inference has been the compositional rule of inference. The conventional inference methods cannot prove the convexity of deduced consequences. Moreover, they tend to deduce consequences with excessively large fuzziness and excessively small specificity. In order to solve the problems, α-GEM (α-level-set and generalized-mean-based inference) has been proposed, which can prove to deduce consequences in convex forms and can control the fuzziness and specificity of consequences. The scheme of α-GEM leads to α-GEMII (α-level-set and generalized-mean-based inference with the proof of two-sided symmetry of consequences). α-GEMII proves the symmetricity of consequences under some axiomatically derived conditions. Since it has been proposed, α-GEMII has played a central role as the basis of other inference methods. This paper introduces the inference methods originated from α-GEM and clarifies the relations between them in order to inspire research to create other related methods.