Paper:

# Inference Based on α-Cut and Generalized Mean with Fuzzy Tautological Rules

## Kiyohiko Uehara^{*}, Takumi Koyama^{*}, and Kaoru Hirota^{**}

^{*}Ibaraki University, Hitachi 316-8511, Japan

^{**}Tokyo Institute of Technology, Yokohama 226-8502, Japan

Theoretical aspects are provided for inference based on α-cuts and generalized mean (α-GEMII). In order to clarify the basic properties of the inference, fuzzy tautological rules (FTRs) are focused on, which are composed by setting fuzzy sets in consequent parts identical to those in antecedent parts of initially given fuzzy rules. It is mathematically proved that the consequences deduced with FTRs are closer to given facts as the number of FTRs increases. The aspects provided in this paper are appropriate from axiomatic viewpoints and can contribute to interpretability in fuzzy systems constructed with α-GEMII. They are not obtained in conventional methods based on the compositional rule of inference. Simulations are performed by evaluating difference (mean square errors) between given facts and deduced consequences under the condition that convex and symmetric fuzzy sets are given as facts. Their results show that the difference becomes smaller as the number of FTRs increases. Thereby, it is confirmed that α-GEMII has an advantage in the interpretability with respect to FTRs over the conventional methods.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.14, No.1, pp. 76-88, 2010.

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