Fuzzy inference has played a significant role in many applications. Although the simplified fuzzy inference method is currently mostly used, the problem is that the number of fuzzy rules becomes very huge and so the setup and adjustment of fuzzy rules become difficult. On the other hand, Yubazaki et al. have proposed a "single input rule modules connected fuzzy inference method" (SIRMs method) whose final output is obtained by summarizing the product of the importance degrees and the inference results from single input fuzzy rule module. Seki et al. have shown that the simplified fuzzy inference method and the SIRMs method are equivalent when the sum of diagonal elements in rules of the simplified fuzzy inference method is equal to that of cross diagonal elements. This paper clarifies the conditions for the infimum and supremum of the fuzzy inference method using the single input type fuzzy inference method, from the view point of fuzzy inference.

Yubazaki et al. have proposed "single input rule modules connected type fuzzy reasoning method" (SIRMs method, for short) whose final output is obtained by summarizing the product of the importance degrees and the inference results from single input fuzzy rule module. Another type of single input type fuzzy reasoning method proposed by Hayashi et al. (we call it "Single Input Connected fuzzy reasoning method" (SIC method, for short) in this paper) uses rule modules to each input item as well as SIRMs method. We expect that inference results of SIRMs method and SIC method have monotonicity if the antecedent parts and consequent parts of fuzzy rules in SIRMs rule modules have monotonicity. However, this paper points out that even if fuzzy rules in SIRMs rule modules have monotonicity, the inference results do not necessarily have monotonicity. Moreover, it clarifies the conditions for the monotonicity of inference results by SIRMs method and SIC method.

The single input rule modules connected fuzzy inference method (SIRMs method) by Yubazaki et al. can decrease the number of fuzzy rules drastically in comparison with the conventional fuzzy inference methods. Moreover, Seki et al. have proposed a functional-type SIRMs method which generalizes the consequent part of the SIRMs method to function. However, these SIRMs methods can not be applied to XOR (Exclusive OR). In this paper, we propose a "kernel-type SIRMs method" which uses the kernel trick to the SIRMs method, and show that this method can treat XOR. Further, a learning algorithm of the proposed SIRMs method is derived by using the steepest descent method, and compared with the one of conventional SIRMs method and kernel perceptron by applying to identification of nonlinear functions, medical diagnostic system and discriminant analysis of Iris data.