New Fuzzy Modus Ponens and Tollens based on Least Common Multiple

Abstract The fuzzy approximate reasoning is one of the important research branches in the uncertainty inference of Artificial Intelligence (AI) and Computational Intelligence (CI). This fuzzy approximate reasoning consists of two parts, i.e., fuzzy modus ponens (FMP) and fuzzy modus tollens(FMT). FMP and FMT have an important significance in modeling of the human thinking process, design and development of a lot of several intelligence systems and etc. In this paper we proposed a novel original method of fuzzy approximate reasoning that can open a new direction of research in the uncertainty inference of AI and CI. Firstly, our proposed method is based on distance measure, concretely, an extended distance measure by the least common multiple(LCM). The proposed fuzzy approximate reasoning method LCM based on a least common multiple is an inference one for the SISO fuzzy system with discrete fuzzy set vectors of equal or different dimensions between the antecedent and consequent of fuzzy rule. We call it LCM method. LCM method is consisted of two parts, i.e., FMP–LCM, and FMT–LCM. Secondly, in this paper we proposed and proved four theorems with respect to the information loss. In other words we pointed out that our proposed method LCM has no information loss, and then the previous methods, i.e., Compositional Rule of Inference(CRI), Triple Implication Principle(TIP), Quintuple Implication Principle(QIP) and Approximate Analogical Reasoning Scheme(AARS) have some information loss compared with our proposed method. Thirdly, we compared the reductive properties for the five fuzzy reasoning methods, i.e., CRI, TIP, QIP, AARS, and our proposed LCM with respect to FMP and FMT. The theoretical and experimental results highlight that the proposed fuzzy approximate reasoning method LCM is comparatively more clear and effective, and in accordance with human thinking than the previous fuzzy reasoning methods. Fourthly, in this paper we proposed and proved six theorems with respect to fuzzy controllability of the several fuzzy reasoning methods based on the fuzzy relation and our proposed method based on LCM. We pointed out our proposed method LCM has not only higher reductive property but also higher fuzzy controllability than the previous fuzzy reasoning methods.