The Equivalence Between CRI And The Full Implication Triple I Method In Fuzzy Reasoning

Based on the fuzzy sets theory, Zadeh firstly proposed the compositional rule of inference(abbreviated to CRI algorithm) as a fuzzy reasoning method in 1973. Since then, fuzzy control technology based on the method of fuzzy reasoning is widely used in industry areas with achievements that have attracted worldwide attention. Nevertheless, fuzzy reasoning is condemned as lacking strict logic basis. Professor Wang Guo-jun firstly pointed out the imperfections of CRI algorithm, and proposed the famous full implication triple I method. Full implication triple I method improves the traditional CRI algorithm effectively, and brings fuzzy reasoning within the framework of logical semantic implication. Nowadays, there's a large amount of effort to discuss and explore full implication triple I method in fuzzy logic, fuzzy reasoning and fuzzy control. However, contrasting with CRI algorithm, full implication triple I method is more complicated and has no uniform form of its solutions.In order to overcome the imperfections of both CRI algorithm.and full implication triple I method, this paper investigates when the inference obtained from CRI algorithm and full implication triple I method become equivalent. Toward this end, we consider two significant properties in classical logic in the framework of fuzzy logicwhere I is fuzzy implication and T is t-norm,x,y,z[0,1].We investigate the conditions under which the four most established and well-studied classes of fuzzy implications, S-, R-, QL- and D-implications, satisfy both Ordering property and Law of importation. Based on the conclusions, we explore the equivalence between CRI algorithm and full implication triple I method in FMP model and FMT model and propose the unified full implication algorithms of fuzzy reasoning.The main conclusions in this paper are:1.We proposed and verified the necessary and sufficient conditions under which both Ordering property and Law of importation holds for S-,R-implications and some specific families of QL-, D-implications.2.We proposed and verified the sufficient condition under which the equivalence between CRI and full implication triple I method for both FMP model and FMT model can be established.3.We proposed and verified a further generalized algorithm for the uniform of the full implication triple I method for both FMP model and FMT model where diverse fuzzy implications can be employed under the same way.