The Study Of(G,N)-Implication Satisfying T Condition And U Condition

As an important binary operation,fuzzy implication is widely used in fuzzy control,fuzzy measure,fuzzy reasoning and data mining.Because of the diversity of practical problems,fuzzy implication needs to meet different properties,so it is very important to study the generation method and properties of fuzzy implication deeply.The first chapter introduces the research status of fuzzy implication operators.(G,N)-implication is an implication generated by the combination of the grouping function G and fuzzy negation N.The in-depth study of(G,N)implication is not only of theoretical significance,but also of practical significance.The main purpose of this paper is to study the conditions for(G,N)-implication to satisfy the relevant inference rules,including the conditions for(G,N)implication to satisfy MP、MT、HS and U-MP、U-MT、U-HS.The main contents of the full text are as follows:Chapter 1 mainly introduces the research background and some common results of fuzzy implication satisfying basic inference rules.The second chapter mainly introduces some basic concepts which will be used in this paper,including triangular norm,fuzzy implication,grouping function,uniform and so on.In Chapter 3,we study the conditions of(G,N)-implication satisfying common inference rules.The first part introduces the condition that(G,N)implication satisfies MP inference rule when triangular norms T are TM,TP,TL and sequential order triangular t-norms respectively.The second part discusses the condition that(G,N)-implication satisfies MT inference rule when N in(G,N)-implication is the same N as N in T(N(y),I(x,y))≤N(x).If N in(G,N)-implication is inconsistent with N in T(N(y),I(x,y))≤N(x),(G,N)implication satisfies the condition of MT inference rule.The third part discusses whether(G,N)-implication satisfies HS inference rule when N is minimum fuzzy negation and maximum fuzzy negation respectively.The necessary condition for(G,N)-implication to satisfy HS inference rule is given when T is TP and TL respectively.In Chapter 4,the conditions satisfying U-MP,U-MT and UHypothetical Syllogism are studied.This chapter first introduces what are MP,MT and Hypothetical Syllogism conditions.Secondly,since the uniform module U is divided into three categories:U∈umin,U∈uIdem,U∈uRep.Therefore,we discuss whether(G,N)-implication satisfies U-MP,U-MT and U-HS in these three cases respectively.The satisfied cases are proved and the unsatisfied cases are illustrated with counter-examples.Finally,the necessary conditions for satisfying U-MP,U-MT and U-HS are given respectively by means of multiplication generator of grouping function and addition generator of uniform module.The fifth chapter summarizes the main content of this paper and explains the future research direction and the next step.