Fuzzy Implication Induced By Copulas And Co-copulas

Fuzzy implication as a generalization of the classical implication, is a very important conjunction in fuzzy logic. They are applied to many fields, such as fuzzy control, expert systems, fuzzy system models, etc. People have researched all aspects of the fuzzy implication induced by triangular norms (t-norms for short), triangular conforms (t-conorms for short) and uninorms. Since1959, Sklar introduced a new concept.That is copula. It has become a good risk management tool. At the same time, copulas are also conjunction operators in the unit interval. Fuzzy implications can be introduce by copulas. However, there are few people research it. Based on this, this paper defines six new kinds of operators, respectively RC-implication,(CC,N)-implication, CQL-operator, CD-operator, cf-implication and ccg-implication. This paper also studies some properties of the six new kinds of operators. The paper contains four chapters as follows:Chapter1introduces the research background and significance, reviewes several basic concepts. The last of this chapter introduces the main works of the paper.Chapter2is preliminaries. This chapter mainly reviews the concept and properties of fuzzy implication, fuzzy negation, copula, co-copula.Chapter3defines six new kinds of operators, respectively RC-implication,(CC,N)-implication, CQL-operator, CD-operator, cf-implication and ccg-implication. This chapter also studies some properties of the six new kinds of operators. The main conclusion can be summarized as follows: (1) The RC-implication,(CC.N)-implication, cf-implication and ccg-implication are fuzzy implication. CQL-operator and CD-operator are not sure fuzzy im-plication.(2) All of the six new kinds of operators satisfies (NP)and condition sat-isfies (CP).(3) If copula and co-copula satisfies the law of associative, then the RC-implication,(CC.N)-implication, cf-implication and ccg-implication satisfies (EP).(4) The φ-conjugate of the six new kinds of operators are also the same type operators and they are introduced by the φ-conjugate of copula, co-copula,f, g and fuzzy negation.(5) The RC-implication satisfies (IP),(RP) and (OP). The (CC,N)-implication condition satisfies (OP) and (IP), ect.Chapter4studies the importation law of cf-implication and ccg-implcation(LI).The main conclusion can be summarized as follows:(1)Let C:[0,1]2â†'[0,1] be a copula, f:[0.1]â†'[0,1] be a strictly decreasing and continuous function with f(0)=1.Ic,f an f-implication.Then the following statements are equivalent:(i) C satisfies the law of associative and the couple of functions Ic,f and T satisfies (LI) for all x.y,z∈[0,1].(ii)T=C.(2)Let CC:[0,1]2â†'[0,1] be a co-copula, g:[0,1]â†'[0,1] be a strictly increasing and continuous function with g(0)=0,g(1)=1.Icc,N,g an ccg-implication.Then the following statements are equivalent:(i) CC satisfies the law of associative and the couple of functions Icc.N.g and T satisfies (LI) for all x,y,z∈[0,1].(ii)T is the N-dual of the co-copula CC.The innovations in this paper can be summarized as follows:(1)Based on copula, co-copula and fuzzy negation, six new kinds of fuzzy implication is defined.(2)The law of importation of cf-implication and ccg-implication are in- vestigated.Necessary and sufficient conditions when they satisfy the law of importation are given.