Fuzzy Implications And Coimplications Based On Unary Operators And The Research About Generalized Tautologies

As the generalization of implications in classical logic, which takes val-ues in{0,1}, to fuzzy logic, fuzzy implications have been successfully used in many fields such as fuzzy control, approximate reasoning, computing with words, fuzzy image processing and etc. Hence, researchers have shown great interest to these operators. Now, the construction and the study of general-ized tautologies about fuzzy implications have become the research hotspots. However, as the dual operators of fuzzy implications, coimplications gained far less attention than implications did. At present, people have realized that coimplications do play an important role in the theories and applications of fuzzy logic. Nevertheless, works about fuzzy coimplications are still limited to the duality of residual implications, S-implications and QL-implications, and there are few works on other classes of coimplications. Furthermore, the research about generalized tautologies that involve fuzzy coimplcations can-not be found at home and aboard. In this paper we will construct a series of fuzzy implications and coimplications based on several classes of unary op-erators, and investigate the generalized classical tautologies. This study not only enriches the theories of fuzzy logic operators, but also provides much choices for operators to the technologists and engineers. The main content of this paper is as follows:The first chapter is the introduction. We introduce the concept of fuzzy implications and coimplications, present the current research state of these operators, and then point out our main works of this paper.In chapter2, by using the additive generators of continuous Archimedean triangular norms f, we introduce the concept of u-operator and then con-struct a new class of fuzzy implications called (f, u)-implications. Further, we discuss the properties of (f,u)-implications such as the left neutrality property, the exchange principle, the identity principle, the order property and the consequent boundary. Finally, three generalized classical tautologies that involve fuzzy implications, viz., the law of importation, contrapositive symmetry and the distributive equations are investigated. The advantage of this construction is that people can get an ideal implication by select-ing proper f-generator and u-operator according to the practical application environment.Chapter3introduces (f,L)-implications by using the additive gener-ators of continuous Archimedean triangular norms f. In this chapter we analyze the properties of these implications and discuss the relationships be-tween these implications and the well known classes of fuzzy implications such as (S, N)-, R-, QL-and Yager’s f-and g-implications. In addition, we investigate the conditions under which the law of importation, contra-positive symmetry and the distributive equations are satisfied by the (f, L)-implications. Specially, we get the non-trivial solution of triangular norms that satisfy the distributive equations with (f, L)-implications under some certain conditions.In chapter4, by using the additive generators of continuous Archimedean triangular conorms g, we introduce the concept of U-operator and then con-struct a new class of fuzzy implications called (g,U)-implications. Basic properties and characterization of these implications are discussed. It is pointed out that (g,U)-implications equivalent to implications that satis-fy (NP). Furthermore, the distributive equations of these implications are investigated. We give the general solutions for distributive equation with continuous Archimedean t-norms and idempotent t-conorm, and find the t-norms that satisfy distributive equation other than TM when I is Yager’s g-implication Ig. This work will bring benefit for approximate reasoning, fuzzy control and other application areas.Chapter5introduces (g, L)-implications by using the additive generators of continuous Archimedean triangular conorms g. we analyze the properties of these implications and discuss the relationships between these implications and the well known classes of fuzzy implications such as (S, N)-, R-, QL-and Yager’s f-and g-implications. Finally, we investigate the conditions under which the law of importation, contrapositive symmetry and the distributive equations are satisfied by the (g,L)-implications. In chapter6, we introduced a new class of fuzzy coimplications called (h,N)-coimplications, by using the generalized additive generators of repre-sentable uninorms and fuzzy negations. We point out that the N-dual class of h-implications proposed in [50] is a subclass of (h,N)-coimplications. Basic properties of this class of coimplications are discussed. Furthermore, the gen-eralized tautologies that involve coimplications, such as law of importation, contrapositive symmetry and distributive law, are investigated.The innovations of this paper are as follows:1. By using the additive generators of continuous Archimedean triangu-lar norms and triangular conorms, we introduce the concept of u-operator and U-operator, and then we construct (f,u)-implications and (g,U)-implications respectively. At the same time, we discuss the properties of these classes of implications such as the left neutrality property, the exchange principle, the identity principle, the order property and the consequent boundary. Finally, we investigate the generalized tautologies that involve the fuzzy implications such as law of importation, contrapositive symmetry and distributive law.2. We introduce (f, L)-and (g,L)-implications, by using the addi-tive generators of continuous Archimedean triangular norms and triangular conorms, respectively. The properties of these classes of fuzzy implications and the relationships between these implications and the well known classes of fuzzy implications such as (S, N)-, R-, QL-and Yager’s f-and g-implications are discussed. The generalized tautologies that involve these implications are also investigated.3. We introduce a new class of fuzzy coimplications called (h, N)-coimplications, by using the generalized additive generators of representable uninorms and fuzzy negations. We discuss this class of fuzzy coimplica-tions and investigate the conditions under which the law of importation, contrapositive symmetry and the distributive equations are satisfied by the (h,N)-coimplications.