Researching On Regular Implications And Fuzzy Logical Systems

In recent years, the application fuzzy control has obtained a great success. But fuzzy reasoning, the core of fuzzy control is suspected because of lacking firm mathematics foundation. In order to solve this problem, a new research field has been built up and more and more scholars devoted themselves to this field. Applications of fuzzy logic range from the field of computer science, including automated theorem proving or approximate reasoning etc, to the field of pure mathematics, including independence or consistency proofs or the theory of particularly algebraic structures (e.g. MV-algebras, BL-algebras, and R₀-algebras). According to L.Zadeh, fuzzy logic is distinguished into broad and narrow senses.In narrow sense, fuzzy logic is the extension of many valued logic and in broad sense, fuzzy logic is the theory of fuzzy sets.This paper belongs to the narrow sense fuzzy logic.Chapter 1: The Regular ImplicationIt is well known that the choice of implication is very important to the fuzzy logical systems,for example the Lukasiewicz system use the Lukasiewicz implication, Godel system use the Godel implication, and L^* system use the R₀-implication etc.The great character of this kind implication is they are ajointed with some triangular norms.We call them regular implication. In this chapter, we simplify the definition of regular implication in paper [1] and study the elementary properties of regular implication firstly. In order to learn the analytical properties, we receive the sufficient and necessary condition that the implicator is continuous or has the only discontinous point (0,0) respectively. Furthermore a new concept of separable element of the regular implicator is given and the character theorem of separable element is discussed. Moreover, in order to get more regular implications, we introduce some new methods to construct them and study the problem of converging to some discontinuous implication operator by a sequence continuous implication. Finally,the concept of the weak-regular implication and the implicator on the completed latticeis are given and the basic properties of them are studied.Chapter 2: The NML Logical System...