On The Distributivity Of Logical Operators On A Finite Chain

Fuzzy logic is a generalization of classical logic. In classical logic, the evaluation scale is the two-element set{0,1}. While in fuzzy logic, the eval-uation scale is the continuous unite interval [0,1]. In classical logic, several propositions are combined to form a new one by logical operators. The most frequently used logical operators are conjunctions, disjunctions, implications and negations. In fuzzy logic, these operators are generalized into t-norms, t-conorms, fuzzy implications and fuzzy negation.When trying to translate some tautologies in classical logic to the case of fuzzy logic, we find that some of the corresponding functional equations of the fuzzy logical operators are not equal everywhere any more. So it is essential to discuss the functional equations of the fuzzy logical operators. Among these functional equations, the distributivity equation is an important one, for its great role in eliminating combinatorial rule explosion in fuzzy systems.There has been a lot of discussion centered the distributivity equation, and the discussion is on [0,1]. However, the study of operators defined on a finite chain is an area of increasing interest. For one reason, the numerical interpretations of the linguistic terms or labels can be avoided by using the finite chain in expert’s reasonings and fuzzy control. For the other reason, the computation on a finite chain is much easier than on [0,1].The main concern of this paper is to characterize the solutions of the distributivity equation on a finite chain where F and G are logical operators on a finite chain.In Chapter1, we discuss the background and the significance of the study on the distributivity equation containing logical operators on a finite chain. In Chapter2, we discuss the distributivity between t-norms and t-conorms on a finite chain, and prove that the distributivity equation only has trivial solu-tions. In Chapter3, discussion on the distributivity of implication operators over t-norms and t-conorms is given on a finite chain, mainly including the S-implication,R-implication and QL-implication. We characterize the solu-tions of four different kinds of distributivity equations of these three classes of implication operators over t-norms and t-conorms respectively. In the end of this chapter, we generalize these implication operators to get the general form-s of the foregoing distributivity equation, and characterize their solutions. In Chapter4, we discuss the distributivity condition for uninorms and t-operators on a finite chainF(x, G(y, z))=G(F(x,y), F(x, z)), and characterize all the solutions in the four possible cases:(a) when F and G are uninorms on a finite chain,(b) when F is a uninorm and G is a t-operator on a finite chain,(c) when F is a t-operator and G is a uninorm on a finite chain,(d) when F and G are t-operators on a finite chain. We need to point out that the uninorms discussed in this chapter are restricted in Umin∪Umax. In Chapter5, enlight-ened by the definition of the representable uninorms on [0,1], we define the commutative representable semi-uninorms on a finite chain. After discussing the properties which they satisfy, we study the distributivity condition for this new kind of operators and t-operators.