Approximate Reasoning Based On Vague Sets

In 1965, Zadeh presented the theory of fuzzy sets, hi resent years, fuzzy set theory has been used for handing fuzzy decision-making system, fuzzy expert system, fuzzy control system and et al. Roughly speaking, a fuzzy set is a class with fuzzy boundries.A fuuzy set A of the universe of discourse U,U = ,is a set of ordered pairs, where is the membership function of the fuzzy set A , [0,1], and JUA (ut) indicates the grade of membership of, in A', VM, e U, the membership value /AA (M .) is a single value between zero and one. Gau et al. pointed out that this single value combines the evidence for ui e U and the evidence against u, € U, withoutindicating how much there is of each. They also pointed out that the single number tells us nothing about its accuracy. Thus . Gau et al. presented the concepts of vaguesets. They used a truth-membership function tA and false-membership function fA to characterize the lower bounds on u 4. These lower bounds are used to creste a subinterval on [0,1], namely \tA (ut ),1 - fA (u,)], to generalize the /f.,() of fuzzy sets, where t., (u,) < f.4 (u.) < 1 - fA (u.) . For example , let A be a vague set with truth-membership function t4 and false-membership function fA, respectively. If [t. (u, ),1 - fA (ut)] =[0.6.0.8], then we can see that 1A (u.) =0.6, 1 - fA (u,) =0.8, f. (w,)=0.2. It can be interpreted as ume degree that object w. belongs to the vague set A is 0.6, the degree rhat object u, dose not belong to the vague set A is 0.2,.As another example ,in a voting model, the vague value [0.6,0.8] can be interpreted as the vote for ; esolution is 6 in favor, 2 against, and 2 abstentions.In this paper, we give t-norm and t-conorm of interval-valued set, and prove a theory that t-norm and t-conorm of interval-valued set may be constructed by t-norm and t-conorm of point-valued set. We give a new definition about intersection and union for vague set based on t-norm and t-conorm of interval-valued set, and draw some results about intersection and union for vague set. So, we give a fuzzy reasoning approach based on vague set.We give two methods of similarity measures between vague sets, then propose a method of approximate reasoning based on similarity measures. In approximate reasoning based on similarity measures, because variation of membership value of each element in fuzzy concept of the result is the same, it will make the degree of belief in the result falling. So we provide a method of approximate reasoning based on similarity measures between elements of vague sets. This method is very good to avoid the problems resulting from the method of approximate reasoning based on similarity measures, and makes the result satisfied.In approximate reasoning method based on fuzzy sets, the uses of composite rule inference (CRI) method and reasoning method based on similarity measures are more than the other methods, but we will meet the following problems when using the abovetwo methods. In the CRI method, it is very difficult to check a satisfied implication operator. In the reasoning method of later, because variation of membership value of each element in fuzzy concept of the result is the same, it will make the degree of belief in the result falling. On the contrary, we are very difficult to give a good similarity measures based on vague sets. In this paper, we provide three kinds of linear interpolation approximate reasoning methods based on vague sets. The following describe are the thoughts of these three methods. When the membership functions (represented with fuzzy set) of fuzzy terms are continuous functions, we give a one-to-one mapping between the universe of discourse of fuzzy terms in antecedent of rule and that in consequent of rule, and we give a method of interpolation approximate reasoning based on fuzzy sets in this case; when the membership functions of fuzzy set are discrete functions, after giving one-to-one mapping between the universe of discourse of fuzzy terms in antecedent of rule and that in consequent of rule, we present a method o...