Fuzzy Implication Operators Generated By Given Fuzzy Implications And Monotone Functions

Fuzzy implication operators are widely used as an important binary operator in fuzzy reasoning,image processing,fuzzy control,fuzzy morphology,data mining and other fields.Because of the complexity and diversity of practical problems,different fuzzy implication operators need to be used to achieve specific purposes.So it is particularly important to construct different fuzzy implication operators and analyze the properties of the implication operators.It is an important generation method to generate new fuzzy implication operators from given fuzzy implication.Whether starting from a single fuzzy implication operator or from two fuzzy implication operators,the basic principle of generating the corresponding fuzzy implication is to keep the conditions satisfied by the definition of the fuzzy implication operator unchanged.That is to say,under the condition that the boundary conditions hold,it is necessary to ensure that the fuzzy implication I(x,y)decreases with respect to the first variable and increases with respect to the second variable.Based on this,two new fuzzy implication operators are proposed in this thesis and their properties are analyzed.The main contents of the full thesis are as follows:Firstly,starting from two given fuzzy implicants I1 and I2,we use the monotone function f on the interval[0,1]to generate fuzzy implicants II1,I2f (II1,I2f(x,y)=I1(x,f(I2(x,y)))).Then we discuss the properties of fuzzy implication II1,I2f in two cases of monotone function f(x)=x and f(x)≠x,such as the properties of the left neutrality,identity,law of importation,generalized Frege’s law and so on.In addition,based on f(x)=x,it is divided into two cases:I1=I2 and I1≠I2,the relationships between II1,I2f and iterative Boolean law I(x,y)＝I(x,I(x,y)),fuzzy implication(I1?I2)(x,y)=I1(x,I2(x,y))are studied respectively.Secondly,starting from the two given fuzzy implications I1 and I2,we use the fuzzy negation N on the[0,1]interval to generate the fuzzy implication II1,I2N(II1,I2N(x,y)＝I1(N(I2(x,y)),y)).Then we discuss the properties of fuzzy implication II1,I2N,such as the properties of the left neutrality,continuity,law of importation,generalized Frege’s law and so on.At the same time,we study whether II1,I2N is(S,N)-implication(or R-implication)when both I1 and I2 take(S,N)-implication(or R-implication).Finally,the conditions under which II1,I2f and II1,I2N satisfy the MP,MT inference rules are studied.For the implication operator II1,I2f,conditions are given for II1,I2f to satisfy the MP,MT inference rules when f(x)=x and the triangular norms T is taken as TM,TD,TP and TLK respectively.For implication operator II1,I2N,we discuss the sufficient condition that II1,I2N satisfies MP and MT when T is TP and TLK.