Some Researches On Implication Operator And Triangular Norm

In recent years, with the rapid development of fuzzy theory, fuzzy mathematics has been achieved a great success in many application fields. Especially in the field of automatic control, the fuzzy control theory based on fuzzy sets and fuzzy inference has been applied successfully.The core of fuzzy control is fuzzy inference which represented by the implication operator. Hence the research of the construction, property and feature of the implication operator and its family becomes important.Meanwhile, the multi-valued logic and fuzzy logic which closely related to the fuzzy inference have also become the hot research. The logic systems based on left-continuous triangular norms and their residuum regular implication operators have been gradually recognized.At the beginning of this century, Professor Wang Guojun established the truth degree theory based on even probability measure in classical two-valued propositional logic systems. Since then the quantitative logic theory was put forward and a new approximate reasoning model was established. Then the domestic and foreign counterparts have done a widely research, similar results have been extended to various logical systems.These three interrelated issues, namely, the construction of fuzzy implication operators, fuzzy logic system based on left-continuous triangular norm, quantitative logic theory, are closely related to fuzzy implication operator. This paper tries to do some research on implication operator and triangular norm from the above three aspects respectively.The main results of this paper are as follows,Firstly, the weighted average method to construct the family of implication operator is proposed. It is proved that the family of implication operator constructed by formal implication operator through weighted average method is still the formal implication. Then some examples about the family of implication operators based on four basic regular implication operators are given.Secondly, the structure of F-λ, triangular norm is discussed and several new families of triangular norms and their residuums are given. Demonstrated that the logic system corresponded to such triangular norms is the WNM logic system.Thirdly, the existing quantitative logic theory is generalized. And the evaluation domain of Lukasiewicz logic system is extended to the MV algebras determined by the type of Lukasiewicz implication operators. With the MV sub-algebra and MV algebra as the evaluation domain respectively, the quantitative logic theory of the Lukasiewicz n-valued logic system and Lukasiewicz fuzzy logic system were discussed. Meanwhile, a new definition of pseudo-metric and its scope of application are proposed and the limit theorem under this framework was proved. Then one can give more choices for the study of semantic and formula evaluation and approximate reasoning. This idea and method can also be applied to other propositional logic, such as NM logic system.