Several Logic Metric Space Reflection Transformation

The study of mathematical logic is formal deduction.With the mathematical logic we can research the conclusion from perspective. Up till the present moment, it has been widely applied in the Artificial Intelligence a few related areas and form the theoretical foundations of computer science.But the mathematical logic stress on formal deduction and rigorous argument. However the computational mathematics were striving after numerical computation and permits approximate solving. I t can be said that there is a great deal of dif-ference between the mathematical logic and the computational mathematics.There has any direct link between the two. Professor Wang Guojun established the theory of quantitative logic by introducing the method of probability to the mathematical logic in order to establish the mathematical logic with the computational mathe-matics. Professor Wang Guojun introducing the the concepts of truth degrees of formulas, the similarity degree between two formulas and pseudometric among for-mulas in quantitative logic.So that we can constructive the logic metric space. Many scholars,such as Lukasiewicz, Godel and L* constructive the corresponding logic metric space in manry kinds of logical system. This includes professor Wang Guo-jun and Wang wei investigated that there is no isolated point in continuous logic in document[10]. The Topological Property of the 3-valued logic system has been in-vestigated in document[17].Hu Mingdi introducing the reflective transformation for the frist time,she investigated the properties of reflective transformation in classical logic metric space. This paper was suggested by document[1]. introducing the re-flective transformation to the 3-valued Lukasiewicz logic metric space and pro-rough logic metric space.Chapter 1. Preliminaries We mainly recall the semantic theory and the truth degrees of formulas, thesimilarity degree between two formulas and pseudometrie among formulas of the Lukasiewicz 3-valued logic system and the pre-rough logic system.Chapter 2. The properties of reflective transformation in 3-valued Lukasiewicz logic netric space.In this chapter we investiguated the properties of reflective trans-formation on the Lukasiewicz's 3-valued logic. It is proved that the reflexive trans-formationφ:F(s)�'F(s) on the Lukasiewicz's 3-valued logic metric space is a homomorphic mapping. Moreover, it keeps the logic equivalence relation and pseudo-symmetric logic formula unchanged. And studied the properties of a reflex-ive transformationφ*:([F(S)],ρ*)�'([F(S)],ρ*) on the Lindenbaum algebra in-duced by which is an automorphie and isometric transformation of the Lindenbaum algebra. At last we investigated the properties of the fixted points. Then we proved that [T] and [0] are the fixed point of ip*. Then the four special forms of fixed points have been obtained, those are [A]∧φ*([A]), [A]∨p*([A]),[A] (?)φ* ([A]),[A](?)φ*([A])Chapter 3. The reflective transformation on the pre-rough logic metric space. On investigated of the above, we found that reflective transformation on the two logic systems have the same properties. So the morphology of the fixed point are changed because the logic symbols are different. [T] and [(?)] are still the fixed point ofφ*. Then the two special forms of fixed points have been obtained, those are and [A](?)φ*([A]).