几种逻辑度量空间中的反射变换
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摘要
数理逻辑是研究形式推理的数学学科,有了数理逻辑,我们就可以研究如何从已知前提推出所需的结论。目前,数理逻辑已经广泛的应用于人工智能等一些相关领域,形成了现代计算机科学的理论基础。
然而,数理逻辑重视的是形式推理和严格论证,计算数学却追求的是数值计算,它允许近似求解。可以说数理逻辑与数值计算相差甚远,之间好像并没有什么直接的联系。为了将数理逻辑和计算数学建立联系,王国俊教授将概率方法引入数理逻辑,建立了计量逻辑学的概念,在计量逻辑学中,王国俊教授引入了命题的真度,以及命题之间的相似度和伪距离。这样我们就可以来构造逻辑度量空间,许多学者已经在Lukasiewicz, Godel和L*等多种逻辑系统中构造了相应的逻辑度量空间,并研究了它们的良好性质。其中,王国俊,王伟在文献[10]中讨论了一般的连续值逻辑度量空间没有孤立点,文献[17]讨论了三值Luk系统的拓扑性质,胡明娣在文献[1]中首次将反射变换引入到逻辑度量空间中,并讨论了逻辑度量空间中的反射变换的良好性质。本文就是受到文献[1]的启发,将反射变换引入到Lukasiewicz三值逻辑度量空间和预粗糙逻辑度量空间中。
在第一章,我们主要介绍了Lukasiewicz三值逻辑系统和预粗糙逻辑系统的语义理论和真度、相似度、伪距离的概念。
第二章主要讨论了Lukasiewicz三值逻辑度量空间中的反射变换的性质,证明了反射变换φ:F(s)→F(s)是同态变换、保逻辑等价、保逻辑(准)对称。并证明了由逻辑等价关系≈诱导的商代数F(s)/≈上的反射变换φ:([F(S)].ρ*)→([F(S)],ρ*)是等距变换。最后讨论了反射变换φ*的不动点的性质。证明了[T]和[O]都是φ*的不动点,并且(?)A∈F(s),[A](?)φ*([A]).[A](?)φ*([A]).[A](?)φ*([A]).[A](?)φ*([A])都是φ*的不动点。
第三章主要讨论了预粗糙逻辑度量空间中的反射变换的性质,经过讨论我们发现,这两个逻辑度量空间中的反射变换具有相同的性质,只是由于这两种逻辑系统中所定义的连接词不同,所以不动点的形态发生了改变, [T]和[O]仍是的不动点,VA∈F(s),[A](?)φ*([A])和[A]??φ*([A])是反射变换φ*的不动点
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]).
然而,数理逻辑重视的是形式推理和严格论证,计算数学却追求的是数值计算,它允许近似求解。可以说数理逻辑与数值计算相差甚远,之间好像并没有什么直接的联系。为了将数理逻辑和计算数学建立联系,王国俊教授将概率方法引入数理逻辑,建立了计量逻辑学的概念,在计量逻辑学中,王国俊教授引入了命题的真度,以及命题之间的相似度和伪距离。这样我们就可以来构造逻辑度量空间,许多学者已经在Lukasiewicz, Godel和L*等多种逻辑系统中构造了相应的逻辑度量空间,并研究了它们的良好性质。其中,王国俊,王伟在文献[10]中讨论了一般的连续值逻辑度量空间没有孤立点,文献[17]讨论了三值Luk系统的拓扑性质,胡明娣在文献[1]中首次将反射变换引入到逻辑度量空间中,并讨论了逻辑度量空间中的反射变换的良好性质。本文就是受到文献[1]的启发,将反射变换引入到Lukasiewicz三值逻辑度量空间和预粗糙逻辑度量空间中。
在第一章,我们主要介绍了Lukasiewicz三值逻辑系统和预粗糙逻辑系统的语义理论和真度、相似度、伪距离的概念。
第二章主要讨论了Lukasiewicz三值逻辑度量空间中的反射变换的性质,证明了反射变换φ:F(s)→F(s)是同态变换、保逻辑等价、保逻辑(准)对称。并证明了由逻辑等价关系≈诱导的商代数F(s)/≈上的反射变换φ:([F(S)].ρ*)→([F(S)],ρ*)是等距变换。最后讨论了反射变换φ*的不动点的性质。证明了[T]和[O]都是φ*的不动点,并且(?)A∈F(s),[A](?)φ*([A]).[A](?)φ*([A]).[A](?)φ*([A]).[A](?)φ*([A])都是φ*的不动点。
第三章主要讨论了预粗糙逻辑度量空间中的反射变换的性质,经过讨论我们发现,这两个逻辑度量空间中的反射变换具有相同的性质,只是由于这两种逻辑系统中所定义的连接词不同,所以不动点的形态发生了改变, [T]和[O]仍是的不动点,VA∈F(s),[A](?)φ*([A])和[A]??φ*([A])是反射变换φ*的不动点
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]).
引文
[1]胡明娣,王国俊.经典逻辑度量空间中的反射变换[J].陕西师范大学大学学报：自然科学版,2009,37(6)：1-6.
[2]王庆平,王国俊.对称逻辑公式在L3*逻辑度量空间中的分布[J].计算机学报,2011,34(1)：105-114.
[3]王国俊.数理逻辑引论与归结原理[M](第二版).北京：科学出版社,2006.
[4]王国俊.非经典数理逻辑与近似推理[M](第二版).北京：科学出版社,2008
[5]折延宏.计量逻辑学与粗糙逻辑的计量化研究[D].西安：陕西师范大学,2010
[6]BANERJEE M.Rough sets through algebraic logic [J].Fundaments Informaticae,1996,28(3-4):479-500
7] BANERJEE M.Roufh sets and 3-valued Lukasiewicz logic[J].fundamenta Informaticae,1997,32:213-220
[8]胡可云,陆玉昌,石纯一.粗糙集理论及其应用进展[J].清华大学学报(自然科学版)2001.41(1)：64-68
[9]徐余法.粗糙集理论及应用[J].上海机电学院学报,2005.8(2)：39-43
[10]王国俊,王伟.逻辑度量空间[J].数学学报,2001,44(1)： 159-168
[11]王国俊.计量逻辑学[J].工程数学学报.2006,23(2：191-215)
[12]裴道武.关于模糊逻辑与模糊推理逻辑基础问题的十年研究综述[J].工程数学学报,2004.21(2)：249-258
[13]陈世联.粗糙集代数与蕴涵格[J].计算工程与科学,2008.30(5)：115-117
[14]王艳平,徐义,窦金培.模糊粗糙逻辑语义及其推理[J].辽宁工业大学学报(自然科学版),2010.30(4)：258-264
[15]张文修,吴伟志.粗糙集理论介绍和研究综述[J].模糊系统与数学,2000.14(4)：1-12.
[16]王艳平,王志强,佟绍成.模糊粗糙集与模糊粗糙逻辑算子[J].大连海事大学学报,2005.31(2)：103-108
[17]高菲菲.三值Luk命题逻辑系统中逻辑理论的拓扑刻画[J].纺织高校基础科学学报,2008.21(1)：26-28
[18]李丽,陈永胜.模糊粗糙逻辑的语义[J].辽宁工业大学学报.2008.28(2)：135-137
[2]王庆平,王国俊.对称逻辑公式在L3*逻辑度量空间中的分布[J].计算机学报,2011,34(1)：105-114.
[3]王国俊.数理逻辑引论与归结原理[M](第二版).北京：科学出版社,2006.
[4]王国俊.非经典数理逻辑与近似推理[M](第二版).北京：科学出版社,2008
[5]折延宏.计量逻辑学与粗糙逻辑的计量化研究[D].西安：陕西师范大学,2010
[6]BANERJEE M.Rough sets through algebraic logic [J].Fundaments Informaticae,1996,28(3-4):479-500
7] BANERJEE M.Roufh sets and 3-valued Lukasiewicz logic[J].fundamenta Informaticae,1997,32:213-220
[8]胡可云,陆玉昌,石纯一.粗糙集理论及其应用进展[J].清华大学学报(自然科学版)2001.41(1)：64-68
[9]徐余法.粗糙集理论及应用[J].上海机电学院学报,2005.8(2)：39-43
[10]王国俊,王伟.逻辑度量空间[J].数学学报,2001,44(1)： 159-168
[11]王国俊.计量逻辑学[J].工程数学学报.2006,23(2：191-215)
[12]裴道武.关于模糊逻辑与模糊推理逻辑基础问题的十年研究综述[J].工程数学学报,2004.21(2)：249-258
[13]陈世联.粗糙集代数与蕴涵格[J].计算工程与科学,2008.30(5)：115-117
[14]王艳平,徐义,窦金培.模糊粗糙逻辑语义及其推理[J].辽宁工业大学学报(自然科学版),2010.30(4)：258-264
[15]张文修,吴伟志.粗糙集理论介绍和研究综述[J].模糊系统与数学,2000.14(4)：1-12.
[16]王艳平,王志强,佟绍成.模糊粗糙集与模糊粗糙逻辑算子[J].大连海事大学学报,2005.31(2)：103-108
[17]高菲菲.三值Luk命题逻辑系统中逻辑理论的拓扑刻画[J].纺织高校基础科学学报,2008.21(1)：26-28
[18]李丽,陈永胜.模糊粗糙逻辑的语义[J].辽宁工业大学学报.2008.28(2)：135-137