命题逻辑公式集上的相似度、伪距离与近似推理
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摘要
人工智能是一个受到广泛重视的研究领域,而人工智能中的推理研究又是最为活跃的。传统的计算机的推理系统是基于二值逻辑的,而能够真正模拟人类思维方式的智能计算机采用的是合情推理,即是一种近似推理。在近似推理中,衡量公式之间的距离非常重要,用来确定公式在近似推理中的作用。本文从命题逻辑公式的相似度入手,研究命题逻辑公式集上的伪距离和近似推理。
第一章基础知识。介绍命题逻辑系统的五个组成部分以及二值逻辑系统L、Lukasiewicz多值命题逻辑系统、R_0型命题逻辑系统L_n~*。
第二章介绍真度、相似度ξ_1、ξ_2和二值逻辑系统中的正则相似度。简化了正则相似度的定义,引入半正则相似度,并定义了相似度ξ_0。证明在二值逻辑中,ξ_0与ξ_1是正则的,在2_n值逻辑度量空间,ξ_0是半正则的。
第三章证明三种相似度可以诱导出三种伪距离空间。给出了三种伪距离之间的大小关系。证明三种伪距离空间是拓扑等价的。
第四章介绍命题逻辑公式集上的发散度。针对王国俊教授所提出的各系统中的三种近似推理的误差的关系问题,在二值系统中,将误差定义为三种距离,计算出三种距离是相等的,从而证明三种近似推理的误差是等价的。部分地解决了这一公开问题。
Artificial Intelligence is a valued research field and inference is active in it. The inference of traditional computer is designed by two-value logic system. It is the approximate reasoning that the intelligence computer should use. It is very important to compute the distance between two formulas in approximate reasoning. This dissertation begins with the research of similarity degree, then pseudo-metric and approximate reasoning in propositional logic system are studied.
Chapter one: Preliminary. The five components of propositional logic system and three logic systems, namely, classical two-value logic system, Lukasiewicz n-value logic system, R_0 n-value logic system are introduced.
Chapter two: The concept of truth degree, similarity degreeξ_1、ξ_2 and regular similarity degree in classical two-value logic system are introduced. The definition of regular similarity is simplified and the definition of the semi-regular similarity degree and a new similarity degreeξ_0 are given. Also, we proved the fact thatξ_0 andξ_1 are regular similarity degree in classical two-value logic system, andξ_0 is semi-regular similarity degree in 2n-value logic system.
Chapter three: First, we induce three pseudo-metric spaces from the three similarity degree. Then, an order was defined on the three different pseudo-metrics. Finally the fact that the three different pseudo-metrics are topological isomorphic is proved.
Chapter four: We give the concept of divergence degree in L , L_n, L*_n. In order to solve Professor Wang Guojun's open question of the relationship between the three errors of approximate reasoning in n-value logic system, we first change the three errors to three distances in propositional logic system, then we figure out the result that the three distances are equal, finally we prove the equivalence of the three errors in propositional logic system. Hence, the open question is partly solved.
第一章基础知识。介绍命题逻辑系统的五个组成部分以及二值逻辑系统L、Lukasiewicz多值命题逻辑系统、R_0型命题逻辑系统L_n~*。
第二章介绍真度、相似度ξ_1、ξ_2和二值逻辑系统中的正则相似度。简化了正则相似度的定义,引入半正则相似度,并定义了相似度ξ_0。证明在二值逻辑中,ξ_0与ξ_1是正则的,在2_n值逻辑度量空间,ξ_0是半正则的。
第三章证明三种相似度可以诱导出三种伪距离空间。给出了三种伪距离之间的大小关系。证明三种伪距离空间是拓扑等价的。
第四章介绍命题逻辑公式集上的发散度。针对王国俊教授所提出的各系统中的三种近似推理的误差的关系问题,在二值系统中,将误差定义为三种距离,计算出三种距离是相等的,从而证明三种近似推理的误差是等价的。部分地解决了这一公开问题。
Artificial Intelligence is a valued research field and inference is active in it. The inference of traditional computer is designed by two-value logic system. It is the approximate reasoning that the intelligence computer should use. It is very important to compute the distance between two formulas in approximate reasoning. This dissertation begins with the research of similarity degree, then pseudo-metric and approximate reasoning in propositional logic system are studied.
Chapter one: Preliminary. The five components of propositional logic system and three logic systems, namely, classical two-value logic system, Lukasiewicz n-value logic system, R_0 n-value logic system are introduced.
Chapter two: The concept of truth degree, similarity degreeξ_1、ξ_2 and regular similarity degree in classical two-value logic system are introduced. The definition of regular similarity is simplified and the definition of the semi-regular similarity degree and a new similarity degreeξ_0 are given. Also, we proved the fact thatξ_0 andξ_1 are regular similarity degree in classical two-value logic system, andξ_0 is semi-regular similarity degree in 2n-value logic system.
Chapter three: First, we induce three pseudo-metric spaces from the three similarity degree. Then, an order was defined on the three different pseudo-metrics. Finally the fact that the three different pseudo-metrics are topological isomorphic is proved.
Chapter four: We give the concept of divergence degree in L , L_n, L*_n. In order to solve Professor Wang Guojun's open question of the relationship between the three errors of approximate reasoning in n-value logic system, we first change the three errors to three distances in propositional logic system, then we figure out the result that the three distances are equal, finally we prove the equivalence of the three errors in propositional logic system. Hence, the open question is partly solved.
引文
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[2]胡世华,陆钟万.数理逻辑基础[M].北京:科学出版社,1983.
[3]罗铸楷,胡谋,陈廷愧.多值逻辑的理论与应用[M].北京:科学出版社,1992.
[4]王国俊.Fuzzy命题演算的一种形式演绎系统[J].科学通报,42(1997),10:1041-1045.
[5]Ying Mingsheng.A Logic for Approximate Reasoning[J].J.Symbolic logic 1994,59(3):830-837.
[6]Wang Guojun.Fuzzy Continuous input-output controllers are universal approximators[J].Fuzzy Sets and System,1998,13(1):95-100.
[7]王国俊.非经典数理逻辑与近似推理[M].北京:科学出版社,2000.
[8]王国俊,王伟.逻辑度量空间[J].数学学报,2001,44(1):159-168.
[9]王国俊,傅丽,宋建社.二值命题逻辑中命题的真度理论[J].中国科学(A辑),2001,11:998-1008。
[10]王国俊.计量逻辑学(Ⅰ)[J].工程数学学报,2006,4:191-215。
[11]王国俊.数理逻辑引论与归结原理(第二版)[M].北京:科学出版社,2006.
[12]宋庆燕,杨兴忠.命题逻辑公式集上的正则相似关系[J].陕西师范大学学报,2002,12:23-27.
[13]刘艳,郑慕聪.Lukasiewicz多值逻辑系统中的相似度及伪距离[J].西安科技大学学报,2005,6:263-266.
[14]吴洪博,文秋梅.积分语义学中的积分相似度与伪距离[J].陕西师范大学学报,2000,9:15-19。
[15]Wang Guojun,Lueng Yi.Intergrated semantics and logic spaces[J].Fuzzy Sets and System,2003,136(1):71-91.
[16]李骏,黎锁平,夏亚峰.n值Lukasiewic逻辑中命题的真度理论[J].数学学报,2004,47(4):769-780.
[17]裴道武,王三民.形式系统L_n~*的完备性[J].高校应用数学学报A辑,2001,16(3):253-262.
[18]裴道武,王国俊.逻辑系统L~*的完备性及其应用[J].中国科学(E)辑,2002,32:55-64.
[19]韩诚,周红军.关于形式系统L~*(强)完备性的注记[J].陕西师范大学学报,2005,33(2):9-12.
[20]王国俊,李璧镜.Lukasiewic n值命题逻辑中公式的真度理论与积分理论[J].中国科学(E辑),2005,11:561-569.
[21]Wang Guojun.Comparison of deduction theories in diverse logic systems[J].NM&NC,2005,1(1):65-78。
[22]李骏,王国俊.n值Lukasiewicz命题逻辑中命题的α-真度理论[J].计算机工程与应用2006,31:16-18.
[23]Zhou Hongjun,Wang Guojun.Generalized consistency degrees of theories w.r.t,formulasin Several standard complete logic systems[J].Fuzzy Sets and System,2006,157:2058-2073.
[24]Zhou Hongjun,Wang Guojun,Zhou Wei.Consistency degrees of theories and methods of graded reasoning in n-valued R_0-logic[J].Fuzzy Sets and System,2006,43:117-132.
[25]Zhou Hongjun,Wang Guojun.A new theory consistency index based on deduction theorem in several logic systems[J].Fuzzy Sets and System,2006,157(3):427-443.
[26]Pei Daowu.On equivalent forms of fuzzy logic systems NM and IMTL[J].Fuzzy Sets and System,2003,138:187-195.
[27]王国俊,刘保翠.四种命题逻辑中公式的相对Γ一重言度理论[J].工程数学学报,2007,4:598-610.
[28]惠小静,王国俊.经典推理模式的随机化研究及其应用[J].中国科学(E)辑,2007,37(6):801-812。
[29]王国俊,折延宏.二值命题逻辑中理论的发散性、相容性及其拓扑刻画[J].数学学报,2007,4:841-850。
[30]吴洪博.基础R_0代数的性质及在L~*系统中的应用[J].数学研究与评论,2003,23(3):557-563.
[2]胡世华,陆钟万.数理逻辑基础[M].北京:科学出版社,1983.
[3]罗铸楷,胡谋,陈廷愧.多值逻辑的理论与应用[M].北京:科学出版社,1992.
[4]王国俊.Fuzzy命题演算的一种形式演绎系统[J].科学通报,42(1997),10:1041-1045.
[5]Ying Mingsheng.A Logic for Approximate Reasoning[J].J.Symbolic logic 1994,59(3):830-837.
[6]Wang Guojun.Fuzzy Continuous input-output controllers are universal approximators[J].Fuzzy Sets and System,1998,13(1):95-100.
[7]王国俊.非经典数理逻辑与近似推理[M].北京:科学出版社,2000.
[8]王国俊,王伟.逻辑度量空间[J].数学学报,2001,44(1):159-168.
[9]王国俊,傅丽,宋建社.二值命题逻辑中命题的真度理论[J].中国科学(A辑),2001,11:998-1008。
[10]王国俊.计量逻辑学(Ⅰ)[J].工程数学学报,2006,4:191-215。
[11]王国俊.数理逻辑引论与归结原理(第二版)[M].北京:科学出版社,2006.
[12]宋庆燕,杨兴忠.命题逻辑公式集上的正则相似关系[J].陕西师范大学学报,2002,12:23-27.
[13]刘艳,郑慕聪.Lukasiewicz多值逻辑系统中的相似度及伪距离[J].西安科技大学学报,2005,6:263-266.
[14]吴洪博,文秋梅.积分语义学中的积分相似度与伪距离[J].陕西师范大学学报,2000,9:15-19。
[15]Wang Guojun,Lueng Yi.Intergrated semantics and logic spaces[J].Fuzzy Sets and System,2003,136(1):71-91.
[16]李骏,黎锁平,夏亚峰.n值Lukasiewic逻辑中命题的真度理论[J].数学学报,2004,47(4):769-780.
[17]裴道武,王三民.形式系统L_n~*的完备性[J].高校应用数学学报A辑,2001,16(3):253-262.
[18]裴道武,王国俊.逻辑系统L~*的完备性及其应用[J].中国科学(E)辑,2002,32:55-64.
[19]韩诚,周红军.关于形式系统L~*(强)完备性的注记[J].陕西师范大学学报,2005,33(2):9-12.
[20]王国俊,李璧镜.Lukasiewic n值命题逻辑中公式的真度理论与积分理论[J].中国科学(E辑),2005,11:561-569.
[21]Wang Guojun.Comparison of deduction theories in diverse logic systems[J].NM&NC,2005,1(1):65-78。
[22]李骏,王国俊.n值Lukasiewicz命题逻辑中命题的α-真度理论[J].计算机工程与应用2006,31:16-18.
[23]Zhou Hongjun,Wang Guojun.Generalized consistency degrees of theories w.r.t,formulasin Several standard complete logic systems[J].Fuzzy Sets and System,2006,157:2058-2073.
[24]Zhou Hongjun,Wang Guojun,Zhou Wei.Consistency degrees of theories and methods of graded reasoning in n-valued R_0-logic[J].Fuzzy Sets and System,2006,43:117-132.
[25]Zhou Hongjun,Wang Guojun.A new theory consistency index based on deduction theorem in several logic systems[J].Fuzzy Sets and System,2006,157(3):427-443.
[26]Pei Daowu.On equivalent forms of fuzzy logic systems NM and IMTL[J].Fuzzy Sets and System,2003,138:187-195.
[27]王国俊,刘保翠.四种命题逻辑中公式的相对Γ一重言度理论[J].工程数学学报,2007,4:598-610.
[28]惠小静,王国俊.经典推理模式的随机化研究及其应用[J].中国科学(E)辑,2007,37(6):801-812。
[29]王国俊,折延宏.二值命题逻辑中理论的发散性、相容性及其拓扑刻画[J].数学学报,2007,4:841-850。
[30]吴洪博.基础R_0代数的性质及在L~*系统中的应用[J].数学研究与评论,2003,23(3):557-563.