n值逻辑系统中命题的绝对真度及其随机化理论
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摘要
无论是二值逻辑还是多值逻辑,都注重形式推理而不大关心数值计算。数理逻辑侧重严密的逻辑推理,而缺乏数值计算的灵活性和广泛的应用范围。通过把数值计算引入到数理逻辑中,就可以建立起符号化与数值计算之间的联系。
“指派真值”的做法或多或少已经反映出了数理逻辑概念的程度化思想。程度化的思想也正是人脑智能的反映。王国俊教授从逻辑理论基本概念的程度化入手,提出了计量逻辑学,通过把数值计算融入到数理逻辑系统中,使得数理逻辑更加灵活,应用也更加广泛。
计量逻辑学研究的逻辑系统主要包括经典的二值命题逻辑系统L、Lukasiewicz多值逻辑系统L_n以及命题演算系统(?)_n~*。王国俊教授在文献[9]中通过引入逻辑公式的的真度τ(A),进一步,引入了公式间的相似度和伪距离,从而建立了一套基于程度化理论的近似推理框架。
本文的研究目的和主要工作:
1.对于逻辑系统的代数结构的研究,是一个重要的研究课题,已成为非经典逻辑系统基础研究和应用研究的重要方向之一。本文将主要研究剩余格在几类常见逻辑系统中的基本应用和特征性质。
2.文献[8]基于测度论的思想,通过将赋值集向测度空间中的可测子集的转化,利用均匀概率空间的无穷乘积测度,在经典命题逻辑中建立了公式的真度理论。此后,文献[15,16]也基于测度论的思想,将公式的真度理论推广至n值Lukasiewicz命题逻辑系统中。文献[15]中公式的真度定义是通过均匀概率空间的无穷乘积测度给出的,其中涉及到无穷维的向量,因此不便于通过计算机编程来实现真度的数值计算。本文将回避均匀概率空间的无穷乘积测度,借助公式所诱导出的函数,给出一种新的绝对真度定义,是对原有方法的改进和简化。从而使得任意一个公式的绝对真度都可在有限步内利用计算机计算出来,使得本文的方法在算法上实现成为可能。因此也将具有更好的应用价值。
3.文献[8]在经典二值命题逻辑系统中,利用测度论的方法,引入了逻辑公式的真度,相似度,伪距离等概念,从等概率的观点出发,研究了均匀概率空间中的情形,即让每个原子公式p赋值为0和1的概率测度均为1/2的情形。文献[10,15,16]将公式真度的概念推广至n值Lukasiewicz逻辑系统中,研究了每个原子公式p赋值为i/(n-1)的概率测度均为1/n的情形。这些研究的一个显著特点就是每个原子公式的发生都是等概率的。这种等概率的观点不能很好地体现概率具有随机性的特点。从而极大地限制了计量逻辑学理论的应用范围。本文将在非均匀概率空间中,利用概率测度的方法,研究了四种常见n值逻辑系统中公式赋值为i/(n-1)的概率测度服从某一类离散型随机分布列的情形,引入了逻辑公式概率真度的概念和表达式,讨论了其性质。在非均匀概率空间中,每个原子公式的赋值不一定相等,这样就更接近实际生活中的应用,具有更好的应用价值。然后在剩余格上引入了L-模糊相似的定义,给出了四种n值逻辑系统中公式间相似度的定义和统一表达式,并研究了相似度的若干特征性质;最后引入了公式间的伪距离,为n值命题逻辑的近似推理理论提供了另外一种可能的框架。
The characteristic of mathematical logic lies in symbolization and formalization, which is quite different from computational mathematics. The former emphasises on formal reasoning and strict proof, while the later concerns with numberical computation and approximate solutions. Computational mathematics is more flexible than mathematical logic. It can build the connection between mathematical logic and computational mathematics by bringing the numberical calculation into mathematical logic.
The mean of "designated truth values" in mathematical logic had reflected the idea of graded method. Also, graded method is the reflection of ideology. Professor Wang Guojun established the quantitative logic based on basic logic concepts. Consequently, numberical calculation was introduced in the mathematics logic. It makes the mathematic logic more flexible and useable.
In Wang's paper[9], the truth degree was introduced.At the same time, similarity and pseudo-distance were also defined based on the concept of truth degree which provides a possible framework for approximate.
The main work of this paper includes:
Chapter One: The study of algebra framework in fuzzy logic systems is an important task. And it had become an embranchment in the research of non-classical logic systems. In this chapter, some preliminaries are recalled and the characteristics of t-norm and residuated lattices are investigated. And also put out the characteristic of t-norm and residuated lattices.
Chapter Two: In [10], by means of infinite product measure of uniformly distributed probability spaces, the author made mention of the truth degree theory, which was based on measure theory. Later, the truth degree theory was extended to the n-valued Lukasiewicz logic system by using this method. But, all the above works run in the uniformly distributed probability spaces. This definition of truth degree was built on infinite product measure, therefore, the calculation of truth degree is very difficult and can not be realized by algorithm. This chapter will avoid the concept of infinite product measure, and give a new definition of truth degree-absolute truth degree. It's a reformative and laconic definition which makes it possible to compute truth degrees of formulas by computer .
Chapter Three: In classical two-valued propositional logic system, truth degrees of formulas were defined by measure theory. Also the similarity and pseudo-distance were from the viewpoint of equal probability. The possibility that each atomic propositionholds is uniform and equals to (1/2). The randomization characteristic of probability was not reflected under this method. In this chapter, firstly, based on the method of probability measure, the concept of probability truth degrees and introduced its general expression in unevenly probability space is given. Secondly, the similarity degree between two formulas on a residuated lattice is defined in terms of L-fuzzy similarity. And the character properties are discussed in four types of n—valued logic systems based on the left-continuous t-norms. Lastly, a pseudo-distance was introduced by means of probability truth degree, which provides another possible framework for approximate reasoning in n-valued propositional logics. It can provide a kind of ingerence frame for the approximate reasoning theory.
“指派真值”的做法或多或少已经反映出了数理逻辑概念的程度化思想。程度化的思想也正是人脑智能的反映。王国俊教授从逻辑理论基本概念的程度化入手,提出了计量逻辑学,通过把数值计算融入到数理逻辑系统中,使得数理逻辑更加灵活,应用也更加广泛。
计量逻辑学研究的逻辑系统主要包括经典的二值命题逻辑系统L、Lukasiewicz多值逻辑系统L_n以及命题演算系统(?)_n~*。王国俊教授在文献[9]中通过引入逻辑公式的的真度τ(A),进一步,引入了公式间的相似度和伪距离,从而建立了一套基于程度化理论的近似推理框架。
本文的研究目的和主要工作:
1.对于逻辑系统的代数结构的研究,是一个重要的研究课题,已成为非经典逻辑系统基础研究和应用研究的重要方向之一。本文将主要研究剩余格在几类常见逻辑系统中的基本应用和特征性质。
2.文献[8]基于测度论的思想,通过将赋值集向测度空间中的可测子集的转化,利用均匀概率空间的无穷乘积测度,在经典命题逻辑中建立了公式的真度理论。此后,文献[15,16]也基于测度论的思想,将公式的真度理论推广至n值Lukasiewicz命题逻辑系统中。文献[15]中公式的真度定义是通过均匀概率空间的无穷乘积测度给出的,其中涉及到无穷维的向量,因此不便于通过计算机编程来实现真度的数值计算。本文将回避均匀概率空间的无穷乘积测度,借助公式所诱导出的函数,给出一种新的绝对真度定义,是对原有方法的改进和简化。从而使得任意一个公式的绝对真度都可在有限步内利用计算机计算出来,使得本文的方法在算法上实现成为可能。因此也将具有更好的应用价值。
3.文献[8]在经典二值命题逻辑系统中,利用测度论的方法,引入了逻辑公式的真度,相似度,伪距离等概念,从等概率的观点出发,研究了均匀概率空间中的情形,即让每个原子公式p赋值为0和1的概率测度均为1/2的情形。文献[10,15,16]将公式真度的概念推广至n值Lukasiewicz逻辑系统中,研究了每个原子公式p赋值为i/(n-1)的概率测度均为1/n的情形。这些研究的一个显著特点就是每个原子公式的发生都是等概率的。这种等概率的观点不能很好地体现概率具有随机性的特点。从而极大地限制了计量逻辑学理论的应用范围。本文将在非均匀概率空间中,利用概率测度的方法,研究了四种常见n值逻辑系统中公式赋值为i/(n-1)的概率测度服从某一类离散型随机分布列的情形,引入了逻辑公式概率真度的概念和表达式,讨论了其性质。在非均匀概率空间中,每个原子公式的赋值不一定相等,这样就更接近实际生活中的应用,具有更好的应用价值。然后在剩余格上引入了L-模糊相似的定义,给出了四种n值逻辑系统中公式间相似度的定义和统一表达式,并研究了相似度的若干特征性质;最后引入了公式间的伪距离,为n值命题逻辑的近似推理理论提供了另外一种可能的框架。
The characteristic of mathematical logic lies in symbolization and formalization, which is quite different from computational mathematics. The former emphasises on formal reasoning and strict proof, while the later concerns with numberical computation and approximate solutions. Computational mathematics is more flexible than mathematical logic. It can build the connection between mathematical logic and computational mathematics by bringing the numberical calculation into mathematical logic.
The mean of "designated truth values" in mathematical logic had reflected the idea of graded method. Also, graded method is the reflection of ideology. Professor Wang Guojun established the quantitative logic based on basic logic concepts. Consequently, numberical calculation was introduced in the mathematics logic. It makes the mathematic logic more flexible and useable.
In Wang's paper[9], the truth degree was introduced.At the same time, similarity and pseudo-distance were also defined based on the concept of truth degree which provides a possible framework for approximate.
The main work of this paper includes:
Chapter One: The study of algebra framework in fuzzy logic systems is an important task. And it had become an embranchment in the research of non-classical logic systems. In this chapter, some preliminaries are recalled and the characteristics of t-norm and residuated lattices are investigated. And also put out the characteristic of t-norm and residuated lattices.
Chapter Two: In [10], by means of infinite product measure of uniformly distributed probability spaces, the author made mention of the truth degree theory, which was based on measure theory. Later, the truth degree theory was extended to the n-valued Lukasiewicz logic system by using this method. But, all the above works run in the uniformly distributed probability spaces. This definition of truth degree was built on infinite product measure, therefore, the calculation of truth degree is very difficult and can not be realized by algorithm. This chapter will avoid the concept of infinite product measure, and give a new definition of truth degree-absolute truth degree. It's a reformative and laconic definition which makes it possible to compute truth degrees of formulas by computer .
Chapter Three: In classical two-valued propositional logic system, truth degrees of formulas were defined by measure theory. Also the similarity and pseudo-distance were from the viewpoint of equal probability. The possibility that each atomic propositionholds is uniform and equals to (1/2). The randomization characteristic of probability was not reflected under this method. In this chapter, firstly, based on the method of probability measure, the concept of probability truth degrees and introduced its general expression in unevenly probability space is given. Secondly, the similarity degree between two formulas on a residuated lattice is defined in terms of L-fuzzy similarity. And the character properties are discussed in four types of n—valued logic systems based on the left-continuous t-norms. Lastly, a pseudo-distance was introduced by means of probability truth degree, which provides another possible framework for approximate reasoning in n-valued propositional logics. It can provide a kind of ingerence frame for the approximate reasoning theory.
引文
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[2]王元元.计算机科学中的逻辑学[M].北京:科学出版社,1989.
[3]胡世华,陆钟万.数理逻辑基础[M].北京:科学出版社,1983.
[4]Hongjun Zhou,Guojun Wang.Generalized consistency degrees of theories w.r.t,formulas in several standard complete logic systems]J].Fuzzy Sets and Systems,2006,157(15):2058-2073.
[5]Pei Daowu.R_0 implication:characteristics and applications[J].Fuzzy Sets and Systems,2002,131:297-302.
[6]王国俊.数理逻辑引论与归结原理(第2版)[M].北京:科学出版社,2006.
[7]李骏,王国俊.Godel n值命题逻辑中命题的α-真度理论[J].软件学报,2007,18(1):33-39.
[8]王国俊,傅丽,宋建社.二值命题逻辑中的真度理论[J].中国科学,A辑,2001,31(11):998-1008.
[9]王国俊,王伟.逻辑度量空间[J].数学学报,2001,44(1):159-168.
[10]王国俊,李璧镜.Lukasiewicz n值命题逻辑中公式的真度理论和极限定理[J].中国科学,E辑,2005,35(6):561-569.
[11]王国俊.三Ⅰ方法与区间值模糊推理[J].中国科学(E辑),2000,30(4):331-340.
[12]王国俊.广义MP规则[J].陕西师范大学学报,2000,28(3):1-9.
[13]王国俊.计量逻辑学(Ⅰ)[J].工程数学学报,2006,2:191-215.
[14]王国俊.非经典数理逻辑于近似推理[M].北京:科学出版社,2003.
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