【作者】 李璧镜；

【导师】 王国俊；

【作者基本信息】 陕西师范大学 ， 基础数学， 2006， 硕士

【摘要】 近年来,模糊控制技术在应用方面取得了举世瞩目的成功。然而,作为其核心的模糊推理在数学基础上却并未无懈可击。所以,以研究模糊推理的数学基础为核心的模糊逻辑,作为一个全新的数学领域,引起了世界上许多著名学者的关注,并且取得了一系列重要的研究成果。模糊逻辑的应用范围十分广泛。一方面,它在机器自动证明理论,近似推理及模糊控制等领域有广泛的应用。另一方面,它也丰富和发展了纯粹数学理论的研究,例如,在证明理论的独立性和相容性方面的研究,在特殊的代数结构（如MV-代数,BL-代数,R₀代数）方面的研究,都引起了人们的极大兴趣并取得了成功。 模糊推理是模糊控制的理论基础,而各种各样的蕴涵算子又是模糊推理的数学工具。完备的逻辑系统的建立与蕴涵算子的选择密切相关,比如Lukasiewicz系统选择的就是Lukasiewicz蕴涵算子,Godel系统选择的就是Godel蕴涵算子,（?）^*系统选择的是R₀蕴涵算子等等。而这些蕴涵算子的最大特点就是可以与某个特定的三角模构成伴随对,我们将这类算子成为正则蕴涵算子。现在已经有了很多的研究工作都是关于蕴涵算子在模糊推理中的应用,然而却很少涉及其逻辑语义方面的性质。本文就主要从两个不同的角度研究蕴涵算子在语义方面的性质。 第一章 基础知识。本章扩充了原来公式集的范围,定义了新的公式集以及与之对应的赋值域。正则蕴涵算子和积分语义学的基本概念和性质也有所介绍。这些内容都为后面关于语义方面的讨论打好基础。 第二章 在Lukasiewicz n值命题逻辑中引入了公式的真度理论,得到了一个极限定理。表明当n趋于无穷时由公式的真度决定的真度函数τ_n收敛于积分真度τ,从而架起了离散值Lukasiewicz逻辑与连续值Lukasiewicz逻辑之间的桥梁。 第三章 针对能建立逻辑度量空间的这类系统,进行了语义方面的统一研究。首先,证明了全体逻辑公式在赋值域上是可积的,从而借助程度化的思想在这类系统中定义了统一形式的公式间的伪距离。同时讨论了逻辑度量空间中孤立点分布的部分情形,得到了在此类系统中,各种逻辑运算均连续,积分推理规则均成立的重要结论,从而使得在此类系统中建立统一的近似推理成为可能。更多还原

【Abstract】 In recent years, the applications of fuzzy control has obtained a great success. But fuzzy reasoning, the core of fuzzy control is suspected because of lacking firm mathematics foundation. In order to solve this problem, a new research field has been built up and more scholars devoted themselves to this field. Applications of fuzzy logic range from the field of computer science, including automated theorem proving or consistency proofs or the theory of particularly algebraic structures（e.g. MV-algebras, BL- algebras and R₀-algebras）.The theoretic foundation of fuzzy control is fuzzy reasoning. The variety of implication operators is one of the most important mathematical tools for fuzzy reasoning. It is well known that the choice of implication is very important to the fuzzy logical systems, for example the Lukasiewicz system use the Lukasiewicz implication, Godel system use the Godel implication and L^* system use R₀ implication etc. The great character of this implication is that they are adjoined with some definition of triangular norms. We call them regular implication. Much of the research has been done on the implications’ applications. However, among the reaserch only a little is related to semantic properties. The present paper tries to find some semantic properties of the logic systems corresponding to regular implication from two different aspects. The arrangement of this paper is as follows:Chapter One: Basic knowledge. In this chapter, the concept of the set of all formed formulas is enlarged, and the corresponding concept of valuational region is also redefined. Moreover, some important properties of the regular implications and integral semantic are introduced. All above is the basis of the later chapters.Chapter Two: The concept of truth degrees of formulas in Lukasiewicz n-valued propositional logic is proposed. A limit theorem is obtained, which says that the truth function induced by truth degrees converges to the integrated truth function when n converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Lukasiewicz logic and the continuous valued Lukasiewicz logic.Chapter Three: Towards to the class of logic syetems which could construct logic pseudo-metric spaces, we do some research on their semantic properties wholely. First of all, we prove that all of the formed formulas are measurable on the valuational region. Therefore, in these logics, we can define a pseudo-metric between every two formulas conformably using the graded idea naturally. Then we discuss only part situation of thedistribution of isolated points in the spaces. At last, we obtain the fact that in logic metric space all of the logic operators are continuous and the regulations of integral reasoning hold, which are important conclusions. So, it is made possible to construct an unified framework of approximate reasoning in these logic systems.更多还原