逻辑等价算子在模糊推理中的应用

发布时间：2018-05-23 21:44

本文选题：逻辑等价算子 + 模糊推理　；参考：《陕西师范大学》2015年博士论文

【摘要】：模糊推理是模糊控制的理论基础,鲁棒性是评判模糊推理的重要标准.在讨论鲁棒性时,扰动参数的选取极为关键.我们常用的扰动参数大多是建立在[0,1]单位区间上通常度量的基础之上.然而模糊推理的结果很大程度上取决于它的内蕴结构,蕴涵算子和模糊连接词.逻辑等价算子由蕴涵算子生成,因此用逻辑等价算子构造的扰动参数讨论鲁棒性,与逻辑推理会更为和谐.本文的第一个研究目的在于借助逻辑等价算子构造一系列的扰动参数,进而讨论三I推理方法的鲁棒性.另一个研究目的在于借助拓扑学工具,对这些扰动参数作以比较,并对一些逻辑等价算子导出的相关拓扑性质作以探讨.全文共分四章：第一章首先介绍模糊推理的基本概念和若干重要的模糊推理方法,其次介绍剩余格和拓扑学的相关知识,为后面章节的研究作必要的准备.第二章首先借助逻辑等价算子定义了模糊集之间的平均逻辑相似度,并以这种平均逻辑相似度作为扰动参数讨论了逻辑连接词的鲁棒性和三I推理方法的鲁棒性.其次,将极小逻辑相似度和平均逻辑相似度从拓扑的角度做了比较,得出这两种相似度导出的度量空间是等价的.最后,分析了不同的蕴涵算子导出的度量空间中孤立点的分布情况及连通性、稠密子集等性质.在此基础上,对不同的相似度从扰动参数的角度作了比较.第三章首先将模糊集之间的极小逻辑相似度推广到格值模糊集上,在此基础上构造了FL(X)上的拓扑空间,其中FL(X)代表论域X上取值于格L的格值模糊集的全体.指出了由R0蕴涵算子和Godel蕴涵算子所确定的拓扑空间中凝聚点是正规模糊集或其补集,但反之不真且构造了反例.其次,当L是剩余格时,借助逻辑等价算子和三角模,构造出两个格值相似度E1,E2,证明了这两者均是L-等式,并指出Ei-Cauchy列是Ei-收敛列,i=1,2.再次,为了更直观地表现两个格值模糊集的相似程度,借助逻辑等价算子m和[0,1]MV上的态算子定义了取值于[0,1]的Ⅰ-Ⅳ型相似度：Sm,Sm,Sm*,和Sm,证明了它们满足格值相似度的三条公理,并将Sm作为扰动参数,讨论了RL-型三I算法的鲁棒性.在以上各种相似度的构造中,剩余格上的两种逻辑等价算子起着关键作用,我们从这两个逻辑等价算子出发建立了剩余格上的-致拓扑和商剩余格.特别地,对[0,1]剩余格中不同的逻辑等价算子导出的度量空间的紧性和序列的收敛性做了详细讨论.第四章首先在完备格上分析了两对剩余算子复合后的性质,给出了反例说明这种复合不具有还原性.讨论了聚合双极信息的双极t-模和双极蕴涵可以分解为两个单级算子的条件,并给出了双极信息在决策方面的一个应用实例.表示双极信息的工具之一就是双极模糊集,双极模糊集又称为直觉模糊集.直觉模糊集作为特殊的格值模糊集有着特殊的应用背景和性质.其次,借助逻辑等价算子,建立了直觉模糊集上的四类相似度并且详细讨论了其性质,继而以这些相似度作为扰动参数,从鲁棒性分析的角度,对它们作了比较.最后,给出了一个模式识别问题的应用实例.
[Abstract]:Fuzzy reasoning is the theoretical basis of fuzzy control. Robustness is an important criterion for evaluating fuzzy reasoning. When the robustness is discussed, the selection of disturbance parameters is very important. Most of the disturbance parameters we commonly use are based on the usual measurement in the [0,1] unit interval. However, the results of fuzzy reasoning largely depend on its internal parameters. Implication structure, implication operator and fuzzy connectives. Logic equivalent operators are generated by implication operators. Therefore, the disturbance parameters constructed by logical equivalence operators are more robust and more harmonious with logical reasoning. The first study of this paper is to construct a series of perturbation parameters with the aid of logical equivalence operator, and then discuss the three I reasoning method. Another research aim is to compare these disturbance parameters with the tools of topology and discuss the related topological properties derived from some logical equivalent operators. The full text is divided into four chapters. The first chapter first introduces the basic concepts of fuzzy reasoning and some important fuzzy reasoning methods. Secondly, the residual lattice and topology are introduced. In the second chapter, the average logical similarity between fuzzy sets is defined by the use of logical equivalence operators. The robustness of logical connectives and the robustness of the three I reasoning method are discussed with this average logical similarity as a disturbance parameter. Secondly, the minimal logic similarity is similar. The degree and the average logical similarity are compared from the topological point of view, and the measurement space derived from the two similarities is equivalent. Finally, the distribution of the isolated points in the metric spaces derived from the different implication operators and the properties of the dense subsets are analyzed. On this basis, the different similarity degrees are from the angle of the disturbance parameters. In the third chapter, the third chapter generalize the minimal logical similarity between the fuzzy sets to the Lattice valued fuzzy set. On this basis, the topological space on FL (X) is constructed, in which the FL (X) represents the Lattice valued fuzzy set of lattice L on the domain X. The aggregation points in the topological space determined by the R0 implication operator and Godel implication operator are pointed out. A normal fuzzy set or its complement, but vice versa is not true. Secondly, when L is a residual lattice, two lattice values similarity E1, E2 are constructed with the aid of logical equivalence operator and trigonometric model. It is proved that both of these are L- equations, and that the Ei-Cauchy column is a Ei- convergence column and i=1,2. again, in order to more intuitively show the similarity of the two Lattice valued fuzzy sets. Degree, by means of the state operators on logical equivalent operators m and [0,1]MV, I define the type I IV similarity degrees of value in [0,1]: Sm, Sm, Sm*, and Sm, and prove that they satisfy three axioms of the lattice value similarity, and discuss the robustness of the RL- type three I algorithm by using Sm as a perturbation parameter. In the construction of the above similarity, two kinds of logic on the residual lattice are constructed. The key role plays a key role. We set up the topological and quotient residual lattices on the remaining lattices from the two logical equivalence operators. In particular, the tightness and the convergence of the metric spaces derived from the different logical equivalent operators in the [0,1] residual lattices are discussed in detail. The fourth chapter first analyzes two pairs in the complete lattice. A counterexample is given to show that the compound is not reductive. The conditions for the bipolar t- mode and the bipolar implication to be decomposed into two single level operators are discussed, and an application example of the bipolar information in decision making is given. One of the tools for the bipolar information is a bipolar fuzzy set, The bipolar fuzzy sets are also called intuitionistic fuzzy sets. The intuitionistic fuzzy sets have special application background and properties as special Lattice valued fuzzy sets. Secondly, four kinds of similarity degrees on intuitionistic fuzzy sets are established with the help of logical equivalent operators and their properties are discussed in detail. Then, the angle of similarity is used as a perturbation parameter, from the angle of robustness analysis. Finally, a practical example of pattern recognition is given.
【学位授予单位】：陕西师范大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O159;O231

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