Lipschitz聚合算子的模糊推理鲁棒性研究

模糊推理中,输入和推理规则发生摄动时,内部连接算子的选择是影响推理输出的主要因素。给出了模糊Lipschitz聚合算子的定义,论证了满足Lipschitz条件的三角模算子和蕴涵算子,研究了一类稳定的Lipschitz聚合算子对模糊推理的鲁棒性影响,指出了当系统发生输入摄动和规则摄动时,内部连接算子为1-Lipschitz算子,能有效地抑制模糊推理的输出摄动,特别是当内部连接算子既是1-k∞-Lipschitz又是quasi-copulas时,模糊推理输出更稳定安全可行,模糊推理的鲁棒性得到更好调控;另外,从实验结果看,规则摄动对推理输出影响较大。实验部分既是对文中所提理论的很好验证,同时也是该理论在图像处理和人脸联想方面的具体应用。     

规则摄动时模糊蕴涵算子对模糊推理的鲁棒性的影响

列举了模糊规则发生摄动的常见情形,建立了一般性的模糊推理算法对规则摄动的鲁棒性的概念;就多重、链式和多维模糊推理情形,重点研究了一般性的模糊蕴涵算子对几个重要的模糊推理算法的这种鲁棒性的影响,并分别给出了相应的充分必要条件;初步尝试了通过一定的摄动制约来改善这种鲁棒性;同时指出了很多现有的模糊蕴涵算子使得所讨论的这些推理算法拥有好的鲁棒性,此时,即使规则中的隶属度有适度的粗糙或摄动,推理仍是可行的、安全的．文中工作对模糊系统的分析、模糊蕴涵算子的选择以及规则获取过程有一定的指导意义．     

逻辑等价度量下的模糊推理系统的鲁棒性

给出了基于模糊逻辑等价度量的模糊集的扰动的定义,讨论了模糊集扰动与模糊连接词及蕴涵算子扰动之间的关系,针对若干特殊的模糊连接词及蕴涵算子的扰动情形,给出了模糊推理系统的扰动的估计,并讨论了模糊推理系统的鲁棒性。     

Robustness of fuzzy reasoning via logically equivalence measure

In this paper, we discuss robustness of fuzzy reasoning. After proposing the definition of perturbation of fuzzy sets based on some logic-oriented equivalence measure, we present robustness results for various fuzzy logic connectives, fuzzy implication operators, inference rules and fuzzy reasoning machines, and discuss the relations between the robustness of fuzzy reasoning and that of fuzzy conjunction and implication operators. The robustness results are presented in terms of δ-equalities of fuzzy sets based on some logic-oriented equivalence measure, and the maximum of δ (which ensures the corresponding δ-equality holds) is derived.     

Robustness of fuzzy reasoning and δ-equalities of fuzzy sets

Fuzzy reasoning methods (or approximate reasoning methods) are extensively used in intelligent systems and fuzzy control. In this paper the author discusses how errors in premises affect conclusions in fuzzy reasoning, that is, he discusses the robustness of fuzzy reasoning. After reviewing his previous work (1996), he presents robustness results for various implication operators and inference rules. All the robustness results are formulated in terms of δ-equalities of fuzzy sets. Two fuzzy sets are said to be δ-equal if they are equal to an extent of δ     

An approach to measure the robustness of fuzzy reasoning

Fuzzy reasoning is intensively used in intelligent systems including fuzzy control, classification, expert systems, and networks to name a few dominant categories of such architectures. As being a fundamental construct permeating so many diverse areas, fuzzy reasoning was studied with respect to its fundamental properties such as robustness. The notion of robustness or sensitivity becomes of paramount importance by leading to a more comprehensive understanding of the way in which reasoning processes are developed. In this study, we introduce and study properties of some measures of robustness (or sensitivity) of fuzzy connectives and implication operators and discuss their relationships with perturbation properties of fuzzy sets. The results produced here are compared and contrasted with the previous findings available in the literature. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 393–413, 2005.     

基于Lukasiewicz三角模及其剩余蕴涵的模糊粗糙集

讨论基于Lukasiewicz三角模及其剩余蕴涵的模糊粗糙集模型,研究了相应模糊粗糙集的代数性质,证明了自反模糊关系下该模型中的下近似集构成一个模糊拓扑,且上、下近似算子恰为其闭包及内部算子。     

Reasoning with truth values on compacted fuzzy chained rules

In this paper, we consider the problem of executing a fuzzy knowledge base (FKB) with rule chaining. The inference process used as starting point is the one based on forward reasoning functions which, obtained from the compositional rule of inference, permits performing the execution of rules in the truth space. This way the process is totally independent from the universes of discourse in which the different variables are defined, allowing a homogeneous treatment for all the variables in the FKB. The execution of the rules is interpreted as the "propagation" of linguistic truth values of the linguistic truth variable that reflect the linguistic degree of fulfillment of each of the propositions in the rules. This execution process is analyzed in two fields of application: control systems, where it is customary to assume t-norm operators as implication functions, and the aggregation process is implemented through the maximum operator and expert systems applications, where other implication functions may be needed and t-norm operators are generally used as aggregation operators. For both of these situations, we present a compaction mechanism which allows a noticeable part of the operations to be performed a priori, thus achieving an important computation time saving.     

直觉模糊推理机的设计及其推理算子选择研究*

首先引入直觉模糊集下的模糊蕴涵算子运算方法,设计应用直觉模糊蕴涵算子的直觉模糊推理机模型的设计,阐述模型的运行机理,通过实例验证直觉模糊推理机模型的有效性和正确性.提出选择模糊蕴涵算子的贴近度法,在具体应用领域中可根据比较结果作具体选择.     

基于强正则蕴涵算子的加权模糊度量空间

在强正则蕴涵算子的统一框架下给出了加权正则度量的定义,建立了基于强正则蕴涵算子的加权正则模糊度量空间,并且研究了该模糊度量空间的性质,分析了常用的逻辑连接词所对应的映射关于加权正则度量的连续性,最后证明了基于Lukasiewicz蕴涵的加权正则模糊度量空间是最适宜于展开模糊推理的模糊度量空间。     

模糊蕴含运算和模糊推理研究

1 引言模糊推理是一种利用模糊集合论研究不确定性问题的方法。实际上多输入多输出系统的推理问题的关键是单输入单输出系统的模糊推理问题,因此本文只考虑单输入单输出系统的模糊推理问题。模糊推理中涉及到两个主要的问题:一是模糊蕴涵算子的选择,二是推理合成方式的选择。模糊蕴涵算子将规则转换成     

Applicability analysis of fuzzy inference by means of generalizedDempster-Shafer theory

The generalization of Dempster-Shafer (D-S) theory to fuzzy sets provides means for evidential reasoning on basis of fuzzy information. Fuzzy implication operators are used to turn partial fuzzy knowledge about a physical system into an inference engine. Presently there are more than 72 fuzzy implication operators known and investigated, evoking demand for strategies of finding the best performing fuzzy implication operator for the physical system under investigation. In this paper the authors propose such a concept based on generalized D-S theory. A convention to update prior knowledge regarding the applicability of fuzzy implication operators, with respect to fuzzy evidences pertaining to the physical system under consideration, is presented     

Transitive approximation of fuzzy relations by alternating closures and openings

 In this paper, a new method is proposed for approximating a fuzzy relation on a finite universe by a min-transitive fuzzy relation that is `close' to it. The method consists of a cascade of T-transitive closure and opening operations, where the t-norm T gradually progresses from the Lukasiewicz t-norm W to the minimum operator M. The underlying T-transitive opening heuristic is particularly interesting for t-norms T that belong to the class of copulas.     

Fuzzy reasoning Petri nets

This paper presents a fuzzy reasoning Petri net (FRPN) model to represent a fuzzy production rule-based system. The issues of how to represent and reason about rules containing negative literals are addressed in the proposed PN model. The execution rules based on the model are defined formally using the operators in max-algebra. Then, a fuzzy reasoning algorithm is proposed to perform fuzzy reasoning automatically. The algorithm is consistent with the matrix equation expression method in the traditional PNs and allows one to exploit the maximum parallel reasoning potential embedded in the model. The legitimacy and feasibility of the proposed approach are proved and validated through a turbine fault diagnosis expert system.     

一种蕴涵算子下FMP模型的新型反向三I算法

针对Fuzzy推理中提出的“过半可信”原则,证明了[RG]蕴涵算子满足这一原则,并讨论了[RG]蕴涵算子下FMP模型的新型反向三I支持算法,新型[α-]反向三I支持算法,新型[α-]反向三I约束算法的计算公式,且给予证明。     

Theoretical analysis of crisp-type fuzzy logic controllers usingvarious t-norm sum-gravity inference methods

The input-output parametric relationship of a class of crisp-type fuzzy logic controllers (FLCs) using various t-norm sum-gravity inference methods is studied. Four most important t-norms are used to calculate the matching level of each control rule and the explicit mathematical forms of reasoning surfaces obtained by using these four t-norms are addressed. Reasoning surfaces of these crisp-type FLCs are proved to be composed of a two-dimensional multilevel relay no matter which t-norm is used and a local position-dependent nonlinear compensator with output pattern influenced by the t-norms is selected. By analyzing the intrinsic operation of the four t-norms, the authors find that both standard intersection and algebraic product are suitable operators to perform the inference of the FLC. However, bounded difference and drastic intersection are disqualified because they cannot satisfy some important criteria. A measure of relative degree-of-nonlinearity is defined to examine the output figures of these crisp-type FLCs. The ultimate behavior of these crisp-type FLCs as the number of linguistic terms approaches infinity is also explored. The local stability criteria for the proportional-integral (PI)-type fuzzy control systems and the natural global stability characteristic for the proportional-derivative (PD)-type fuzzy control systems are also examined     

Fuzzy Interpolative Reasoning for Sparse Fuzzy Rule-Based Systems Based on ${bm alpha}$-Cuts and Transformations Techniques

In sparse fuzzy rule-based systems, the fuzzy rule bases are usually incomplete. In this situation, the system may not properly perform fuzzy reasoning to get reasonable consequences. In order to overcome the drawback of sparse fuzzy rule-based systems, there is an increasing demand to develop fuzzy interpolative reasoning techniques in sparse fuzzy rule-based systems. In this paper, we present a new fuzzy interpolative reasoning method via cutting and transformation techniques for sparse fuzzy rule-based systems. It can produce more reasonable results than the existing methods. The proposed method provides a useful way to deal with fuzzy interpolative reasoning in sparse fuzzy rule-based systems.      

A Survey on Fuzzy Implication Functions

One of the key operations in fuzzy logic and approximate reasoning is the fuzzy implication, which is usually performed by a binary operator, called an implication function or, simply, an implication. Many fuzzy rule based systems do their inference processes through these operators that also take charge of the propagation of uncertainty in fuzzy reasonings. Moreover, they have proved to be useful also in other fields like composition of fuzzy relations, fuzzy relational equations, fuzzy mathematical morphology, and image processing. This paper aims to present an overview on fuzzy implication functions that usually are constructed from t-norms and t-conorms but also from other kinds of aggregation operators. The four most usual ways to define these implications are recalled and their characteristic properties stated, not only in the case of [0,1] but also in the discrete case.     

Atanassov’s intuitionistic fuzzy submodules with respect to a t-norm

In this paper, t-norm based intuitionistic fuzzy submodules are defined, and their various properties are investigated. We also show that if an intuitionistic fuzzy set of a module is an intuitionistic fuzzy submodule with respect to a t-norm, then it is also an intuitionistic fuzzy submodule with respect to the annihilation t-norm. We investigate the nature of intuitionistic fuzzy submodules of a module with the help of their respective cut sets. We obtain some results based on (α, β)-level sets and t-norm based α-cut sets of an intuitionistic fuzzy submodule with respect to a t-norm T.     

Fuzzy backward reasoning using fuzzy Petri nets

Chen, Ke and Chang (1990) have presented a fuzzy forward reasoning algorithm for rule-based systems using fuzzy Petri nets. In this paper, we extend the work of Chen, Ke and Chang (1990) to present a fuzzy backward reasoning algorithm for rule-based systems using fuzzy Petri nets, where the fuzzy production rules of a rule-based system are represented by fuzzy Petri nets. The system can perform fuzzy backward reasoning automatically to evaluate the degree of truth of any proposition specified by the user. The fuzzy backward reasoning capability allows the computers to perform reasoning in a more flexible manner and to think more like people.