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.     

Fuzzy conditional inference under max-⊙ composition

This paper shows that the majority of fuzzy inference methods for a fuzzy conditional proposition “If x is A then y is B,” with A and B fuzzy concepts, can infer very reasonable consequences which fit our intuition with respect to several criteria such as modus ponens and modus tollens, if a new composition called “max-⊙ composition” is used in the compositional rule of inference, though reasonable consequences cannot always be obtained when using the max-min composition, which is used usually in the compositional rule of inference. Furthermore, it is shown that a syllogism holds for the majority of the methods under the max-⊙ composition, though they do not always satisfy the syllogism under the max-min composition.     

基于直觉模糊逻辑的近似推理方法

针对直觉模糊逻辑及命题演算，提出了利用隶属度和犹豫度计算直觉模糊逻辑命题真值的合成方法．给出了直觉模糊逻辑命题的运算规则，重点研究了基于直觉模糊逻辑的近似推理方法．该方法包括直觉模糊取式推理，直觉模糊拒武推理及直觉模糊假官推理．井推导了相关的推理合成运算公式．以具体算例验证和表明了所提出的推导方法的正确性和有效性，以及对方法进行验证的详细步骤．     

一种基于区间规则的条件证据网络推理决策方法

针对证据网络推理方法无法对区间规则进行表示和推理的问题, 提出一种基于区间规则的条件证据网络推理决策方法. 该方法针对模糊规则的条件概率或信度为不确定区间的情况, 可同时表达不确定性和模糊性; 并将区间不确定规则转化为区间条件信度函数作为证据网络的结点参数, 通过条件推理和证据融合得到条件证据网络中各结点幂集空间中焦元的随机分布作为决策依据. 最后, 通过空中目标态势评估实例, 验证了所提出方法的有效性.

     

The construction of a fuzzy inference network by extension of the rule inference network

Fuzzy logic can bring about inappropriate inferences as a result of ignoring some information in the reasoning process. Neural networks are powerful tools for pattern processing, but are not appropriate for the logical reasoning needed to model human knowledge. The use of a neural logic network derived from a modified neural network, however, makes logical reasoning possible. In this paper, we construct a fuzzy inference network by extending the rule–inference network based on an existing neural logic network. The propagation rule used in the existing rule–inference network is modified and applied. In order to determine the belief value of a proposition pertaining to the execution part of the fuzzy rules in a fuzzy inference network, the nodes connected to the proposition to be inferenced should be searched for. The search costs are compared and evaluated through application of sequential and priority searches for all the connected nodes.     

一种基于真度变换观点的近似推理

以全新的思想和视角,把蕴涵式p→q看作一种真度变换,并提出了真度变换率和真度变换差的概念,然后在此基础上给出了一组称为肯定前件式和否定后件式真度假言推理的推理规则,从而得到了一种命题近似推理的新方法。把该方法推广到谓词逻辑,就得到一种基于谓词逻辑的近似推理新方法。因此,文章的思想和方法可作为模糊推理的理论基础。     

Towards more realistic (e.g., non-associative) “and”- and “or”-operations in fuzzy logic

How is fuzzy logic usually formalized? There are many seemingly reasonable requirements that a logic should satisfy: e.g., since A B and B A are the same, the corresponding and-operation should be commutative. Similarly, since A A means the same as A, we should expect that the and-operation should also satisfy this property, etc. It turns out to be impossible to satisfy all these seemingly natural requirements, so usually, some requirements are picked as absolutely true (like commutativity or associativity), and others are ignored if they contradict to the picked ones. This idea leads to a neat mathematical theory, but the analysis of real-life expert reasoning shows that all the requirements are only approximately satisfied. we should require all of these requirements to be satisfied to some extent. In this paper, we show the preliminary results of analyzing such operations. In particular, we show that non-associative operations explain the empirical 7±2 law in psychology according to which a person can normally distinguish between no more than 7 plus minus 2 classes.     

DF控制推理模型的研究

针对控制系统中对象的模糊性和动态性,基于动态模糊集(Dynamic Fuzzy Sets)及动态模糊逻辑(Dynamic FuzzyLogic)系统理论,给出DF控制推理模型的相关概念,如DF向量、DF语言变量、DF语言规则和DF蕴涵关系等,并在此基础上探讨基于DF语言规则的DF推理方法,最后通过实例说明这些概念和方法的应用。     

Interval type-2 fuzzy expert system for prediction of carbon monoxide concentration in mega-cities

This paper presents an indirect approach to interval type-2 fuzzy logic system modeling to forecaste the level of air pollutants. The type-2 fuzzy logic system permits us to model the uncertainties among rules and the parameters related to data analysis. In this paper, we propose an indirect method to create an interval type-2 fuzzy logic system from a historical data, where Footprint of Uncertainties of fuzzy sets are extracted by implementation of an interval type-2 FCM algorithm and based on an upper and lower value for the level of fuzziness m in FCM. Finally, the proposed model is applied for prediction of carbon monoxide concentration in Tehran air pollution. It is shown that the proposed type-2 fuzzy logic system is superior in comparison to type-1 fuzzy logic systems in terms of two performance indices.     

A fuzzy expert system shell using both exact and inexact reasoning

This paper presents a comprehensive expert system shell which can deal with both exact and inexact reasoning. A prototype of this proposed shell, code named as SYSTEM Z-IIe, has been implemented successfully. It is a rule-based system which employs fuzzy logic and numbers for its reasoning. Two basic inexact concepts, fuzziness and uncertainty, are both used and distinct from each other clearly in the system. Moreover, these two concepts have been built into two levels for inexact reasoning, i.e. the level of the rules and facts, and the level of the values of the objects of these rules and facts. Other features of Z-IIe include multiple fuzzy propositions in rules and dual fact input mechanisms. It also allows any combinations of fuzzy and normal terms and uncertainties. Fuzzy numeric comparison logic control is also available for the rules and facts. Its natural language interface which uses English with restricted syntax improves the efficiency of knowledge engineering. Z-IIe is also coupled to a Database Management System for supplying facts from existing databases if appropriate. All these features can be combined to build very powerful expert systems and are illustrated by an example.     

Is there a need for fuzzy logic?

“Is there a need for fuzzy logic?” is an issue which is associated with a long history of spirited discussions and debate. There are many misconceptions about fuzzy logic. Fuzzy logic is not fuzzy. Basically, fuzzy logic is a precise logic of imprecision and approximate reasoning. More specifically, fuzzy logic may be viewed as an attempt at formalization/mechanization of two remarkable human capabilities. First, the capability to converse, reason and make rational decisions in an environment of imprecision, uncertainty, incompleteness of information, conflicting information, partiality of truth and partiality of possibility - in short, in an environment of imperfect information. And second, the capability to perform a wide variety of physical and mental tasks without any measurements and any computations [L.A. Zadeh, From computing with numbers to computing with words - from manipulation of measurements to manipulation of perceptions, IEEE Transactions on Circuits and Systems 45 (1999) 105-119; L.A. Zadeh, A new direction in AI - toward a computational theory of perceptions, AI Magazine 22 (1) (2001) 73-84]. In fact, one of the principal contributions of fuzzy logic - a contribution which is widely unrecognized - is its high power of precisiation.Fuzzy logic is much more than a logical system. It has many facets. The principal facets are: logical, fuzzy-set-theoretic, epistemic and relational. Most of the practical applications of fuzzy logic are associated with its relational facet.In this paper, fuzzy logic is viewed in a nonstandard perspective. In this perspective, the cornerstones of fuzzy logic - and its principal distinguishing features - are: graduation, granulation, precisiation and the concept of a generalized constraint.A concept which has a position of centrality in the nontraditional view of fuzzy logic is that of precisiation. Informally, precisiation is an operation which transforms an object, p, into an object, p^∗, which in some specified sense is defined more precisely than p. The object of precisiation and the result of precisiation are referred to as precisiend and precisiand, respectively. In fuzzy logic, a differentiation is made between two meanings of precision - precision of value, v-precision, and precision of meaning, m-precision. Furthermore, in the case of m-precisiation a differentiation is made between mh-precisiation, which is human-oriented (nonmathematical), and mm-precisiation, which is machine-oriented (mathematical). A dictionary definition is a form of mh-precisiation, with the definiens and definiendum playing the roles of precisiend and precisiand, respectively. Cointension is a qualitative measure of the proximity of meanings of the precisiend and precisiand. A precisiand is cointensive if its meaning is close to the meaning of the precisiend.A concept which plays a key role in the nontraditional view of fuzzy logic is that of a generalized constraint. If X is a variable then a generalized constraint on X, GC(X), is expressed as X isr R, where R is the constraining relation and r is an indexical variable which defines the modality of the constraint, that is, its semantics. The primary constraints are: possibilistic, (r = blank), probabilistic (r = p) and veristic (r = v). The standard constraints are: bivalent possibilistic, probabilistic and bivalent veristic. In large measure, science is based on standard constraints.Generalized constraints may be combined, qualified, projected, propagated and counterpropagated. The set of all generalized constraints, together with the rules which govern generation of generalized constraints, is referred to as the generalized constraint language, GCL. The standard constraint language, SCL, is a subset of GCL.In fuzzy logic, propositions, predicates and other semantic entities are precisiated through translation into GCL. Equivalently, a semantic entity, p, may be precisiated by representing its meaning as a generalized constraint.By construction, fuzzy logic has a much higher level of generality than bivalent logic. It is the generality of fuzzy logic that underlies much of what fuzzy logic has to offer. Among the important contributions of fuzzy logic are the following:

1.

FL-generalization. Any bivalent-logic-based theory, T, may be FL-generalized, and hence upgraded, through addition to T of concepts and techniques drawn from fuzzy logic. Examples: fuzzy control, fuzzy linear programming, fuzzy probability theory and fuzzy topology.

2.

Linguistic variables and fuzzy if-then rules. The formalism of linguistic variables and fuzzy if-then rules is, in effect, a powerful modeling language which is widely used in applications of fuzzy logic. Basically, the formalism serves as a means of summarization and information compression through the use of granulation.

3.

Cointensive precisiation. Fuzzy logic has a high power of cointensive precisiation. This power is needed for a formulation of cointensive definitions of scientific concepts and cointensive formalization of human-centric fields such as economics, linguistics, law, conflict resolution, psychology and medicine.

4.

NL-Computation (computing with words). Fuzzy logic serves as a basis for NL-Computation, that is, computation with information described in natural language. NL-Computation is of direct relevance to mechanization of natural language understanding and computation with imprecise probabilities. More generally, NL-Computation is needed for dealing with second-order uncertainty, that is, uncertainty about uncertainty, or uncertainty² for short.

In summary, progression from bivalent logic to fuzzy logic is a significant positive step in the evolution of science. In large measure, the real-world is a fuzzy world. To deal with fuzzy reality what is needed is fuzzy logic. In coming years, fuzzy logic is likely to grow in visibility, importance and acceptance.     

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     

Fuzzy set-based methods in instance-based reasoning

A formal framework of instance-based prediction is presented in which the generalization beyond experience is founded on the concepts of similarity and possibility. The underlying extrapolation principle is formalized within the framework of fuzzy rules. Thus, instance-based reasoning can be realized as fuzzy set-based approximate reasoning. More precisely, our model makes use of so-called possibility rules. These rules establish a relation between the concepts of similarity and possibility, which takes the uncertain character of similarity-based inference into account: inductive inference is possibilistic in the sense that predictions take the form of possibility distributions on the set of outcomes, rather than precise (deterministic) estimations. The basic model is extended by means of fuzzy set-based modeling techniques. This extension provides the basis for incorporating domain-specific (expert) knowledge. Thus, our approach favors a view of instance-based reasoning according to which the user interacts closely with the system     

Fuzzy arithmetic-based interpolative reasoning for nonlinear dynamic fuzzy systems

FAIR (fuzzy arithmetic-based interpolative reasoning)—a fuzzy reasoning scheme based on fuzzy arithmetic, is presented here. Linguistic rules of the Mamdani type, with fuzzy numbers as consequents, are used in an inference mechanism similar to that of a Takagi–Sugeno model. The inference result is a weighted sum of fuzzy numbers, calculated by means of the extension principle. Both fuzzy and crisp inputs and outputs can be used, and the chaining of rule bases is supported without increasing the spread of the output fuzzy sets in each step. This provides a setting for modeling dynamic fuzzy systems using fuzzy recursion. The matching in the rule antecedents is done by means of a compatibility measure that can be selected to suit the application at hand. Different compatibility measures can be used for different antecedent variables, and reasoning with sparse rule bases is supported. The application of FAIR to the modeling of a nonlinear dynamic system based on a combination of knowledge-driven and data-driven approaches is presented as an example.     

随机模糊环境下的命题逻辑真度理论*

在实单位区间[0,1]具有一定概率分布的基础上,引入命题逻辑公式的随机模糊意义下的真度概念,指出随机真度是已有文献中各种命题逻辑真度的共同推广.利用随机模糊真度定义公式间的随机模糊相似度,导出全体公式集上的一种伪距离——随机模糊逻辑伪距离,证明在随机模糊逻辑伪距离空间无孤立点.利用概率论中的积分收敛定理,证明一个关于随机模糊真度的极限定理.研究已有各种真度之间的联系.证明随机逻辑伪距离空间中逻辑运算的连续性,并将概率逻辑学基本定理推广至多值命题逻辑.在随机逻辑伪距离空间中提出2种不同类型的近似推理模式并应用于实际问题的近似推理.     

Interpreting and extracting fuzzy decision rules from fuzzy information systems and their inference

Information systems, which contain only crisp data, precise and unique attribute values for all objects, have been widely investigated. Due to the fact that in realworld applications imprecise data are abundant, uncertainty is inherent in real information systems. In this paper, information systems are called fuzzy information systems, and formalized by (objects; attributes; f), in which f is a fuzzy set and expresses some uncertainty between an object and its attribute values. To interpret and extract fuzzy decision rules from fuzzy information systems, the meta-theory based on modal logic proposed by Resconi et al. is modified. The modified meta-theory not only expresses uncertainty between objects and their attributes, but also uncertainty in the process of recognizing fuzzy information systems. In addition, according to perception computing (proposed by Zadeh), granules of fuzzy information systems can be represented by fuzzy decision rules, so that, fuzzy inference methods can be used to obtain the decision attribute of a new object. Finally, a novel way of combining evidences based on the modified meta-theory is introduced, which extends the concept of combining evidences based on Dempster-Shafer theory.     

Formalizing Default Reasoning

Fuzzy set systems can be used to solve the problem with uncertain knowledge,and default logic can be used to solve the problem with incomplete knowledge,in some sense.In this paper,based on interval-valued fuzzy sets we introduce a method of inference which combines approximate reasoning an default ogic,and give the procedure of transforming monotonic reasoning into default reasoning.     

归结与调解方法的有效性和完备性——基于扩充模糊逻辑

模糊集与模糊逻辑是处理大量存在的不确定性与模糊性信息的重要数学工具,在近似推理等领域有着广泛的应用。该文将王家兵等人提出的真值取在[0,1]区间上的带有相似性关系的模糊逻辑,扩充到很一般的与滋可比的有余完全分配格值逻辑中,将王家兵等人的许多结论进行了推广。首先对带有相似性关系的模糊逻辑的语义描述进行了扩充,然后讨论了在这种模糊推理中归结式与调解式的有效性,最后通过证明一个子句集在扩充模糊逻辑中的不可满足性与它在带有相等关系的二值逻辑中的不可满足性是等价的,得到了基于归结与调解方法对这种广义模糊演算的完备性。     

Complex fuzzy logic

A novel framework for logical reasoning, termed complex fuzzy logic, is presented in this paper. Complex fuzzy logic is a generalization of traditional fuzzy logic, based on complex fuzzy sets. In complex fuzzy logic, inference rules are constructed and "fired" in a manner that closely parallels traditional fuzzy logic. The novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets, characterized by complex-valued membership functions. The range of these membership functions is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane, thus providing a method for describing membership in a set in terms of a complex number. Several mathematical properties of complex fuzzy sets, which serve as a basis for the derivation of complex fuzzy logic, are reviewed in this paper. These properties include basic set theoretic operations on complex fuzzy sets - namely complex fuzzy union and intersection, complex fuzzy relations and their composition, and a novel form of set aggregation - vector aggregation. Complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic, while benefiting from the properties of complex numbers and complex fuzzy sets. The introduction of complex-valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties, and considerable potential for further research and application.