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.     

Evolution of fuzzy logic: from intelligent systems and computation to human mind

Inspired by human’s remarkable capability to perform a wide variety of physical and mental tasks without any measurements and computations and dissatisfied with classical logic as a tool for modeling human reasoning in an imprecise environment, Lotfi A. Zadeh developed the theory and foundation of fuzzy logic with his 1965 paper “Fuzzy sets” (Zadeh in Inf Control 8:378–53, 1965) and extended his work with his 2005 paper “Toward a generalized theory of uncertainty (GTU)—an outline” (Zadeh in Inf Control, 2005). Fuzzy logic has at least two main sources over the past century. The first of these sources was initiated by Peirce in the form what he called a logic of vagueness in 1900s, and the second source is Lotfi’s A. Zadeh work, fuzzy sets and fuzzy Logic in the 1960s and 1970s.     

Modeling and implementation of Z-number

Computing with words provides symbolic and semantic methodology to deal with imprecise information associated with natural languages. It encapsulates various fuzzy logic techniques developed in past decades and formalizes them. Z-number is an emerging paradigm that has been utilized in computing with words among others. The concept of a Z-number is intended to provide a basis for computation with numbers, specifically with reliability of information. Z-numbers are in confluence between the two most prominent approaches to uncertainty, probability and possibility, that allow computations on complex statements. Certain computations related to Z-numbers are ambiguous and complicated leading to its slow adaptation into areas such as computing with words. The biggest contributing factor to the complexity is the usage of probability distributions in the computations. This paper seeks to provide an applied model of Z-number based on certain realistic assumptions regarding probability distributions. Algorithms are presented to implement this model and integrate it into an expert system shell for computing with words called CWShell. CWShell is a software tool that abstracts the underlying computation required for computing with words, and provides a convenient way to represent and compute with unstructured natural language using specialized language called Generalized constraint language (GCL). This paper introduces new constructs for Z-numbers to GCL and provides detailed inference mechanism and computation strategy on those constructs. We present two case studies to demonstrate the working and feasibility of the approach.     

模糊交互时态逻辑的模型检测

交互时态逻辑已被广泛应用于开放系统的规范描述,交互时态逻辑的模型检测技术是一个比较重要的验证方法。为了形式化描述和验证具有模糊不确定性信息的开放系统的性质,提出了一种模糊交互时态逻辑,并讨论了它的模型检测问题。首先,引入了模糊交互时态逻辑的基于路径和基于不动点的两种语义,证明了其等价性。然后,基于其等价性,给出了模糊交互时态逻辑的模型检测算法和复杂性分析。     

Information Compression by Multiple Alignment, Unification and Search as a Unifying Principle in Computing and Cognition

This article presents an overview ofthe idea that information compression bymultiple alignment, unification and search(ICMAUS) may serve as a unifying principle incomputing (including mathematics and logic) andin such aspects of human cognition as theanalysis and production of natural language,fuzzy pattern recognition and best-matchinformation retrieval, concept hierarchies withinheritance of attributes, probabilisticreasoning, and unsupervised inductive learning.The ICMAUS concepts are described together withan outline of the SP61 software model in whichthe ICMAUS concepts are currently realised. Arange of examples is presented, illustratedwith output from the SP61 model.     

THE ARCHITECTURE OF SYSTEMS PROBLEM SOLVING by George J. Klir. Plenum Press,New York, 1985, xvi + 540 pages.

This is an overview paper presenting the main results obtained by the author and his colleagues in the field of fuzzy logic and modelling of natural language semantics and is composed of an introduction followed by two main parts. Section 2 discusses our results in terms of fuzzy logic, namely that of prepositional and first-order fuzzy logic based on a residuated lattice of truth values. Section 3 presents the concept of the Alternative mathematical Model of natural Language semantics and pragmatics (AML) the development of which is based on a philosophical approach rather than the approaches usually adopted in most classical papers.     

Automatic fuzzy ontology generation for semantic Web

Ontology is an effective conceptualism commonly used for the semantic Web. Fuzzy logic can be incorporated to ontology to represent uncertainty information. Typically, fuzzy ontology is generated from a predefined concept hierarchy. However, to construct a concept hierarchy for a certain domain can be a difficult and tedious task. To tackle this problem, this paper proposes the FOGA (fuzzy ontology generation framework) for automatic generation of fuzzy ontology on uncertainty information. The FOGA framework comprises the following components: fuzzy formal concept analysis, concept hierarchy generation, and fuzzy ontology generation. We also discuss approximating reasoning for incremental enrichment of the ontology with new upcoming data. Finally, a fuzzy-based technique for integrating other attributes of database to the ontology is proposed.     

Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge

Abstract: Just like rough set theory, fuzzy set theory addresses the topic of dealing with imperfect knowledge. Recent investigations have shown how both theories can be combined into a more flexible, more expressive framework for modelling and processing incomplete information in information systems. At the same time, intuitionistic fuzzy sets have been proposed as an attractive extension of fuzzy sets, enriching the latter with extra features to represent uncertainty (on top of vagueness). Unfortunately, the various tentative definitions of the concept of an ‘intuitionistic fuzzy rough set’ that were raised in their wake are a far cry from the original objectives of rough set theory. We intend to fill an obvious gap by introducing a new definition of intuitionistic fuzzy rough sets, as the most natural generalization of Pawlak's original concept of rough sets.     

广义可能性决策过程的计算树逻辑模型检测

模型检测作为一种形式化验证技术,已被广泛应用于各种并发系统的正确性验证。针对具有非确定性选择和广义可能性分布的并发系统,引入广义可能性决策过程作为此类系统的模型;给出描述其性质的规范语言广义可能性计算树逻辑的概念;研究此类系统的广义可能性计算树逻辑模型检测问题。结论表明,其模型检测算法的时间复杂度也为多项式时间。所获得的结果扩大了广义可能性测度在模型检测中的应用范围。     

Generalized and group-based generalized intuitionistic fuzzy soft sets with applications in decision-making

Intuitionistic fuzzy soft set (IFSS) theory acts as a fundamental tool for handling the uncertainty in the data by adding a parameterizing factor during the process as compared to fuzzy and intuitionistic fuzzy set (IFS) theories. In this paper, an attempt has been made to this effect to describe the concept of generalized IFSS (GIFSS), as well as the group-based generalized intuitionistic fuzzy soft set (GGIFSS) in which the evaluation of the object is done by the group of experts rather than a single expert. Based on this information, a new weighted averaging and geometric aggregation operator has been proposed by taking the intuitionistic fuzzy parameter. Finally, a decision-making approach based on the proposed operator is being built to solve the problems under the intuitionistic fuzzy environment. An illustrative example of the selection of the optimal alternative has been given to show the developed method. Comparison analysis between the proposed and the existing operators have been performed in term of counter-intuitive cases for showing the superiority of the approach.     

Fuzzy logic = computing with words

As its name suggests, computing with words (CW) is a methodology in which words are used in place of numbers for computing and reasoning. The point of this note is that fuzzy logic plays a pivotal role in CW and vice-versa. Thus, as an approximation, fuzzy logic may be equated to CW. There are two major imperatives for computing with words. First, computing with words is a necessity when the available information is too imprecise to justify the use of numbers, and second, when there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost, and better rapport with reality. Exploitation of the tolerance for imprecision is an issue of central importance in CW. In CW, a word is viewed as a label of a granule; that is, a fuzzy set of points drawn together by similarity, with the fuzzy set playing the role of a fuzzy constraint on a variable. The premises are assumed to be expressed as propositions in a natural language. In coming years, computing with words is likely to evolve into a basic methodology in its own right with wide-ranging ramifications on both basic and applied levels     

Applications of a novel fuzzy expert system shell

Abstract: Much of the information resident in the knowledge base of a typical expert system is imprecise, incomplete or not totally reliable. The special features of a novel expert system shell based on fuzzy logic and numbers are presented. This rule-based system can deal with exact, inexact (fuzzy) and combined reasoning as well as uncertainty represented by fuzzy numbers. Natural language interface is built in naturally using fuzzy logic representation. Several application areas, namely, classification, risk analysis and information retrieval, are discussed with four appropriate sample expert systems actually built using this shell. Through these expert systems, the additional power and advantages over traditional expert systems are illustrated. It has been demonstrated that the introduction of fuzzy concepts into expert systesms is not feasible but highly desirable.     

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。     

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.     

基于组合逻辑的故障诊断方法

针对复杂系统故障传播和故障分析的模糊性和不确定性,首先,在逻辑Petri网和模糊Petri网的理论基础上,根据逻辑Petri网的传值不确定性以及模糊Petri网对模糊信息的表示和推理能力的特点,提出模糊逻辑Petri网的概念及推理规则,考虑不同故障源对故障的影响程度,将概率信息引入模糊逻辑Petri网,对故障源赋予置信度,使故障诊断过程更符合实际。其次,利用模糊逻辑Petri网对故障诊断系统进行建模,用模糊逻辑Petri网描述了系统故障状态组合的逻辑关系,并进一步简化了系统模型的表达形式,具有良好的封装性、重构性和可维护性,在一定程度上缓解了状态组合空间爆炸问题。针对故障的传播性,采用可达性分析方法对故障信息的传播路径进行模拟论证,提高了故障诊断效率。最后,通过离心式压缩机故障诊断过程实例分析,验证了该方法的有效性和可行性,提高了故障诊断过程的准确性和高效性。     

Variable precision intuitionistic fuzzy rough set model and applications based on conflict distance

Due to the complexity and uncertainty of the physical world, as well as the limitation of human ability to comprehend, it is very difficult for any single method of uncertainty to effectively deal with the decision‐making problem that exists in real life. So, it is natural for us to think about incorporating the advantages of various theories of uncertainty to develop a more powerful hybrid method of soft decision‐making. In view of this recognition, the thought and method of intuitionistic fuzzy sets and variable precision rough sets are used to construct a novel intuitionistic fuzzy rough set model. With respect to the fact that the information system is intuitionistic fuzzy, the idea of measuring intuitionistic fuzzy similarity is used to define conflict distance. After that, this concept is combined with the variable precision rough sets so that a variable precision intuitionistic fuzzy rough set model is established, and its properties are investigated. After proposing an attribute reduction algorithm based on variable precision intuitionistic fuzzy rough sets, a case study is used to verify the feasibility and effectiveness of our novel model. The results show that our model indeed improves the classification ability of earlier models and possesses some ability to tolerate faults through adjusting the parameter λ and the confidence threshold β; it realizes the correct classification and extracts the decision rules.     

Human centricity and information granularity in the agenda of theories and applications of soft computing

Soft computing is an interdisciplinary area that focuses on the design of intelligent systems to process uncertain, imprecise and incomplete information. It mainly builds on fuzzy sets theory, fuzzy logic, neural computing, optimization, evolutionary algorithms, and approximate reasoning et al. Information granularity is in general regarded as a crucial design asset, which helps establish a better rapport of the resulting granular model with the system under modeling. Human centricity is an inherent property of people's view on a system, a process, a machine or a model. Information granularity can be used to reflect people's level of uncertainty and this makes its pivotal role in soft computing. Indeed, the concept of information granularity facilitates the development of theory and application of soft computing immensely. A number of papers pertaining to some recent advances in theoretical development and practical application of information granularity in soft computing are highlighted in this special issue. The main objective of this study is to collect as many as possible researches on human centricity and information granularity in the agenda of theories and applications of soft computing, review the main idea of these literatures, compare the advantages and disadvantages of their methods and try to find the relationships and relevance of these theories and applications.     

A new expert system for learning management systems evaluation based on neutrosophic sets

There has been a sudden increase in the usage of Learning Management Systems applications to support learner's learning process in higher education. Many studies in learning management system evaluation are implemented under complete information, while the real environment has uncertainty aspects. As these systems were described by development organizations with uncertainty terms such as vague, imprecise, ambiguity and inconsistent, that is why traditional evaluation methods may not be effective. This paper suggests neutrosophic logic as a better option to simulate human thinking than fuzzy logic because unlike fuzzy logic, it is able to handle indeterminacy of information which expresses the percentage of unknown parameters. As previous studies suggested neutrosophic decision maker and neutrosophic expert systems as future work in ecommerce and e‐learning applications, this paper presents neutrosphic expert system for learning management systems evaluation. Information for building and validating the neutrosophic expert system is collected from five experts using surveys, and then analysis is done by using Fuzzytech 5.54d software. Finally, the comparison between fuzzy expert system and neutrosophic expert system results show that the neutrosophic logic is capable of representing uncertainty in human thinking for evaluating Learning Management Systems.     

A review on type-2 fuzzy logic applications in clustering,classification and pattern recognition

In this paper a review of type-2 fuzzy logic applications in pattern recognition, classification and clustering problems is presented. Recently, type-2 fuzzy logic has gained popularity in a wide range of applications due to its ability to handle higher degrees of uncertainty. In particular, there have been recent applications of type-2 fuzzy logic in the fields of pattern recognition, classification and clustering, where it has helped improving results over type-1 fuzzy logic. In this paper a concise and representative review of the most successful applications of type-2 fuzzy logic in these fields is presented. At the moment, most of the applications in this review use interval type-2 fuzzy logic, which is easier to handle and less computational expensive than generalized type-2 fuzzy logic.