Zadeh's Fuzzy Set Research Articles

Fully implicational methods for interval-valued fuzzy reasoning with multi-antecedent rules

Based on the fully implicational idea, we investigate the interval-valued fuzzy reasoning with multi-antecedent rules. First, we construct a class of interval-valued fuzzy implications by means of a type of implications and a parameter on the unit interval, then use them to establish three kinds of fully implicational reasoning methods for the interval-valued fuzzy reasoning with multi-antecedent rules which are called triple I method, hierarchical triple I method and URC triple I method respectively. At the same time, we analyze the Modus-Ponens (MP for short) and continuity properties of these methods and investigate the equivalence between the first method and the latter two methods. We also restrict our discussion to Zadeh fuzzy set theory and obtain a series of corresponding results.

The normal numbers of the fuzzy systems and their classes

ℝ-fuzzy set is defined in this paper, which is regarded as the generalization of the Zadeh fuzzy set. By means of CRI method, some fuzzy systems are constructed by suitably using several kinds of ℝ-fuzzy sets as fuzzy inference antecedents, such as interpolation fuzzy system, Bernstein fuzzy system, Lagrange fuzzy system and Hermite fuzzy system. A notion of the normal number of the fuzzy system is defined here, we have shown that all fuzzy systems are able to be classified as three classes such as the normal fuzzy systems, the regular fuzzy systems and the singular fuzzy systems under the significance of the normal numbers of fuzzy systems. Finally, the generalized Bernstein polynomial is obtained by constructing Bernstein fuzzy system, it is proved that the generalized Bernstein polynomial is uniformly convergent in C[a, b] under a weaker condition, and it is pointed out that there exist generalized Bernstein polynomials to be not convergent in C[a, b] by use of constructing a counterexample.

Diagnosis of Arthritis Through Fuzzy Inference System

Expert or knowledge-based systems are the most common type of AIM (artificial intelligence in medicine) system in routine clinical use. They contain medical knowledge, usually about a very specifically defined task, and are able to reason with data from individual patients to come up with reasoned conclusion. Although there are many variations, the knowledge within an expert system is typically represented in the form of a set of rules. Arthritis is a chronic disease and about three fourth of the patients are suffering from osteoarthritis and rheumatoid arthritis which are undiagnosed and the delay of detection may cause the severity of the disease at higher risk. Thus, earlier detection of arthritis and treatment of its type of arthritis and related locomotry abnormalities is of vital importance. Thus the work was aimed to design a system for the diagnosis of Arthitis using fuzzy logic controller (FLC) which is, a successful application of Zadeh's fuzzy set theory. It is a potential tool for dealing with uncertainty and imprecision. Thus, the knowledge of a doctor can be modelled using an FLC. The performance of an FLC depends on its knowledge base which consists of a data base and a rule base. It is observed that the performance of an FLC mainly depends on its rule base, and optimizing the membership function distributions stored in the data base is a fine tuning process.

The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets

In this paper, the cut sets, decomposition theorems and representation theorems of intuitionistic fuzzy sets and interval valued fuzzy sets are researched indail. First, new definitions of four kinds of cut sets on intuitionistic fuzzy sets are introduced, which are generalizations of cut sets on Zadeh fuzzy sets and have the same properties as that of Zadeh fuzzy sets. Second, based on these new cut sets, the decomposition theorems and representation theorems on intuitionistic fuzzy sets are established. Each kind of cut sets corresponds to two kinds of decomposition theorems and representation theorems. Thus eight kinds of decomposition theorems and representation theorems on intuitionistic fuzzy sets are obtained, respectively. At last, new definitions of cut sets on interval valued fuzzy sets are given based on the theory of cut sets on intuitionistic fuzzy sets, and eight kinds of decomposition theorems and representation theorems on interval valued fuzzy sets are also obtained. These results provide a fundamental theory for the research of intuitionistic fuzzy sets and interval valued fuzzy sets.

The n-dimensional fuzzy sets and Zadeh fuzzy sets based on the finite valued fuzzy sets

The connections among the n -dimensional fuzzy set, Zadeh fuzzy set and the finite-valued fuzzy set are established in this paper. The n -dimensional fuzzy set, a special L -fuzzy set, is first defined. It is pointed out that the n -dimensional fuzzy set is a generalization of the Zadeh fuzzy set, the interval-valued fuzzy set, the intuitionistic fuzzy set, the interval-valued intuitionistic fuzzy set and the three dimensional fuzzy set. Then, the definitions of cut set on n -dimensional fuzzy set and n -dimensional vector level cut set of Zadeh fuzzy set are presented. The cut set of the n -dimensional fuzzy set and n -dimensional vector level set of the Zadeh fuzzy set are both defined as n + 1 -valued fuzzy sets. It is shown that a cut set defined in this way has the same properties as a normal cut set of the Zadeh fuzzy set. Finally, by the use of these cut sets, decomposition and representation theorems of the n -dimensional fuzzy set and new decomposition and representation theorems of the Zadeh fuzzy set are constructed.

A Type-2 Fuzzy Approach to Linguistic Summarization of Data

This paper introduces an application of type-2 fuzzy sets in data linguistic summarization. The original approach by Yager (1982) based on representing natural language statements via type-1, i.e., the Zadeh fuzzy sets, is generalized with type-2 fuzzy sets applied as models of linguistically expressed quantities and/or properties of objects. Type-2 sets extend the known summarization procedures by handling fuzzy values stored in databases, and allow to represent a linguistic term via a few different membership functions (e.g., provided by different experts), which makes the method more general and human-consistent. Furthermore, quality measures for type-2 summaries are discussed in order to evaluate the informativeness of the messages generated. Finally, two prototype applications are presented and the success of the new method is discussed.

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.

A PROBABILITY-LIKE NEW FUZZY SET THEORY

In this paper, the reasons for the shortcoming of Zadeh's fuzzy set theory — it cannot correctly reflect different kinds of fuzzy phenomenon in the natural world — are discussed. In addition, the proof of the error of Zadeh's fuzzy set theory — it incorrectly defined the set complement that cannot exist in Zadeh's fuzzy set theory — is proposed. This error of Zadeh's fuzzy set theory causes confusion in thinking, logic and conception. It causes the seriously mistaken belief that logics of fuzzy sets necessarily go against classical and normal thinking, logic and conception. Two new fuzzy set theories, C-fuzzy set theory and probability-like fuzzy set theory, the N-fuzzy set theory, are proposed. The two are equivalent, and both overcome the error and shortcoming of Zadeh's fuzzy set theory, and they are consistent with normal, natural and classical thinking, logic and concepts. The similarities of N-fuzzy set theory with probability theory are also examined.

EVIDENCE SETS: MODELING SUBJECTIVE CATEGORIES

Zadeh's Fuzzy Sets are extended with the Dempster-Shafer Theory of Evidence into a new mathematical structure called Evidence Sets, which can capture more efficiently all recognized forms of uncertainty in a formalism that explicitly models the subjective context dependencies of linguistic categories. A belief-based theory of Approximate Reasoning is proposed for these structures. Evidence sets are then used in the development of a relational database architecture useful for the data mining of information stored in several networked databases. This useful data mining application establishes an Artificial Intelligence model of Cognitive Categorization with a hybrid architecture that possesses both connectionist and symbolic attributes.

Using fuzzy set theory to assess basis set quality

An algorithm, based on Zadeh's fuzzy set theory, to make an adequate basis set choice to compute simultaneously a number of molecular properties within Roothaan's algebraic approximation is presented. Its usefulness is illustrated by analyzing fuzzy set predictions and actual results from molecular calculations on NiH2, a system for which it has been recently shown that the choice of an adequate basis set is important.

A fuzzy set approach to the problem of sustainability

Evaluating sustainability in forest management requires decision makers to consider multiple, conflicting definitions in an environment of risk, uncertainty, and incomplete or non-quantitative information. Zadeh's fuzzy set theory provides a rigorous, flexible approach to the problem of defining and evaluating sustainability and comparing alternative scenarios. We discuss how fuzzy set concepts can be used to better manage for goals such as sustainability, equity, and ecosystem health. We then show how a simple tabular technique using fuzzy sets can be used to compare management alternatives, incorporate multiple objectives, and identify knowledge gaps and areas of disagreement. The simplicity and flexibility of this technique provides needed support for more explicit, objective decisions.

A survey of recent advances in fuzzy logic in telecommunications networks and new challenges

The use of fuzzy logic in telecommunication systems and networks is recent and limited. Fundamentally, Zadeh's fuzzy set theory provides a robust mathematical framework for dealing with real-world imprecision and nonstatistical uncertainty. Given that the present day complex networks are dynamic, that there is great uncertainty associated with the input traffic and other environmental parameters, that they are subject to unexpected overloads, failures and perturbations, and that they defy accurate analytical modeling, fuzzy logic appears to be a promising approach to address many important aspects of networks. This paper reviews the current research efforts in fuzzy logic-based approaches to queuing, buffer management, distributed access control, load management, routing, call acceptance, policing, congestion mitigation, bandwidth allocation, channel assignment, network management, and quantitative performance evaluation in networks. The review underscores the future potential and promise of fuzzy logic in networks. The paper then presents a list of key research efforts in the areas of fuzzy logic-based algorithms and new hardware and software architectures that are necessary both to address new challenges in networking and to help realize the full potential of fuzzy logic in networks.

Generalized formulation of yield, variability, minimax and Taguchi circuit optimization problems

A generalized formulation of several circuit design problems, such as manufacturing yield optimization, circuit performance variability minimization, deterministic and statistical minimax design, Income Index maximization, Taguchi approach, etc., is developed. Several other “intermediate” problems can be defined in a sense similar to the one used in Zadeh's fuzzy set theory. A specific problem is identified by the selection of a generalized membership function ω(·) of the acceptability region, and a sequence of the values of the “smoothing” parameter β. Generalized gradient formulas are developed, and various possible algorithmic implementations discussed. As a result, trade-offs between different design strategies can be investigated by circuit designers, within one coherent methodology.

A comparative study of fuzzy versus "fixed" thresholds for robust queue management in cell-switching networks

High-performance cell-based communications networks have been conceived to carry asynchronous traffic sources and support a continuum of transport rates ranging from low bit-rate to high bit-rate traffic. When a number of bursty traffic sources add cells, the network is inevitably subject to congestion. Traditional approaches to congestion management include admission control algorithms, smoothing functions, and the use of finite-sized buffers with queue management techniques. Most queue management schemes, reported in the literature, utilize fixed thresholds to determine when to permit or refuse entry of cells into the buffer. The aim is to achieve a desired tradeoff between the number of cells carried through the network, propagation delays of the cells, and the number of discarded cells. While binary thresholds are excessively restrictive, the rationale underlying the use of a large number of priorities appears to be ad hoc, unnatural, and unclear. The paper introduces the notion of cell-blocking, wherein a fuzzy thresholding function, based on Zadeh's (1965) fuzzy set theory, is utilized to deliberately refuse entry to a fraction of incoming cells from other switches. The blocked cells must be rerouted by the sending switch to other switches and, in the process, they may incur delays. The fraction of blocked cells is a continuous function of the current buffer occupancy level unlike the abrupt. The fuzzy cell-blocking scheme is simulated on a computer. Fuzzy queue management adapts superbly to sharp changes in cell arrival rates and maximum burstiness of bursty traffic sources. >

Proof methods for reasoning about possibility and necessity

Possibilistic logic is a quantitative method for uncertainty reasoning that is closely related to Zadeh's fuzzy set theory. In this paper, we formulate it as a kind of multimodal logic and develop some proof methods for it, including tableau method and two styles of natural deduction methods. The completeness and soundness of these methods are proved. Finally, some potential applications and the possible research directions are pointed out.

A fuzzy GIS modeling approach for Urban land evaluation

Present analytical functions and conventional cartographic modeling techniques in Geographic Information Systems (GIS) are based on Boolean logic, which implicitly assumes that objects in a spatial database and their attributes can be uniquely defined. The inherent constraint of the classical set theory does not allow for partial set membership conditions and imprecise information in GIS. The inadequacy of the Boolean logic for the representation and manipulation of spatial data is a major obstacle toward realistic GIS modeling. This paper demonstrates the usefulness of Zadeh's fuzzy set theory in GIS modeling for urban land evaluation. The results indicate that incorporating fuzzy set theory into GIS modeling can provide more details about the gradual transition of urban land value than the traditional cartographic modeling approach. Fuzzy GIS modeling can also reduce the information loss by obtaining membership grade for each individual land parcel. The membership function allows identification of the extent to which a particular area belongs to a valuation class based on given criteria.

Fuzzy integrals on L-fuzzy sets

The fuzzy measures and fuzzy integrals for Zadeh's fuzzy sets were studied by Qiao and Wang [5, 6, 7, 12]. In this paper we consider the real-valued fuzzy measures and fuzzy integrals for L-fuzzy sets (where L is a pseudocomplemented infinitely distributive complete lattice). On a fuzzy σ-algebra of L-fuzzy sets, the concepts of fuzzy measure and fuzzy integral are introduced, some equivalent forms of fuzzy integrals on L-fuzzy sets are given. Moreover we discuss several properties of fuzzy integrals on L-fuzzy sets, and prove some convergence theorems for a sequence of fuzzy integrals on L-fuzzy sets.

An approach to investigation of complex technological processes in food industry

This paper considers an approach to investigation and automatic control of complex technological processes with significant uncertainties and inaccuracies. It is based on a combined application of the basic merits of Zadeh's fuzzy set theory and methods of non-fuzzy information processing and can be regarded as an attempt to solve the problem of engineering-oriented use of fuzzy set theory. The approach is illustrated by the construction of a mathematical model of the sunflower oil extraction process.

Real-valued multisets and fuzzy sets

We develop a first-order theory MSTR for positive real-valued multisets. By restricting multiplicities of elements to the real interval (0, 1] we obtain a theory MSTF that gives a Zermelo-Fraenkel-like axiomatization for Zadeh's fuzzy sets. Moreover, MSTR and MSTF contain an exact copy of classical ZFC set theory and both are shown to be relatively consistent. MSTF is compared to other axiom systems that have been proposed for fuzzy sets.

An AI view of the treatment of uncertainty

AbstractThis paper reviews many of the very varied concepts of uncertainty used in AI. Because of their great popularity and generality “parallel certainty inference” techniques, so-called, are prominently in the foreground. We illustrate and comment in detail on three of these techniques; Bayes' theory (section 2); Dempster-Shafer theory (section 3); Cohen's model of endorsements (section 4), and give an account of the debate that has arisen around each of them. Techniques of a different kind (such as Zadeh's fuzzy-sets, fuzzy-logic theory, and the use of non-standard logics and methods that manage uncertainty without explicitly dealing with it) may be seen in the background (section 5).The discussion of technicalities is accompanied by a historical and philosophical excursion on the nature and the use of uncertainty (section 1), and by a brief discussion of the problem of choosing an adequate AI approach to the treatment of uncertainty (section 6). The aim of the paper is to highlight the complex nature of uncertainty and to argue for an open-minded attitude towards its representation and use. In this spirit the pros and cons of uncertainty treatment techniques are presented in order to reflect the various uncertainty types. A guide to the literature in the field, and an extensive bibliography are appended.