George J. Klir

Fuzzy sets, uncertainty, and information

A solution to get the problem off, have you found it? Really? What kind of solution do you resolve the problem? From what sources? Well, there are so many questions that we utter every day. No matter how you will get the solution, it will mean better. You can take the reference from some books. And the fuzzy sets uncertainty and information is one book that we really recommend you to read, to get more solutions in solving this problem.

Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers by Lotfi A. Zadeh

From the Publisher: This book consists of papers written by the founder of fuzzy set theory, Lotfi A. Zadeh. Since Zadeh is not only the founder of this field but has also been the principal contributor to its development over the last 30 years, the papers contain virtually all the major ideas in fuzzy set theory, fuzzy logic, and fuzzy systems in their historical context.

Generalized measure theory

Preliminaries.- Basic Ideas of Generalized Measure Theory.- Special Areas of Generalized Measure Theory.- Extensions.- Structural Characteristics for Set Functions.- Measurable Functions on Monotone Measure Spaces.- Integration.- Sugeno Integrals.- Pan-Integrals.- Choquet Integrals.- Upper and Lower Integrals.- Constructing General Measures.- Fuzzification of Generalized Measures and the Choquet Integral.- Applications of Generalized Measure Theory.

MEASURES OF UNCERTAINTY AND INFORMATION BASED ON POSSIBILITY DISTRIBUTIONS

Abstract A measure of uncertainly and information for possibility theory is introduced in this paper The measure is called the U-uncertainty or, alternatively, the U-information. Due to its properties, the U-uncertainty/information can be viewed as a possibilistic counterpart or the Shannon entropy and, at the same time, a generalization or the Hartley uncertainty/information. A conditional U-uncertainty is also derived in this paper, it depends on the U-uncertainties or the joint and marginal possibility distributions in exactly the same way as the conditional Shannon entropy depends on the entropies or the joint and marginal probability distributions. The conditional U-uncertainty is derived without the use of the notion of conditional possibilities, thus avoiding a current controversy in possibility theory. The proposed measures of U-uncertainty and conditional U-uncertainty provide a foundation for developing an alternative theory of information, one based on possibility theory rather than probability...

The paradoxical success of fuzzy logic

Fuzzy logic methods have been used successfully in many real-world applications, but the foundations of fuzzy logic remain under attack. Taken together, these two facts constitute a paradox. A second paradox is that almost all of the successful fuzzy logic applications are embedded controllers, while most of the theoretical papers on fuzzy methods deal with knowledge representation and reasoning. I hope to resolve these paradoxes by identifying which aspects of fuzzy logic render it useful in practice, and which aspects are inessential. My conclusions are based on a mathematical result, on a survey of literature on the use of fuzzy logic in heuristic control and in expert systems, and on practical experience in developing expert systems.<<ETX>>

Fuzzy arithmetic with requisite constraints

Abstract The purpose of this paper is to show that the standard fuzzy arithmetic does not take into account known constraints when applied to states of linguistic variables. These constraints, referred to as requisite constraints, represent additional information. When they are ignored, as in the standard fuzzy arithmetic, the obtained results exhibit, in general, information deficiency. It is thus important, as argued in the paper, to revise fuzzy arithmetic to take relevant requisite constraints into account. Some typical requisite constraints are overviewed, with a more extensive examination of the requisite equality constraint (crisp or fuzzy).

Resolution of finite fuzzy relation equations

Abstract General schemes for solving fuzzy relation equations with finite sets are derived in this paper. The relationship between the results derived here and those of previous publications regarding this subject is also discussed.

A PRINCIPLE OF UNCERTAINTY AND INFORMATION INVARIANCE

The paper introduces a new principle, referred to as the principle of uncertainty and information invariance, for making transformations between different mathematical theories by which situations under uncertainty can be characterized. This principle requires that the amount of uncertainty (and related information) be preserved under these transformations. The principle is developed in sufficient details for transformations between probability theory and possibility theory under interval, log-interval and ordinal scales. Its broader use is discussed only in general terms and illustrated by an example.

On fuzzy-set interpretation of possibility theory

A revised fuzzy-set interpretation of possibility theory is introduced in this paper. Contrary to the standard fuzzy-set interpretation of possibility theory, which is coherent only for normal fuzzy sets, the revised interpretation is shown to be coherent for all fuzzy sets. It is also argued that the revised interpretation, which coincides with the standard one for normal fuzzy sets, is more meaningful on intuitive grounds. Prior to the introduction of the revised interpretation, previous efforts to overcome the well-known difficulties of the standard interpretation are critically examined, and it is demonstrated that none of them results in a coherent and meaningful interpretation of possibility theory.

Where do we stand on measures of uncertainty, ambiguity, fuzziness, and the like?

Abstract It is argued in this paper that the theory of fuzzy sets involves at least four fundamentally different types of uncertainty. Each of these types requires a measure by which the degree of uncertainty of that type can be determined. Two main categories of uncertainty are connected with the terms ‘vagueness’ (or ‘fuzziness’) and ‘ambiguity’. In general, vagueness is associated with the difficulty of making sharp or precise distinctions in the world. Ambiguity, on the other hand, is associated with one-to-many relations, i.e., situations with two or more alternatives that are left unspecified. While the concept of a fuzzy set represents a basic mathematical framework for dealing with vagueness, the concept of a fuzzy measure is a general framework for dealing with ambiguity. Several classes of measures of vagueness, usually referred to as measures of fuzziness, have been proposed in the literature. Each class is based on some underlying conception of the degree of fuzziness. A general set of requirements for measures of fuzziness is formulated, followed by an overview of the measures proposed in the literature. Measures of ambiguity are discussed within the framework of plausibility and belief measures. Although it does not cover all fuzzy measures, this framework is sufficiently broad for most practical purposes, and represents a generalization of both probability theory and possibility theory. It is argued that three complementary measures of ambiguity should be employed. One of them is obtained by generalizing the Hartley measure of uncertainty; it measures the degree of nonspecificity in individual situations described by the various belief and plausibility measures. The other two are obtained by generalizing the well known Shannon measure of uncertainty; they measure the degree of dissonance and the degree of confusion in evidence, respectively. Basic mathematical properties of these measures are overviewed. It is also argued that each of the four types of uncertainty measures, which are fundamentally different from each other, can be used for measuring structural (syntactic) information in the same sense as the Hartley and Shannon measures have been used in this respect. As such, these measures are potentially powerful tools for dealing with systems problems such as systems modelling, analysis, simplification, or design.