Fuzzy Logic Theory and Applications

Part I and Part II

Lotfi A Zadeh, Rafik A Aliev

• Verlag: WSPC
• Jahr: 2018
• ISBN:
  9789813238190
• Thema: Ciencia de la computación
• Thema: Ciencias computacionales general

Nowadays, voluminous textbooks and monographs in fuzzy logic are devoted only to separate or some combination of separate facets of fuzzy logic. There is a lack of a single book that presents a comprehensive and self-contained theory of fuzzy logic and its applications.

Written by world renowned authors, Lofti Zadeh, also known as the Father of Fuzzy Logic, and Rafik Aliev, who are pioneers in fuzzy logic and fuzzy sets, this unique compendium includes all the principal facets of fuzzy logic such as logical, fuzzy-set-theoretic, epistemic and relational. Theoretical problems are prominently illustrated and illuminated by numerous carefully worked-out and thought-through examples.

This invaluable volume will be a useful reference guide for academics, practitioners, graduates and undergraduates in fuzzy logic and its applications.

Contents:

• Part I: Fuzzy Sets and Fuzzy Logic (L Zadeh, R Aliev):
  • Fuzzy Sets
  • Fuzzy Logic
  • Restriction Concept
  • Fuzzy Probabilities
  • Fuzzy Functions
  • Fuzzy Systems
  • Z-Number Theory
  • Generalized Theory of Uncertainty
• Part II: Applications and Advanced Topics of Fuzzy Logic (Lotfi A Zadeh, Rafik A Aliev):
  • Restriction-Based Semantics (L Zadeh)
  • Granular Computing: Principles and Algorithms (W Pedrycz)
  • Complex Fuzzy Sets and Complex fuzzy Logic, an Overview of Theory and Applications (Dan E Tamir, Naphtali D Rishe, Abraham Kandel)
  • Introduction to Fuzzy Logic Control (H Ying and D Filev)
  • Fuzzy Decision Making (R Aliev)
  • Selected Interpretability Aspects of Fuzzy Systems for Classification (L Rutkowski)
  • Fuzzy Reinforcement Learning (H Berenji)
  • ANFIS: Adaptive Neuro-Fuzzy Inference Systems (J-S R Jang, C-T Wu, K-J Zhang)
  • Fuzzy Expert Systems (R Aliev)
  • Application of Logistic Regression Analysis to Fuzzy Cognitive Maps (V Niskanen)
  • Fuzzy Logic in Medicine (Y Hata)

Readership: Researchers, academics, professionals, graduate and undergraduate students in fuzzy logic and its applications.

Part I

Fuzzy Logic Theory

Chapter 1

Fuzzy Sets

A fuzzy set is a class of objects with a continuum of grades of membership. Such set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are described. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint [501].

1.1.Introduction

More often than not, the classes of objects encountered in the real physical world do not have precisely defined criteria of membership. For example, the class of animals clearly includes dogs, horses, birds, etc. as its members, and clearly excludes such objects as rocks, fluids, plants, etc. However, such objects as starfish, bacteria, etc. have an ambiguous status with respect to the class of animals. The same kind of ambiguity arises in the case of a number such as 10 in relation to the “class” of all real numbers which are much greater than 1.

Clearly, the “class of all real numbers which are much greater than 1,” or “the class of beautiful women,” or “the class of tall men,” do not constitute classes or sets in the usual mathematical sense of these terms. Yet, the fact remains that such imprecisely defined “classes” play an important role in human thinking, particularly in the domains of pattern recognition, communication of information, and abstraction.

The purpose of this chapter is to explore in a preliminary way some of the basic properties and implications of a concept which may be used in dealing with “classes” of the type cited above. The concept in question is that of a fuzzy set, that is, a “class” with a continuum of grades of membership. As will be seen in the sequel, the notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables.

We begin the discussion of fuzzy sets with several basic definitions.

1.2.Definitions

Let X be a space of points (objects), with a generic element of X denoted by x. Thus, X = {x}.

A fuzzy set (class) A in X is characterized by a membership (characteristic) function μ_A(X) which associates with each point in X a re...