Fuzzy Set Theory. Basic Concepts, Techniques and Bibliography R. LOWEN SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Transcription

1 Fuzzy Set Theory

2 Fuzzy Set Theory Basic Concepts, Techniques and Bibliography by R. LOWEN Department 0/ Mathematics and Computer Science, University 0/ Antwerp, Antwerp, Belgium SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

3 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper All Rights Reserved 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996 Softcover reprint ofthe hardcover 1st edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inc1uding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

4 To Mom and Dad

5 Contents List of Figures ix Preface xiii Chapter 1 Elementary Set Theory Section 1 Sets and subsets Section 2 Functions and relations Section 3 Partially ordered sets Section 4 The lattice of subsets of a set Section 5 Characteristic functions Section 6 Notes Chapter 2 Fuzzy Sets Section 1 Definitions and examples Section 2 Lattice theoretical operations on fuzzy sets Section 3 Pseudocomplementation Section 4 Fuzzy sets, functions and fuzzy relations Section 5 a-ievels Section 6 Notes Chapter 3 t-norms, t-conorms and Negations Section 1 Pointwise extensions Section 2 t-norms and t-conorms Section 3 Negations Section 4 Notes Chapter 4 Special Types of Fuzzy Sets Section 1 Normal fuzzy sets Section 2 Convex fuzzy sets Section 3 Piecewise linear fuzzy sets Section 4 Compact fuzzy sets Section 5 Notes Chapter 5 Fuzzy Real Numbers Section 1 The probabilistic view vii

6 Seetion 2 Seetion 3 Seetion 4 Chapter 6 Seetion 1 Seetion 2 Seetion 3 Seetion 4 Seetion 5 Seetion 6 Seetion 7 Chapter 7 Seetion 1 Seetion 2 Index The non-probabilistie view Interpolation Notes Fuzzy Logic Conneetives in classieal logie Fundamental classieal theorems Basic prineiples of fuzzy logie Lattiee generated fuzzy eonneetives t-norm generated fuzzy eonneetives Probabilistieally generated fuzzy eonneetives Notes Bibliography Books Artieles viii

7 List of Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Figure 24 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29 Figure 30 Figure 31 Figure 32 An impression of 8 A JL ::; v JL 1\ v and 11 v v Brouwerian complement of JL Pseudocomplement of JL la(jl) = [a,e] and l~(il) =]a,b[ Drastic product Drastic sum Minimum Maximum Bounded product Bounded sum Algebraic product Algebraic sum Einstein product Einstein sum Dombi's t-norm for..\ = Dombi's t-conorm for..\ = Hamacher's t-norm for..\ = Hamacher's t-conorm for..\ = Yager's t-norm for..\ = Yager's t-conorm for..\ = Frank's t-norm for..\ = Frank's t-conorm for..\ = Weber's first t-norm for..\ = Weber's first t-conorm for..\ = Weber's second t-norm for..\ = Weber's second t-conorm for A = Dubois and Prade's t-norm for A =! Dubois and Prade's t-conorm for..\ =! Schweizer's first t-norm for..\ = Schweizer's first t-conorm for A = ix

8 Figure 33 Figure 34 Figure 35 Figure 36 Figure 37 Figure 38 Figure 39 Figure 40 Figure 41 Figure 42 Figure 43 Figure 44 Figure 45 Figure 46 Figure 47 Figure 48 Figure 49 Figure 50 Figure 51 Figure 52 Figure 53 Figure 54 Figure 55 Figure 56 Figure 57 Figure 58 Figure 59 Figure 60 Figure 61 Figure 62 Figure 63 Figure 64 Figure 65 Figure 66 Schweizer's second t-norm for A = Schweizer's second t-conorm for A = Schweizer's third t-norm for A = Schweizer's third t-conorm for A = Mizumoto's first t-norm Mizumoto's first t-conorm Mizumoto's second t-norm Mizumoto's second t-conorm Mizumoto's third t-norm Mizumoto's third t-conorm Mizumoto's fourth t-norm for A = Mizumoto's fourth t-conorm for A = Mizumoto's fifth t-norm for A = e Mizumoto's tifth t-conorm tor A = e Mizumoto's sixth t-norm for A = Mizumoto's sixth t-conorm for A = Mizumoto's seventh t-norm for A = Mizumoto's seventh t-conorm for A = Mizumoto's eighth t-norm for A = e Mizumoto's eighth t-conorm tor A = e Mizumoto's ninth t-norm tor A = 2/ Mizumoto's ninth t-conorm for A = 2/ Mizumoto's tenth t-norm tor A = Mizumoto's tenth t-conorm for A = A convex fuzzy set which is not a convex function A tri angular fuzzy set A trapezoidal fuzzy set Basic connectives of classical logic Truth table for "A and B" Truth table for "A or B" Truth table for "not A" Truth table tor "it Athen B" Truth table for "A if and only if B" Truth table of "if (A and B) then C" x

9 Figure 67 Figure 68 Figure 69 Figure 70 Figure 71 Figure 72 Figure 73 Figure 74 Figure 75 Figure 76 Figure 77 Figure 78 Figure 79 Figure 80 Figure 81 Figure 82 Figure 83 Figure 84 Figure 85 Figure 86 Figure 87 Figure 88 Figure 89 Figure 90 Figure 91 Figure 92 Figure 93 Figure 94 lattice-extension of "and" lattice-extension of "or" lattice-extension of "implies" lattice-extension of "ift" lattice-extensions of "not and" and "not or" Modus Ponens for lattice-extensions Law of Syllogism-Iattice case P- and Toc-extension of "implies" E- and Hw-extension of "implies" y; d S I xt. f... I' " an 2 -e enslon 0 Imp les P-extension of "ift" T oc -extension of "ift" Modus Ponens for P-extensions min-probabilistic extension of "and" P- and Toc-probabilistic extensions of "and" M2- and Wl-probabilistic extensions of "or" D Pl. -probabilistic extension of "or" Y2-probabilistic extension of "implies" D2- and FlO-probabilistic extensions of "implies" 218 P- and Toc-probabilistic extensions of "ift" Y2-probabilistic extension of "ift" (x! y ) n (y! x ) - (x J: y) FlO-probabilistic connective for "implies" Y2- and D2-probabilistic connectives for "implies" D Pl. -probabilistic connective for "ift" E- and Y2-probabilistic connectives for "ift" Modus Ponens for the Y2-probabilistic extension Wf-probabilistic connectives for "not and" and "not or". 232 xi

10 Preface The purpose of this book is to provide the reader who is interested in applications of fuzzy set theory, in the first place with a text to which he or she can refer for the basic theoretical ideas, concepts and techniques in this field and in the second place with a vast and up to date account of the literature. Although there are now many books about fuzzy set theory, and mainly about its applications, e.g. in control theory, there is not really a book available which introduces the elementary theory of fuzzy sets, in what I would like to call "a good degree of generality". To write a book which would treat the entire range of results concerning the basic theoretical concepts in great detail and which would also deal with all possible variants and alternatives of the theory, such as e.g. rough sets and L-fuzzy sets for arbitrary lattices L, with the possibility-probability theories and interpretations, with the foundation of fuzzy set theory via multi-valued logic or via categorical methods and so on, would have been an altogether different project. This book is far more modest in its mathematical content and in its scope. As such it does not really address itself to mathematicians, but rather to researchers working in the areas of engineering, data analysis, control theory, pattern recognition, neural networks, clustering, expert systems, information retrieval, operations research, decision making, image and signal processing, and so on, who wish to apply fuzzy sets but might not be too knowledgeable in the set-theoretical basics on which fuzzy set theory is based. Hence I hope that this book might be a handy companion next to other more specifically application-oriented texts. Of course the choice of what to include and what not to include was strongly inftuenced by personal taste. For this reason I have also tried to provide much information for the reader as to where he or she can find (1) more detailed results related to the concepts introduced, (2) alternative concepts and results and (3) work related to applications. The first chapter gives a review of the basic concepts of set theory. Not only is naive fuzzy set theory built with classical set-theoretical tools (sets and functions) but moreover, in order to justify the various operations which exist in fuzzy set theory, it is necessary to have some background in ordinary set theory. Furthermore a review is given of the basic lattice theory which is required. In the second chapter classical (or naive) fuzzy sets are introduced, as defined by L.A. Zadeh, and the basic properties which hold in the lattice-theoretical framework are given. In the third chapter t-norms and t-conorms are introduced. They form the basis for a wide new variety of operations xiii

11 on fuzzy sets and for the connectives of fuzzy logic. The fourth chapter covers the most important special properties which in certain contexts are often required of fuzzy sets. The fifth chapter deals with the important notion of fuzzy real numbers, both the probabilistic and the non-probabilistic views. These form the basis not only of purely mathematical work in this area, which is not treated in this text, but also of the main applications of fuzzy set theory. The sixth chapter treats "naive" fuzzy logic, as it is being used, mainly in applications in control theory. Here too a review is given of the elementary notions of elassicallogic. The book ends with two chapters which contain a vast account of the literature up to now and which in my opinion should give the reader a starting point making it possible to find almost anything he or she may want in this area. Whereas the contents of chapters 1 to 6 focus on the elementary theoretical ideas, chapters 7 and 8 which contain the biographical data, focus mainly on applications. Necessary references to theoretical work related to the concepts of the first 6 chapters are given in full in the text, mainly in the notes following each chapter. Throughout the text I have taken care to provide many graphs of t-norms, t-conorms and logical operators. To the best of my knowledge this is the first time that these operators are thus presented, and in my opinion, the visual information next to the mathematical formulas is often interesting. I would like to thank my students R. Brys, V. Fest jens, W. Peeters and M. Sioen for their extensive help in collecting the biographical data. Furthermore I would also like to thank N. Blasco and W. Peeters for proofreading the final manuscript. Of course the responsibility for any errors which may remain lies completely and solely with the author. The idea to write this book emerged from talks with Alexander Schimmelpenninck. Paul Roos and Alexander Schimmelpenninck, both editors at Kluwer Academic Publishers, followed the development from elose by. For their much appreciated, friendly encouragement and professional support during the entire period of writing they both have my sincere thanks. R. Lowen xiv