FUNDAMENTALS OF FUZZY SETS

THE HANDBOOKS OF FUZZY SETS SERIES

Series Editors Didier Dubois and Henri Prade

IRIT, Universite Paul Sabatier, Toulouse, France

FUNDAMENT ALS OF FUZZY SETS, edited by Didier Dubois and Henri Prade MATHEMATICS OF FUZZY SETS: Logic, Topology, and Measure Theory, edited

by Ulrich Hahle and Stephen Ernest Rodabaugh FUZZY SETS IN APPROXIMATE REASONING AND INFORMATION SYSTEMS, edited by James C. Bezdek, Didier Dubois and Henri Prade FUZZY MODELS AND ALGORITHMS FOR PATTERN RECOGNITION AND

IMAGE PROCESSING, by James C. Bezdek, James Keller, Raghu Krisnapuram and Nikhil R. Pal

FUZZY SETS IN DECISION ANALYSIS, OPERATIONS RESEARCH AND STATISTICS, edited by Roman Slowinski

FUZZY SYSTEMS: Modeling and Control, edited by Hung T. Nguyen and Michio Sugeno

PRACTICAL APPLICATIONS OF FUZZY TECHNOLOGIES, edited by Hans­JUrgen Zimmermann

FUNDAMENTALS OF FUZZYSETS

edited by

Didier Dubois

and

Henri Prade

IRIT, CNRS & University of Toulouse III

Foreword by Lotfi A. Zadeh

~.

" Springer Science+Business Media, LLC

Library of Congress Cataloging-in-Publication Data

Fundamentals of fuzzy sets / edited by Didier Dubois and Henri Prade ; foreword by Lotfi A. Zadeh.

p. em. -- (The handbooks of fuzzy sets series ; FSHS 7) Includes bibliographieal referenees and index. ISBN 978-1-4613-6994-3 ISBN 978-1-4615-4429-6 (eBook) DOI 10.1007/978-1-4615-4429-6

1. Fuzzy sets. 1. Dubois, Didier. II. Prade, Henri M. III. Series.

QA248.5 .F86 2000 511.3'22--dc21

Copyright ® 2000 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers, New York in 2000 Softcover reprint ofthe hardcover lst edition 2000

99-049471

AII rights reserved. No part of this publieation may be reprodueed, stored in a retrieval system or transmitted in any form or by any means, meehanieal, photo­copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC.

Printed on acid-free paper.

Contents

Foreword by Lotti A. Zadeh

Preface

Series Foreword

Contributing Authors

General Introduction Didier Dubois, Henri Prade

II

III

Fuzzy Sets: From Basic Concepts to Applications

The Role of Fuzzy Sets in Information Engineering

Conclusion: The Legitimacy of Fuzzy Sets

References

PART I FUZZY SETS

1 Fuzzy Sets: History and Basic Notions Didier Dubois, W. Ostasiewicz and Henri Prade

1.1 Introduction

1.2 The Historical Emergence of Fuzzy Sets 1.2.1 Fuzzy-ism 1.2.2 Philosophical Background 1.2.3 From Logic to Fuzzy Logics 1.2.4 From Sets to Fuzzy Sets

1 .3 Basic Notions of Fuzzy Set Theory 1.3.1 Representations of a Fuzzy Set 1.3.2 Scalar Characteristics of a Fuzzy Set 1.3.3 Extension Principles 1 .3.4 Basic Connectives 1.3.5 Set-Theoretic Comparisons Between Fuzzy Sets 1.3.6 Fuzzy Sets on Structured Referentials

1.4 Notions Derived from Fuzzy Sets 1.4.1 Fuzzy Relations

4

9

13

16

21

21

24 25 26 31 36

42 42 47 50 53 58 66

70 70

VI

2

1.4.2 Possibility Measures and Other Fuzzy Set-Based Functions 77

1.5 Generalisations and Variants of Fuzzy Sets 80 1.5.1 L-Fuzzy Sets 81 1 .5.2 Fuzzy Sets as Ordering Relations 82 1.5.3 Toll Sets 84 1.5.4 Interval-Valued Fuzzy Sets 86 1.5.5 Type 2 Fuzzy Sets 88 1 .5.6 Probabilistic Extensions of Fuzzy Sets 89 1.5.7 Level 2 Fuzzy Sets 90 1.5.8 Fuzzy Rough Sets and Rough Fuzzy Sets 91

1.6 Semantics and Measurement of Fuzzy Sets 93 1.6.1 What Membership Grades May Mean 95 1.6.2 Measuring Membership Grades 97 1.6.3 The Semantic Meaningfulness of Fuzzy Logic 100 1.6.4 Membership Grades: Truth Values or Uncertainty

Degrees 102 1.6.5 Towards Membership Function Measurement 104

1.7 Conclusion 106

References 106

Fuzzy Set-Theoretic Operators and Quantifiers Janos Fodor and Ronald R. Yager

125

2.1 Introduction 125

2.2 Complementation 127 2.2.1 Representation of Negations 129 2.2.2 Other Important Results 129

2.3 Intersection and Union 130 2.3.1 Triangular Norms and Conorms 131 2.3.2 The Special Role of Minimum and Maximum 134 2.3.3 Continuous Archimedean t-Norms and t-Conorms 135 2.3.4 Parametered Families of t-Norms and t-Conorms 141 2.3.5 Complementation Defined from Intersection and Union 145

2.4 Inclusion and Difference 146 2.4.1 Fuzzy Implications 147 2.4.2 Fuzzy Implications Defined by t-Norms, t-Conorms and

Negations 148 2.4.3 Negations Defined by Implications 153 2.4.4 Axioms for Fuzzy Inclusions 154 2.4.5 Difference of Fuzzy Sets 156

2.5 Equivalence 158

2.6 Uninorms 2.6.1 Important Classes of Uninorms

2.7 Mean Aggregation Operators

2.8 Ordered Weighted Averaging Operators

2.9 Quantifiers

2.10 Linguistic Quantifiers and OWA Operators

159 160

162

165

172

173

3

2.11 Weighted Unions and Intersections

2.12 Prioritized Fuzzy Operations

2.13 Other Aggregation Operators on Fuzzy Sets 2.13.1 Symmetric Sums 2.13.2 Weak t-Norms 2.13.3 Compensatory Operators

References

Measurement of Membership Functions: Theoretical and Empirical Work Taner Bilgic and I. Burhan TDrksen

3.1 Introduction and Preview

3.2 Interpretations of Grade of Membership 3.2.1 The Likelihood View 3.2.2 Random Set View 3.2.3 Similarity View 3.2.4 View from Utility Theory 3.2.5 View from Measurement Theory

3.3 Elicitation Methods 3.3.1 Polling 3.3.2 Direct Rating 3.3.3 Reverse Rating 3.3.4 Interval Estimation 3.3.5 Membership Exemplification 3.3.6 Pairwise Comparison 3.3.7 Fuzzy Clustering Methods 3.3.8 Neural-Fuzzy Techniques 3.3.9 General Remarks

3.4 Summary

vii

179

181

184 184 185 186

187

195

195

197 198 200 201 202 203

211 211 212 213 213 214 214 215 216 216

218

References 220

Appendix: Ordered Algebraic Structures and their Representations 228

PART II FUZZY RELATIONS

4 An Introduction to Fuzzy Relations Sergei Ovchinnikov

4.1 Introduction

4.2 Basic ConceptlS

4.3 Coverings and Proximity Relations

4.4 Similarity Relations and Fuzzy Partitions

4.5 Fuzzy Orderings

4.6 Representation Theorems

References

233

233

235

238

241

246

254

258

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5 Fuzzy Equivalence Relations: Advanced Material Dionis Boixader, Joan Jacas and Jordi Recasens

261

5.1 Introduction 261

5.2 How to Build Fuzzy Equivalence Relations 263

5.3 Fuzzy Equivalence Relations and Generalized Metrics 267

5.4 The Generators Set: Granularity, Observability and Approximation 270

5.5 Dimension and Basis - Their Calculation 279

References 288

6 Analytical Solution Methods for Fuzzy Relational Equations Bernard De Baets

291

6.1 Introduction

6.2 Images and Compositions 6.2.1 Relational Calculus and Boolean Equations 6.2.2 Fuzzy Relational Calculus

6.3 Types of Inverse Problems

6.4 Sup-'t Equations 6.4.1 The Equation 't(a,x) = b 6.4.2 Greatest Solution - Solvability Conditions 6.4.3 Complete Solution Set 6.4.4 Systems of Sup-'t Equations 6.4.5 Fuzzy Relational Equations

6.5 Left Inf-.9 Equations 6.5.1 The Equation .9(x,b) = a 6.5.2 Greatest Solution - Solvability Conditions 6.5.3 Complete Solution Set 6.5.4 Systems of Left Inf-3 Equations 6.5.5 Fuzzy Relational Equations

6.6 Right Inf-3 Equations 6.6.1 The Equation 3(a,x) = b 6.6.2 Smallest Solution - Solvability Conditions 6.6.3 Complete Solution Set 6.6.4 Systems of Right Inf-21 Equations 6.6.5 Fuzzy Relational Equations

6.7 Approximate Solution Methods

6.8 Further Reading 6.8.1 Various Generalizations 6.8.2 Miscellaneous Problems 6.8.3 Implementations 6.8.4 Applications

References

291

293 293 294

296

297 297 299 301 307 308

314 314 316 317 319 320

321 321 323 324 326 327

329

330 330 330 332 332

333

ix

PART III UNCERTAINTY

7 Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps 343 Didier Dubois, Hu~g T. Nguyen and Henri Prade

7.1 Introduction 343

7.2 Some Misunderstandings Between Fuzzy Sets and Probability 346 7.2.1 Membership Function and Probability Measure 346 7.2.2 Fuzzy Relative Cardinality and Conditional Probability 349 7.2.3 Fuzzy Sets Can Be Cast in Random Set Theory 350 7.2.4. Membership Functions as Likelihood Functions 351

7.3 Possibility Theory 353 7.3.1 The Meaning of Possibility 354 7.3.2 Possibility Distributions 356 7.3.3 Information Content of a Possibility Distribution 358 7.3.4 Possibility and Necessity of Events 360 7.3.5 Joint Possibility, Separability and Non-Interactive

Variables 364 7.3.6 Certainty and Possibility Qualification and the Extension

Problem 367 7.3.7 Conditional Possibility and Possibilistic Independence 368 7.3.8 Combination Rules in Possibility Theory 376

7.4 Quantitative Possibility Theory as a Bridge Between Probability and Fuzzy Sets 378 7.4.1 Possibility Theory and Bayesian Statistics 378 7.4.2 Upper and Lower Probabilities 380 7.4.3 Possibility Distributions as Special Cases of Random

Sets and Belief Functions 381 7.4.4 Possibility-Probability Transformations 383 7.4.5 Possibility Theory and the Calculus of Likelihoods 389 7.4.6 Probabilistic Interpretations of Fuzzy Set Operations 390 7.4.7 Possibility Degrees as Infinitesimal Probabilities 391

7.5 Towards Operational Semantics of Possibility Distributions and Fuzzy Sets 393 7.5.1 Frequentist Possibility 393 7.5.2 Uncertainty Measures and Scoring Rules 394 7.5.3 Betting Possibilities 395 7.5.4 Possibility as Similarity 396 7.5.5 Possibility as Preference and Graded Feasibility 397 7.5.6 Refinements of Qualitative Possibility Theory 401

7.6 Possibility and Necessity of Fuzzy Events: A Tool for Decision Under Uncertainty 402 7.6.1 Possibility and Necessity of Fuzzy Events 402 7.6.2 Sugeno Integrals 405 7.6.3 Quantitative Possibility and Choquet Integrals 406 7.6.4 Decision-Theoretic Foundations of Possibility Theory 408

7.7 Conclusion 413

Mathematical Appendix

References

414

423

x

8 Measures of Uncertainty and Information George J. K/ir

8.1 Introduction

8.2 Measures of Nonspecificity 8.2.1 Classical Set Theory 8.2.2 Fuzzy Set Theory 8.2.3 Possibility Theory 8.2.4 Evidence Theory

8.3 Entropy-Like Measures 8.3.1 Probability Theory 8.3.2 Evidence Theory 8.3.3 Possibility Theory

8.4 Measures of Fuzziness 8.4.1 Fuzzy Set Theory 8.4.2 Fuzzified Evidence Theory

8.5 Conclusions

References

9 Quantifying Different Facets of Fuzzy Uncertainty Nikhil R. Pal and James C. Bezdek

9.1 Introduction

9.2 Different Facets of Fuzzy Uncertainty

9.3 Measuring Fuzziness 9.3.1 Postulates of Measures of Fuzziness 9.3.2 Various Measures of Fuzziness

9.4 Generalized Measure of Fuzziness 9.4.1 Higher Order Measures of Fuzziness 9.4.2 Weighted Fuzziness

9.5 Measuring Non-Specificity

9.6 Conclusions

References

PART IV FUZZY SETS ON THE REAL LINE

10 Fuzzy Interval Analysis Didier Dubois, Etienne Kerre, Radko Mesiar and Henri Prade

10.1 Introduction

10.2 Fuzzy Quantities and Intervals 10.2.1 Definitions 10.2.2 Characteristics of a Fuzzy Interval 10.2.3 Noninteractive Fuzzy Variables

10.3 Basic Principles of Fuzzy Interval Analysis 10.3.1 The Extension Principle

439

439

440 440 443 444 446

447 447 449 451

452 452 453

454

454

459

459

461

462 462 464

473 473 474

475

477

478

483

483

486 486 492 497

498 498

11

10.3.2

10.3.3 10.3.4

xi

Functions on Non-Interactive Fuzzy Variables: Basic Results 501 Application to Usual Operations 505 Proper and Improper Representations of Functions 509

10.4 Practical Computing with Non-Interactive Fuzzy Intervals 511 10.4.1 Parameterized Representations of a Fuzzy Interval 511 10.4.2 Exact Calculation of the Four Arithmetic Operations 514 10.4.3 Approximate Parametric Calculation of Functions of

Fuzzy Intervals 516 10.4.4 Approximate Calculation of Functions of Fuzzy Intervals

Using Level-Cuts 519

10.5 Alternative Fuzzy Interval Calculi 521 10.5.1 Fuzzy Interval Calculations with Linked Variables 521 10.5.2 Additions of Fuzzy Intervals in the Sense of a Triangular

Norm 524 10.5.3 Multidimensional Fuzzy Quantities 530 10.5.4 Fuzzy Equations and the Optimistic Calculus of Fuzzy

Intervals 534

10.6 Comparison of Fuzzy Quantities 539 10.6.1 Positioning a Number with Respect to a Fuzzy Quantity 540 10.6.2 Ranking Fuzzy Intervals via Defuzzification 541 10.6.3 Goal-Driven Ranking Methods 542 10.6.4 Fuzzy Ordering Relations Induced by Fuzzy Intervals 544 10.6.5 Fuzzy Dominance Indices and Linguistic Methods 553 10.6.6 Criteria for Ranking Fuzzy Intervals 554

10.7 Conclusion: Applications of Fuzzy Numbers and Intervals 558

References 561

Metric Topology of Fuzzy Numbers and Fuzzy Analysis Phil Diamond and Peter Kloeden

583

11.1 Introduction

11.2 Calculus of Compact Convex Subsets in ~n 11.2.1 Subsets and Algebraic Operations 11.2.2 The Hausdorff Metric 11.2.3 Compact Subsets of ~n 11.2.4 Support Functions 11.2.5 LP-Metrics 11.2.6 Continuity and Measurability 11 .2.7 Differentiation 11.2.8 Integration 11.2.9 Bibliographical Notes

11.3 The Space 6,'n 11.3.1 Definitions and Basic Properties 11.3.2 Useful Subsets of.0 n and 6,'n 11.3.3 Bibliographical Notes

11.4 Metrics on 6,'n 11.4.1 Definitions and Basic Properties 11.4.2 Completeness 11.4.3 Separability

583

585 585 586 587 588 590 592 594 596 599

600 600 603 604

605 605 607 608

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11.4.4 Convergence Relationships 608 11.4.5 Bibliographical Notes 609

11.5 Compactness Criteria 609 11.5.1 Introduction 609 11.5.2 Compact Subsets in (&,n, dp) 611 11.5.3 Bibliographical Notes 613

11.6 Fuzzy Set Valued Mappings of Real Variables 613 11.6.1 Continuity and Measurability 613 11.6.2 Differentiation 615 11.6.3 Integration 621 11.6.4 Bibliographical Notes 624

11.7 Interpolation and Approximation 625 11.7.1 Interpolation and Splines 625 11.7.2 Bernstein Approximation 628 11.7.3 Bibliographical Notes 629

11.8 Fuzzy Differential Equations 630 11.8.1 Introduction 630 11.8.2 Existence and Uniqueness of Solutions 632 11.8.3 Reinterpreting Fuzzy DEs 632 11.8.4 Bibliographical Notes 637

11.9 Conclusion 637

References 637

Index 643

FOREWORD

The seven volume Handbook of Fuzzy Sets Series is a monumental achievement. Its publication caps years of sustained effort by the principal editors Didier Dubois and Henri Prade -the walking encyclopaedias of fuzzy set theory- working in close collaboration with the editors of individual volumes, the contributors and the publisher, Kluwer. The result is a compendium which is much more than the sum of its parts; it is an integrated, authoritative and up-to-date exposition of the entire body of knowledge centered on fuzzy set theory and its wide-ranging applications. The editors and the contributors come from the ranks of leaders in their fields; the quality of exposition is uniformly high; and the organization of the Handbook reflects the exceptional mastery and deep insight of the principal editors, Didier Dubois and Henri Prade. The vast panorama of methods, techniques and applications which is presented in the Handbook mirrors -as it should- the current perception of the topography of fuzzy set theory, its boundaries and its applications. In this connection, there is a point of semantics which is in need of clarification, namely: What is the difference, if any, between fuzzy set theory and fuzzy logic?

In some of my recent comments on this issue, I pointed out that the label "fuzzy logic" may be interpreted in two different ways: first, in a wide sense, in which case fuzzy logic (FL) is essentially coextensive with fuzzy set theory; and second, in a narrow sense in which fuzzy logic is a logical system which, in contrast to conventional logical systems, is aimed at a formalization of modes of reasoning which are approximate rather than exact.

More recently, I elaborated on my earlier suggestion by proposing a view of fuzzy logic which may be summarized as follows. Fuzzy logic (FL), in its wide sense, has four principal facets: the logical facet, FUL, which is basically fuzzy logic in its narrow sense; the set-theoretic facet, FUS; the relational facet, FUR; and the epistemic facet, FUE. These and other facets of fuzzy logic have unsharp, overlapping boundaries. Briefly, the logical facet, FUL, is a logical system which underlies approximate reasoning and inference from imprecisely defined premises. The set-theoretic facet is focused on the theory of sets which have unsharp boundaries, rather than on issues which relate to logical inference. My 1965 paper on fuzzy sets is in this spirit, as are most of the papers in the realm of what is frequently referred to as fuzzy mathematics. The relational facet, FUR, is concerned in the main with representation and analysiS of imprecise dependencies. Of central importance in FUR are the concepts of a linguistic variable and the calculus of fuzzy if-then rules. Most of the applications of fuzzy logic in control and systems analysiS relate to this facet of fuzzy logic. The epistemic facet of fuzzy logic is focused on knowledge, meaning and imprecise information. Possibility theory is a part of this facet, while possibilistic logic is shared by the logical and epistemic facets of FL. The core of FL, which is shared by all of its facets, is centered on two basic concepts: fuzziness and granularity. The conjunction of these concepts is fuzzy granularity or f-granularity, for short. F-granularity is a concomitant of a fundamental limitation on the cognitive ability of humans to resolve detail, process information

xiv

and store data. In this perspective, Fl may be viewed basically as a methodology for dealing with f-granularity in all of its manifestations, encompassing natural sciences, physical sCiences, cognitive sciences, mathematics and engineering.

What I have described is a perception of the conceptual structure of fuzzy logic which is consistent in the main, but perhaps not in detail, with the organization of the Handbook. Explicitly or implicitly, f-granularity is addressed in all of its incarnations. A perception that emerges from reading the Handbook is that much of fuzzy logic may be viewed as the result of f.g-generalization (fuzzification and fuzzy granulation) of classical concepts, techniques and methodologies. The important pOint is that fuzzy logic complements rather than competes with existing theories. This is a pOint that is frequently a source of misunderstanding and misdirected criticism.

There is what I believe to be a new and important direction in Fl which is in its initial stages of development and thus is not as yet reflected in the contents of the Handbook. What I have in mind is the methodology of computing with words (CW) and the computing-with-words-based methodology of the computational theory of perceptions (CTP). Basically, CW & CTP lay the groundwork for a major enlargement of the role of natural languages in Fl and, eventually, in all branches of science. Linguistic concepts, and especially the concepts of a linguistic variable and linguistic if-then rules have long played a major role in fuzzy logic and its applications. But CW moves much further in this direction by the development of a fuzzy-logic-based machinery for computing and reasoning with a much wider class of propositions drawn from a natural language. Through the use of constraint-centered semantics, such propositions are translated into what is called the generalized constraint language (GCl). In this way, reasoning and computation with propositions drawn from a natural language is reduced to constraint propagation from generalized antecedent constraints to generalized consequent constraints. The rules governing constraint propagation in CW coincide with the rules of inference in FUL.

In the computational theory of perceptions, the objects of computation are perceptions expressed as propositions in a natural language. The importance of CTP derives from the fact that much of human reasoning is perception rather than measurement based. An important concept which is suggested by CW is that of a precisiated natural language, PNL. Basically, PNl is a subset of a natural language which is equipped with constraint-oriented semantics and admits of translation into the generalized constraint language GCL. The concept of a precisiated natural language lays the groundwork for a far more extensive use of natural languages in concept definition, reasoning and computation in all realms of science. Such use would represent a major paradigm shift by adding to existing theories the capability to define and compute with perceptions rather than measurements. The development of CW and CTP, both of which are fuzzy-logic-based, has the effect of greatly amplifying the importance of fuzzy logic in the basic sciences and, especially, in applications involving the conception, design and utilization of information/intelligent systems. In this perspective, the Handbook makes a major contribution to the advancement of fuzzy logic, thereby enlarging its impact on the basic and applied sciences and providing new and powerful tools for the conception and design of systems with a much higher MIQ (Machine IQ) than those we have today. Publication of the Handbook of Fuzzy Sets is an achievement whose importance cannot be exaggerated. The principal editors, the volume editors, the contributors and the publisher deserve our applause and appreciation.

lotfiA.Zadeh

PREFACE

Fuzzy sets are almost thirty five years old. The novelty and the richness of the ideas carried out by the development of fuzzy sets, which have been often challenging previous views and old prevailing dogmas, have raised many doubts, criticisms, controversies cast by opponents or skeptics, especially during the first twenty years when fuzzy sets were still in their infancy or in their adolescence. Nevertheless, they have in the same time been encountering an increasing interest, and gaining more and more recognition. Each new decade has seen the blossoming of a larger set of methods and theoretical results as well as applications. This has led to a truly enormous literature, since there are presently approximately thirty thousands published papers dealing with fuzzy logic and several hundreds of books have appeared on the various facets of the theory and the methodology. There are currently several thousands of papers written every year.

Inevitably, such a gigantic amount of publications, by reporting so many achievements and applications (but sometimes also introducing more confusion or repetition than genuine progress), may leave newcomers, or even more experienced scholars, partially lost in front of this forest of results, in spite of the existence of many good monographs or edited volumes providing introductory or advanced materials to large segments of the field. Thus it was becoming more and more necessary, but also more and more challenging, to realize a general treatise covering, as much as pOSSible, all the aspects of the development of the fuzzy set literature, and providing an organized and rather detailed view of the field and its key notions, with reference to the relevant literature.

For this reason, when Kluwer Academic Publishers came to us in the early nineties for proposing us to prepare a general Handbook of Fuzzy Sets, we enthusiastically considered the project as worth doing, although we already anticipated some of the difficulties of the enterprize. Indeed the extent and the variety of the fuzzy sets developments was forbiding us to repeat what we did once at the end of the seventies, namely writing by ourselves a monograph covering the whole field. The only way of approaching the problem was for us to organize a collective handbook, where each chapter would be prepared by one, or preferably several, recognized researchers, specialists of the covered topic. Quite rapidly we came up with a proposal structured in five volumes with a list of chapters and potential contributors, which was further expanded into six and then seven volumes, as more contributions had been accumulating. The first proposal was then refined, completed, and sometimes substantially revised, under the scientific authority and guidance of the volume editors and the feedbacks we got from the advisory board. But it is only in the mid-nineties that Kluwer gave its final green light to the project. For being honest, it is only about that time that we have started to measure the amount of difficulties for gathering about eighty tutorial introductions and surveys, in a recommended format, for each of the various areas where fuzzy set have brought contributions, in spite of the considerable dedication of the volume editors who had kindly agreed to take part to the project.

xvi

The efforts of the contributors, of the volume editors and of the series editors have finally culminated in a 7 volume handbook of about 4000 pages, covering most of the fields from the Fundamentals of Fuzzy Sets to the Practical Applications of Fuzzy Technologies, including the contributions of fuzzy sets to Mathematics, Approximate Reasoning and Information Systems, Pattern Recognition and Image Processing, Decision Analysis, Operations Research and Statistics, and Modelling and Control of Systems. Inevitably the coverage and the depth in treatment are not perfectly homogeneous through the whole set of volumes, resulting in chapters of variable lengths, somewhat mirroring the present unequal development of the different areas related to fuzzy sets. Depending on the cases, an area may have been covered by a unique, often long, chapter, or by several shorter chapters written by different authors, thus leading to differences in the granulation of the topics in some places. It is also clear that in spite of the long and sometimes huge lists of references provided with each chapter, it is now becoming impossible to cite all the existing literature, and that noticeable and interesting contributions might have been unfortunately missed. We hope that these defects, which we expect, remain limited, will not prevent the reader from finding substantial material in these Handbook volumes. We also strongly hope that the Handbooks of Fuzzy Sets Series will contribute to a better understanding, to a more informed and structured view of what has been achieved, what is going on, and what is still missing in the fuzzy sets and systems research.

Didier Dubois and Henri Prade Toulouse

SERIES FOREWORD

Fuzzy sets were introduced in 1965 by Lotfi Zadeh with a view to reconcile mathematical modeling and human knowledge in the engineering sciences. Since then, a considerable body of literature has blossomed around the concept of fuzzy sets in an incredibly wide range of areas, from mathematics and logics to traditional and advanced engineering methodologies (from civil engineering to computational intelligence). Applications are found in many contexts, from medicine to finance, from human factors to consumer products, from vehicle control to computational linguistics, and so on .... Fuzzy logic is now currently used in the industrial practice of advanced information technology.

As a consequence of this trend, the number of conferences and publications on fuzzy logic has grown exponentially, and it becomes very difficult for students, newcomers, and even scientists already familiar with some aspects of fuzzy sets, to find their way in the maze of fuzzy papers. Notwithstanding circumstancial edited volumes, numerous fuzzy books have appeared, but, if we except very few comprehensive balanced textbooks, they are either very specialized monographs, or remain at a rather superficial level. Some are even misleading, conveying more ideology and unsustained claims than actual scientific contents.

What is missing is an organized set of detailed guidebooks to the relevant literature, that help the students and the newcoming scientist, having some preliminary knowledge of fuzzy sets, get deeper in the field without wasting time, by being guided right away in the heart of the literature relevant for her or his purpose. The ambition of the HANDBOOKS OF FUZZY SETS is to address this need. It will offer, in the compass of several volumes, a full picture of the current state of the art, in terms of the basic concepts, the mathematical developments, and the engineering methodologies that exploit the concept of fuzzy sets.

This collection will propose a series of volumes that aim at becoming a useful source of reference for all those, from graduate students to senior researchers, from

xviii

pure mathematicians to industrial information engineers as well as life, human and social sciences scholars, interested in or working with fuzzy sets. The original feature of these volumes is that each chapter is written by one or several experts in the concerned topic. It provides introduction to the topic, outlines its development, presents the major results, and supplies an extensive bibliography for further reading.

The core set of volumes are respectively devoted to fundamentals of fuzzy set, mathematics of fuzzy sets, approximate reasoning and information systems, fuzzy models for pattern recognition and image processing, fuzzy sets in decision research and statistics, fuzzy systems modeling and control, and a guide to practical applications of fuzzy technologies.

D. Dubois H. Prade Toulouse

Contributing Authors

James C. Bezdek Computer Science Department University of West Florida Pensacola, FL 32514, USA

Taner Bilgie Department of Industrial Engineering University of Toronto Toronto, Ontario M5S 1A4, Canada

Dionis Boixader Sec. Matematiques i Inf., ETSAV Pere Serra 1-15, 08190 Sant Cugat, Spain

Bernard De Baets Department of Applied Mathematics and Computer Science University of Gent Krijgslaan 281 - S9, 9000 Gent, Belgium

Phil Diamond Department of Mathematics University of Queensland st. Lucia, QLD 4067, Australia

Didier Dubois Institut de Recherche en Informatique de Toulouse Universite Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex 4, France

Janos Fodor Department of Mathematics Institute of Mathematics and Computer Science University of Agricultural Sciences (GATE) Pater Karoly u.1., 2103 G6d61i6, Hungary

xx

Joan Jacas Sec. Matematiques i Inf., ETSAB Avda. Diagonal 649, 08028 Barcelona, Spain

Etienne Kerre Fuzziness and Uncertainty Modeling Applied Mathematics and Computer Science University of Gent Krijgslaan 281 - S9, 9000 Gent, Belgium

George J. Klir Center for Intelligent Systems and Dept. of Systems Science & Industrial Eng. Thomas J. Watson School of Engineering and Applied Science Binghamton University - SUNY Binghamton, NY 13902-6000, USA

Peter Kloeden School of Computing and Mathematics Deakin University Australia

Radko Mesiar Slovak Technical University Bratislava Radlinskeho 11, 81368 Bratislava, Slovakia and UTI A AV CR, P.O. Box 18,18204 Prague, Czech Republic.

Hung T. Nguyen Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003, USA

Walenty Ostasiewicz Katedra Statystiki i Cybernetyki Ekonomicznej Akademia Ekonomicsna im. O. Langegn ul. Komandorska 118/120, 53-345 Wroclaw, Poland

Sergei Ovchinnikov Mathematics Department San Francisco State University 1600 Holloway Avenue, San Francisco, CA 94709, USA

Nikhil R. Pal Electronics & Communication Science Unit Indian Statistical Institute Calcutta, India

Henri Prade Institut de Recherche en Informatique de Toulouse Universite Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex 4, France

Jordi Recasens Sec. Matematiques i Inf., ETSAV Pere Serra 1-15, 08190 Sant Cugat, Spain

I. Burhan Turksen Department of Industrial Engineering University of Toronto Toronto, Ontario M5S 1 A4, Canada

Ronald R. Yager Machine Intelligence Institute lona College New Rochelle, NY 10801, USA

ACKNOWLEDGEMENTS

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As editors of the Handbooks of Fuzzy Sets Series, our deepest thanks are due to J.C. Bezdek, U. Hi:ihle, N. Pal, S. Rodabaugh, R. Slowinski, M. Sugeno, H.J. Zimmermann who have accepted the heavy responsibility of preparing volumes of the Series. We also gladly thank the other members of the advisory board for their valuable suggestions at a very early step of the preparation of this huge projet: J. Baldwin, P. Bonissone, B. Bouchon-Meunier, A. Di Nola, D. Gabbay, J. Goguen, K. Hirota, J. Kacprzyk, E.P. Klement, G.J. Klir, R. L6pez de Mantaras, R. Lowen, E.H. Mamdani, H.T. Nguyen, S. Orlowski, M. Roubens, E. Ruspini, E. Sanchez, P. Smets, L. Stout, M. Sugeno, H. Tanaka, E. Trillas, LB. TGrksen, P.Z. Wang, R.R. Yager, L.A. Zadeh and M. Zemankova.

The volume editors also wish to express their sincere gratitude to Alex Greene for his kind, patient and smooth cooperation in the realization of the Handbooks of Fuzzy Sets Series.

Special thanks are due to Agathe Baritaud who kindly and professionally prepared the final manuscript.