Fuzzy Sets and Systems (eBook)

Front Cover 1
Fuzzy Sets and Systems: Theory and Applications 4
Copyright Page 5
Contents 8
Foreword 14
Preface 16
Acknowledgments 18
Part I: Introduction 20
Introduction 20
Appendix: Some Historical and Bibliographical Remarks 23
References 24
Part II: Mathematical Tools 26
Chapter 1. Fuzzy Sets 28
A. Definitions 28
B. Set-Theoretic Operations 30
C. a-Cuts 38
D. Cardinality of a Fuzzy Set 39
E. Inclusions and Equalities of Fuzzy Sets 40
F. Convex Fuzzy Sets and Fuzzy Structured Sets 44
G. L-Fuzzy Sets 46
H. Measures of Fuzziness 51
References 53
Chapter 2. Extension Principle, Extended Operations, and Extended Fuzzy Sets 55
A. Extension Principle 55
B. Extended Real Operations 60
C. Extended Fuzzy Sets 81
References 86
Chapter 3. Fuzzy Relations 87
A. n-Ary Fuzzy Relations 87
B. Binary Fuzzy Relations 91
C. Similarity Relations and Related Topics 96
D. Fuzzy Orderings 99
E. Equations of Fuzzy Relations 103
F. Generalized Fuzzy Relations 109
References 112
Chapter 4. Fuzzy Functions 114
A. Various Kinds of Fuzzy Functions 114
B. Fuzzy Extremum 120
C. Integration of Fuzzy Functions over (Non)Fuzzy Intervals 125
D. Differentiation 135
E. Fuzzy Topology 138
F. Categories of Fuzzy Sets 139
References 141
Chapter 5. Fuzzy Measures, Probabilities/Possibilities 144
A. Fuzzy Measures and Sugeno's Integrals 144
B. Possibility and Probability 155
C. Fuzzy Events 159
D. Fuzzy Distributions 163
References 165
Part III: Fuzzy Models and Formal Structures 168
Chapter 1. Multivalent and Fuzzy Logics 170
A. Fuzzy Switching Logic 170
B. Multivalent Logics 177
C. Fuzzy Modal Logic 188
D. Fuzzy-Valued Logics 190
E. Approximate Reasoning 192
References 204
Chapter 2. Dynamic Fuzzy Systems 207
A. Complexity and Fuzziness in System Theory 208
B. Discrete-Time Fuzzy Systems 210
C. Fuzzy Automata 218
D. Deterministic Systems in a Fuzzy Environment 220
E. Linear Fuzzy Systems 222
F. Other Topics Related to Fuzziness and Systems 223
References 226
Chapter 3. Fuzzy Languages—Fuzzy Algorithms 229
A. Fuzzy Languages and Fuzzy Grammars 229
B. Fuzzy Algorithms 245
References 257
Chapter 4. Fuzzy Models tor Operations Research 261
A. Optimization in a Fuzzy Environment 261
B. Fuzzy Graphs 267
References 270
Part IV: Systems-Oriented Fuzzy Topics 272
Chapter 1. Where Do "They" Come From? 274
A. Informal Preliminary Discussion 274
B. Practical Estimation of Membership Functions 275
C. Identification of Fuzzy Set Operators 280
D. Tolerance Analysis Using Fuzzy Sets 282
References 282
Chapter 2. From Programming Languages to Natural Languages 284
A. Fuzzy Programming Languages 284
B. Modeling Natural Languages 286
C. Deductive Verbal Dynamic Systems 293
References 294
Chapter 3. Fuzzy Sets in Decision-Making 296
A. Fuzzy Rank-Ordering 296
B. Multicriteria Decision-Making 299
C. Aggregation of Opinions in a Social Group. Consensus 304
D. Decision-Making under Randomness and Fuzziness 308
References 314
Chapter 4. Fuzzy Control 316
A. Fuzzy Optimal Control 316
B. Synthesis of Linguistic Controllers 320
References 325
Chapter 5. Fuzzy Sets in Learning Systems 327
A. Learning with Automata or Fuzzy Conditional Measures 327
B. Self-Improvement of Fuzzy Linguistic Controllers 332
References 334
Chapter 6. Pattern Classification with Fuzzy Sets 336
A. Pattern Recognition 336
B. Clustering Algorithms 342
C. Information Retrieval 349
References 350
Chapter 7. Fuzzy Diagnosis 354
A. Discrimination of Fuzzy States in a Probabilistic Environment 354
B. Representation of Causality by Fuzzy Relations 356
References 358
Chapter 8. Fuzzy Sets in the Identification of Structures 360
A. Fuzzy Structural Modeling 360
B. Heuristic Structure Synthesis 362
C. Fuzzy Grammar Inference 364
References 367
Chapter 9. Fuzzy Games 
368 
References 370
Chapter 10. Fuzzy Sets and Catastrophes 
372 
References 374
Part V: A Survey of Potential Applications 376
A. Artificial Intelligence and Robotics 377
B. Image Processing and Speech Recognition 377
C. Biological and Medical Sciences 378
D. Control 378
E. Applied Operations Research 378
F. Economics and Geography 379
G. Sociology 379
H. Psychology 380
I. Linguistics 380
J. Semiotics 380
K. Other Topics 381
References 381
Conclusion 390
References in the Nonfuzzy Literature 392
List of the Most Commonly Used Symbols 396
Author Index 400
Subject Index 408