Table of contents for A first course in fuzzy logic / Hung T. Nguyen, Elbert A. Walker.


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1 The Concept of Fuzziness                                    1
1.1 Examples .................          ...........        1
1.2  Mathematical modeling ...................             2
1.3  Some operations on fuzzy sets . . . . . . . . . . . . . . . .  6
1.4  Fuzziness as uncertainty . . . . . . . . . . . . . . . ...  11
1.5 Exercises ..................          .   ........    14
2 Some Algebra of Fuzzy Sets                                 17
2.1 Boolean algebras and lattices . . . . . . . . . . . . ....  17
2.2  Equivalence relations and partitions . . . . . . .. . . . .  23
2.3  Composing mappings ................. .           .   27
2.4 Isomorphisms and homomorphisms . . . . . . . . . . . . .  29
2.5  Alpha-cuts .......................             . ..32
2.6 Images of alpha-level sets  . . . . . . . . . .. . . . . . . .  34
2.7  Exercises  . . . . . . . . . . . . . . . . . .. . . . . . . . 36
3 Fuzzy Quantities                                           45
3.1 Fuzzy quantities ....... ..        .....  ......      45
3.2  Fuzzy numbers ................... .         ....     52
3.3  Fuzzy  intervals  ................        ....   .   55
3.4  Exercises  . . . . . . . . . . .. . .. . . . . . . . . . . .  56
4 Logical Aspects of Fuzzy Sets                              59
4.1  Classical two-valued logic  . . . . . . .. . . . . . . . . . .  60
4.2  A  three-valued  logic  .................      . .   64
4.3  Fuzzy  logic  ................      ........     .   65
4.4  Fuzzy and Lukasiewicz logics . . . . . ... . . . . . . . .  66
4.5  Interval-valued  fuzzy  logic  ..................    68
4.6  Canonical forms .......................              70
4.7 Notes on probabilistic logic . . . . . . . . . . . . . ..  74
4.8  Exercises  ...................           .......     76
5 Basic Connectives                                          81
5.1  t-norms  .  . . . . . . . . . . . . . . . . . . . ..... .  81
5.2  Generators of t-norms  .  .. ............    . ..    85
5.3  Isomorphisrs of t-norms  ...................        93
5.4  Negations  .  . . . . . . . . . . . . . . . . . . . . . . . . . .  98
5.5  Nilpotent t-norms and negations .. . . . . . . . . . . . 102
5.6  t-conorms  . . .... ... ...  ...  ... .....   .    106
5.7  De Morgan systems .....   .......    ........     .  109
5.7.1  Strict De Morgan systems .. . .. . . . . . .... . 109
5.7.2  Nilpotent De Morgan systems .... . . . .. . . ..113
5.7.3  Nonuniqueness of negations in strict
De) Morgan systems . . ............... . .    116
5.8  Groups and  t-norms  .............      .......     118
5.8.1  The normaiizer ofR . . ............... . 119
5.8.2  Families of strict t-norms  . . . . ..  .  .  .  ... . . 122
5.8.3  Families of nilpotent t-norms  .. . . . . . . . . . .  126
5.9  Interva-val- ued  fuzzy sets  . .................   127
5.9.1  t-norms on interval-valued fuzzy sets . . . . . ...  128
5.9.2  Negations an(  t-conorms  . . . . . . . . . . . ... 30
5.10  Type-2  fuzzy  sets  ...... ... .........   ..   .  134
5.10.1 Pointwise operations and convolutions .. . . ... . 134
5.10.2  Type-2  fuzzy  sets  .......  .  ......... . . 135
5.10.3 The algebra (AMap(J, I),, . 0,.1) . . . . . . . . 136
5.10.4  Two order relations  .. . . . . .  .. .  . . . . . .143
5.10.5 Subalgebras of type-2 fuzzy sets . . . . . . . . . . 145
5.10.6 Convolutions using product . . . . . . . . . . ...  153
5.10.7 T-norms for ttype-2 fuzzy sets . . . . . . . . . ..  157
5.10.8  Commments .......................           163
5.11  Exercises  ..............     ...    . .  .....    163
6 Additional Topics on Connectives                          171
6.1  Fuzzy  im plications  ....................        .  171
6.2  Averaging  operators  . .. . . . . ... ... ...... ....  177
6.2.1  Averaging operators and negations .... .  . . . .. 180
6.2.2  Averaging operators and nilpotent t-norms .... 184
6.2.3  De Morgan systems with averaging operators . . . 187
6.3  Powers of t-norms  . ... .......    ...  .....   . .. 190
6.4  Sensitivity  of connectives  .. .  . . . . . .  . . .  . ... .  194
6.5  Copulas and t-norms .. .... .. .  .  .... . . . ... . 197
6.6  Exercises . . . ............................ 200
7 Fuzzy Relations                                          207
7.1  Definitions and examples . . . . . ... . . . . . . . . . . .  207
7.2 Binary fuzzy relations . . . . . . . . . . . . . . .....  . 208
7.3  Operations on fuzzy relations . . . . . . . . . . . . ...  212
7.4  Fuzzy  partitions  .......................          214
7.5 Fuzzy relations as Chu spaces ..... . . . . . . . . . . .  215
7.6 Approximate reasoning . . . . . . . . . . . . . . .....  . 217
7.7 Approximate reasoning in expert systems ..... . . . .  220
7.7.1  Fuzzy  syllogisms  ...................       226
7.7.2  Truth  qualification  ..................     226
7.7.3  Probability qualification . . . . . . . . . . .....  226
7.7.4  Possibility qualification  . . . . . . . . . . . . ...  227
7.8 A simple form of generalized modus ponens . . . . . . . . 227
7.9  The compositional rule of inference . . . . . . . . . . ...  229
7.10 Exercises ....    ...............      ........ 230
8 Universal Approximation                                  235
8.1 Fuzzy rule bases .................. ....             235
8.2  Design methodologies  . . . . . . . . . . . . ... . . . .  238
8.3  Some mathematical background . . . . . . . . . . . ....  240
8.4  Approximation capability . . . . . . . . . . . . . . . . . . 242
8.5 Exercises .........       ................ 247
9 Possibility Theory                                       251
9.1 Probability and uncertainty . . . . . . . . . . . . . . . . . 251
9.2 Random sets .............        ............ 254
9.3 Possibility measures . . . . . . . . . . . . . . ....... . . 256
9.3.1  Measures of noncompactness . . . . . . . . . .. . 261
9.3.2  Fractal dimensions ..................        262
9.3.3 Information measures . . . . . . . ............. 263
9.4  Exercises  ..................        . ........     267
10 Partial Knowledge                                       271
10.1 Motivation  .................        ......... 271
10.2 Belief functions and incidence algebras . . . . . . . . . .  274
10.3  Monotonicity  .......................           .  278
10.4 Beliefs, densities, and allocations . . . . . . . . . .....  282
10.5 Belief functions on infinite sets . . . . . . . . . . . . . . . 287
10.5.1 Inner measures and belief functions . . . . . . . . . 288
10.5.2 Possibility measures and belief functions . . . . . . 289
10.6 Note on Mobius transforms
of set-functions  ........................          292
10.7 Reasoning with belief functions ...... . . . . . . . . .  293
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10.8 Decision making using belief functions .. . . . . . . . ..295
10.8.1 A minimax viewpoint ... . . . . . . . . . .  . . 296
10.8.2 An expected-value approach . . . . . . . . . ...  297
10.8.3 Maximum entropy principle . . . . . . . . . ...  298
10.9  Rough  sets  ...................         ....  ...  302
10.9.1  An  example  ..........   ..........     .   305
10.9.2 The structure of R .................. 307
10.10  Conditional events  ......................         309
10.11  Exercises  .  . . . . . . . . . . . . . . . . . . . . . . . . . .  311
11 Fuzzy Measures                                           319
11.1 Motivation and definitions .................. 319
11.2 Fuzzy measures and lower probabilities ..... . . . . . ... 321
11.3 Fuzzy measures in other areas . . . . ..... .. .  . . . . 326
11.3.1  Capacities  . . . . . . . . . . . . . . . . . . . . . . .  326
11.3.2 Measures and dimensions . . . . . . . . . . . ...  328
11.3.3  Game  theory  .....................          330
11.4 Conditional fuzzy measures . . . ... . . . . . . . . ... 331
11.5 Exercises  .......... ................. 336
12 The Choquet Integral                                     341
12.1 The Lebesgue integral . . . . . . . . . . . . ..... . . ... 341
12.2  The Sugeno  integral  ...............     ......    343
12.3  The Choquet integral... .  ..................       348
12.3.1  Motivation  .... .. . ..... ...  .. ....  .  348
12.3.2  Foundations  ..................       .  ...  352
12.3.3 Radon-Nikodvm derivatives  . . . . . . . . . . . . 359
12.3.4 Multicriteria decisions with Choquet integrals . . . 363
12.4  Exercises  . . . . . . . . . . . . . . . . . . . . . . . . . . .  365
13 Fuzzy Modeling and Control                               371
13.1 Motivation for fuzzy control ....... . . . . . . . . . .. . 371
13.2 The methodology of fuzzy control . . . . . . . ..... .  .374
13.3  Optimal fuzzy  control  ................. .       .  381
13.4 An analysis of fuzzy control techniques . . . . ..... .. . 382
13.5  Exercises  . . . . .......................          385
Bibliography                                                 387
Answers to Selected Exercises                                401
Index                                                        425



Library of Congress subject headings for this publication: Neural networks (Computer science) Textbooks, Fuzzy logic Textbooks