Dubois / Prade / Yager Readings in Fuzzy Sets for Intelligent Systems

1. Auflage 2014
ISBN: 978-1-4832-1450-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 928 Seiten, Web PDF

Readings in Fuzzy Sets for Intelligent Systems is a collection of readings that explore the main facets of fuzzy sets and possibility theory and their use in intelligent systems. Basic notions in fuzzy set theory are discussed, along with fuzzy control and approximate reasoning. Uncertainty and informativeness, information processing, and membership, cognition, neural networks, and learning are also considered. Comprised of eight chapters, this book begins with a historical background on fuzzy sets and possibility theory, citing some forerunners who discussed ideas or formal definitions very close to the basic notions introduced by Lotfi Zadeh (1978). The reader is then introduced to fundamental concepts in fuzzy set theory, including symmetric summation and the setting of fuzzy logic; uncertainty and informativeness; and fuzzy control. Subsequent chapters deal with approximate reasoning; information processing; decision and management sciences; and membership, cognition, neural networks, and learning. Numerical methods for fuzzy clustering are described, and adaptive inference in fuzzy knowledge networks is analyzed. This monograph will be of interest to both students and practitioners in the fields of computer science, information science, applied mathematics, and artificial intelligence.

Autoren/Hrsg.

Dubois, Didier J.

Prade, Henri

Yager, Ronald R.

Inhaltsverzeichnis

1;Front Cover;1
2;Readings in Fuzzy Sets for Intelligent Systems;4
3;Copyright Page;5
4;Table of Contents;6
5;Acknowledgments;11
6;Chapter 1.
Introduction;14
6.1;The Emergence of Fuzzy Sets and
Possibility Theory;14
6.2;What Fuzzy Sets Are About;16
6.3;Fuzzy Sets Versus Possibility
Distributions;17
6.4;Degrees of Truth and Incomplete
Information;17
6.5;Possibility and Probability;18
6.6;Possibility Theory and Interval Analysis;18
6.7;Fuzzy Relations;18
6.8;Where Are Fuzzy Sets Useful?;19
6.9;Organization of the Readings;21
6.10;Acknowledgments;22
6.11;References;22
6.12;Appendix A:
Papers by L.A. Zadeh since 1965;24
6.13;Appendix B: General English Books and
Proceedings on Fuzzy Sets;28
6.14;Appendix C: Books on Applications of
Fuzzy Sets to Pure Mathematics;31
6.15;Appendix D: References on Fuzzy
Hardwares;31
7;Chapter 2.
Basic Notions in Fuzzy Set Theory;34
7.1;Introduction;34
7.2;Further Readings;35
7.3;FUZZY SETS;40
7.3.1;NOTATION, TERMINOLOGY, AND BASIC OPERATIONS;42
7.3.2;THE CONCEPT OF A FUZZY RESTRICTION AND TRANSLATION RULES
FOR FUZZY PROPOSITIONS;57
7.3.3;REFERENCES;65
7.3.4;BIBLIOGRAPHY;65
7.4;REPRESENTATION THEOREMS FOR FUZZY CONCEPTS;78
7.4.1;1 INTRODUCTION;78
7.4.2;2 PRELIMINARIES;78
7.4.3;3 THE GENERAL REPRESENTATION THEOREM;79
7.4.4;4 REPRESENTATION THEOREM FOR L-TOPO-LOGICAL
SUBSPACES;80
7.4.5;5 L-ALGEBRAIC
SUBSTRUCTURES;81
7.4.6;6 CONCLUSIONS;82
7.4.7;REFERENCES;82
7.5;On Some Logical Connectives for Fuzzy Sets Theory;84
7.5.1;INTRODUCTION;84
7.5.2;1. PRELIMINARIES ON FUZZY CONNECTIVES;84
7.5.3;2. ON STRONG NEGATIONS AND DEMORGAN'S LAWS;86
7.5.4;3. SOME LOGICAL PROPERTIES OF
NONDISTRIBUTIVE CONNECTIVES;87
7.5.5;4. FURTHER CHARACTERIZATION OF THE
CLASSICAL CONNECTIVES;88
7.5.6;REFERENCES;89
7.6;Symmetric Summation: A Class of Operations
on Fuzzy Sets;90
7.6.1;I. COMPLEMENTARY SETS;90
7.6.2;II. SET COMBINATION;90
7.6.3;III. SYMMETRIC SUMS;90
7.6.4;IV. EXAMPLES;91
7.6.5;V. SUMMARY;92
7.6.6;ACKNOWLEDGMENT;92
7.6.7;REFERENCES;92
7.7;On Ordered Weighted Averaging Aggregation
Operators in Multicriteria Decisionmaking;93
7.7.1;INTRODUCTION;93
7.7.2;FORMULATING OF THE AGGREGATION PROBLEM;93
7.7.3;GENERAL "ANDING" AND "ORING" OPERATORS;93
7.7.4;OWA OPERATORS;94
7.7.5;PROPERTIES OF OWA OPERATORS;95
7.7.6;QUANTIFIERS AND OWA OPERATORS;96
7.7.7;MEASURE OF "ANDNESS" AND "ORNESS";97
7.7.8;IN A GENERAL SETTING;98
7.7.9;BUILDING CONSISTENT OWA OPERATORS;99
7.7.10;INCLUDING UNEQUAL IMPORTANCES;99
7.7.11;CONCLUSION;100
7.7.12;REFERENCES;100
7.8;FUZZY POWER SETS AND FUZZY IMPLICATION
OPERATORS;101
7.8.1;1. Towards a theory of fuzzy power sets;101
7.8.2;2. Comparative semantics of fuzzy implication operators;102
7.8.3;3. Height, plinth and crispness of fuzzy sets;105
7.8.4;4. Fuzzy set-inclusions and equalities;106
7.8.5;5. Dis joint ness of fuzzy sets;107
7.8.6;6. A fuzzy set and its complement;108
7.8.7;7. Conservation of crispness, versus the 'bootstrap effect';108
7.8.8;8. Conclusion;109
7.8.9;References;109
7.9;ON IMPLICATION AND INDISTINGUISHABELTTY IN THE
SETTING OF FUZZY LOGIC;110
7.9.1;1. INTRODUCTION;110
7.9.2;2. SOME PRELIMINARIES;111
7.9.3;3.
S-IMLICATIONS;111
7.9.4;4.
R-IMPLICATION;113
7.9.5;REFERENCES;116
7.10;A THEOREM ON IMPLICATION FUNCTIONS
DEFINED FROM TRIANGULAR NORMS;118
7.10.1;0· Introduction;118
7.10.2;1. Background;118
7.10.3;2.
Theorem;120
7.10.4;3. Proof;120
7.10.5;4. Conclusion;123
7.10.6;References;123
7.11;FUZZY NUMBERS: AN OVERVIEW;125
7.11.1;ABSTRACT;125
7.11.2;I. INTRODUCTION;126
7.11.3;II. DEFINITIONS AND FUNDAMENTAL PRINCIPLES;126
7.11.4;III. THE CALCULUS OF FUZZY QUANTITIES WITH NONINTERACTIVE
VARIABLES;134
7.11.5;IV. CALCULATIONS WITH FUZZY INTERVALS IN PRACTICE;141
7.11.6;V. ALTERNATIVE FUZZY INTERVAL CALCULI;145
7.11.7;VI. COMPARISON OF FUZZY QUANTITIES;150
7.11.8;VII. SOME APPLICATIONS OF FUZZY NUMBERS AND INTERVALS;154
7.11.9;VIII. CONCLUSION;155
7.11.10;LIST OF SYMBOLS;156
7.11.11;REFERENCES;157
7.12;A REVIEW OF SOME METHODS FOR RANKING
FUZZY SUBSETS;162
7.12.1;1. Introduction;162
7.12.2;2. Ranking methods;163
7.12.3;3. Comparative examples;166
7.12.4;4. Conclusions;170
7.12.5;References;170
7.13;SOLUTIONS IN COMPOSITE FUZZY RELATION EQUATIONS: APPLICATION TO MEDICAL DIAGNOSIS IN
BROUWERIAN LOGIC;172
7.13.1;PRELIMINARY DEFINITIONS AND RESULTS
OF FUZZY SETS AND FUZZY RELATIONS;172
7.13.2;FUZZY RELATION INEQUATIONS;174
7.13.3;FUZZY RELATIONS;174
7.13.4;MEDICAL DIAGNOSIS AND BROUWERIAN LOGIC;177
7.13.5;CONCLUSION;178
7.13.6;REFERENCES;178
7.14;FUZZY RELATION EQUATION UNDER A CLASS
OF TRIANGULAR NORMS : A SURVEY ANDNEW RESULTS;179
7.14.1;1. Introduction;179
7.14.2;2 . t- nores, conorms and definitions;181
7.14.3;3. Resolution of sup-t fuzzy
equations;184
7.14.4;4. LOWER SOLUTIONS OF EQUATION (4);186
7.14.5;5 . LOWER SOLUTIONS OF EQUATION
(3);189
7.14.6;6. Fuzziness measures based on t-norms and t-conorms;191
7.14.7;8. Approximate solutions;196
7.14.8;9. Some applicationa 1 aspect;197
7.14.9;References;199
8;Chapter 3. Uncertainty and Informativeness;204
8.1;Introduction;204
8.2;Further Readings;205
8.3;A Definition of a Nonprobabilistic Entropy in the
Setting of Fuzzy Sets Theory;210
8.3.1;1. INTRODUCTION;210
8.3.2;2. ENTROPY OF A FUZZY SET;210
8.3.3;3. INTERPRETATION OF d(f);212
8.3.4;ACKNOWLEDGMENTS;215
8.3.5;REFERENCES;215
8.4;On the specificity of a possibility
distribution;216
8.4.1;1. Introduction;216
8.4.2;2. Specificity and negative entropy;217
8.4.3;3. A unifying view of specificity measures;218
8.4.4;4. On the relationship between specificity of
negations;222
8.4.5;5. Specificity in the continuous domain;225
8.4.6;6. Specificity measures modified by a similarity
relationship;227
8.4.7;7. Conclusion;228
8.4.8;References;228
8.5;MEASURES OF UNCERTAINTY AND INFORMATION
BASED ON POSSIBILITY DISTRIBUTIONS;230
8.5.1;1 INTRODUCTION;230
8.5.2;2 HARTLEY'S MEASURE OF
INFORMATION;232
8.5.3;3 POSSIBILITY DISTRIBUTIONS AND
MEASURES;233
8.5.4;4 CHARACTERIZATION OF
POSSIBILISTIC UNCERTAINTY;236
8.5.5;5 PROPOSED MEASURE OF
POSSIBILISTIC UNCERTAINTY;238
8.5.6;6 CONDITIONAL POSSIBILISTIC
UNCERTAINTY;242
8.5.7;7 CONCLUSIONS;243
8.5.8;ACKNOWLEDGEMENT;244
8.5.9;REFERENCES;244
8.6;CONDITIONAL POSSIBILITY MEASURES;246
8.6.1;INTRODUCTION;246
8.6.2;POSSIBILITY DISTRIBUTIONS AND UNCERTAINTY FUNCTIONS;247
8.6.3;DEFINITION OF CONDITIONAL POSSIBILITIES;249
8.6.4;CONDITIONAL AND MARGINAL POSSIBILITIES;250
8.6.5;CONDITIONAL POSSIBILITIES AND INFORMATION DISTANCE;251
8.6.6;COMPARISONS WITH OTHER PROPOSALS;252
8.6.7;REFERENCES;252
8.7;On Modeling of Linguistic Information Using Random Sets;255
8.7.1;I. INTRODUCTION;255
8.7.2;II. GENERALITIES ON RANDOM SETS;256
8.7.3;III. SOME MEASURES OF UNCERTAINTY;257
8.7.4;IV. PROBABILISTIC INTERPRETATION OF POSSIBILITY MEASURES;257
8.7.5;V. CONCLUDING REMARKS;259
8.7.6;REFERENCES;259
8.8;ON THE CONCEPT OF POSSIBILITY-PROBABILITY
CONSISTENCY;260
8.8.1;1. Introduction;260
8.8.2;2. The concept of consistency;260
8.8.3;3. Consistency axioms;262
8.8.4;4. Conclusion;263
8.8.5;References;263
8.9;FUZZY MEASURES AND FUZZY
INTEGRALS—A SURVEY;264
8.9.1;FUZZY MEASURES, FUZZY INTEGRALS AND THEIR SEMANTICS;264
8.9.2;REFERENCES;270
8.10;PROPERTIES OF THE FUZZY EXPECTED VALUE AND THE FUZZY EXPECTED INTERVAL IN FUZZY
ENVIRONMENT;271
8.10.1;1. Introduction;271
8.10.2;2. Fuzzy expected value;271
8.10.3;3. The fuzzy expected interval (FEI);272
8.10.4;4. Addition of fuzzy intervals;274
8.10.5;5. Building a mapping table;275
8.10.6;6. Application to fuzzy expert systems;276
8.10.7;7. Conclusion;277
8.10.8;References;277
8.11;Fuzzy Random Variables;278
8.11.1;1. INTRODUCTION;278
8.11.2;2. INTEGRAL CALCULUS FOR SET-VALUED FUNCTIONS;278
8.11.3;3. FUZZY VARIABLES AND THEIR EXPECTATIONS;280
8.11.4;4. PROPERTIES OF THE EXPECTATION;281
8.11.5;5. COMPUTATION OF E(X);282
8.11.6;6. CONCLUDING REMARKS;283
8.11.7;7. APPENDIX;283
8.11.8;REFERENCES;284
8.12;The Strong Law of Large Numbers for Fuzzy Random Variables;285
8.12.1;1. INTRODUCTION;285
8.12.2;2. FUZZY NUMBERS;285
8.12.3;3. FUZZY RANDOM VARIABLES;286
8.12.4;4. THE LAW OF LARGE NUMBERS;286
8.12.5;REFERENCES;288
9;Chapter 4.
Fuzzy Control;290
9.1;Introduction;290
9.2;Further Readings;291
9.3;An Experiment in Linguistic Synthesis with a Fuzzy
Logic Controller;296
9.3.1;Introduction;296
9.3.2;The Plant to be Controlled;296
9.3.3;The Controller;296
9.3.4;Results and Conclusions;298
9.3.5;Appendix;299
9.3.6;References;302
9.4;ANALYSIS OF A FUZZY LOGIC
CONTROLLER;303
9.4.1;1. Introduction;303
9.4.2;2 . The fuzzy logic controller;303
9.4.3;3. Multilevel relay analogy;304
9.4.4;4. Properties of a multilevel relay: Describing function;305
9.4.5;5. The Prediction of oscillations;306
9.4.6;6. Discussion;308
9.4.7;7. Conclusion;308
9.4.8;8. Appendix;308
9.4.9;Acknowlegdement;310
9.4.10;References;310
9.5;SELECTION OF PARAMETERS FOR A FUZZY LOGIC
CONTROLLER;311
9.5.1;1. Introduction;311
9.5.2;2. A review of the fuzzy logic controller;311
9.5.3;3. Parameters of the fuzzy logic controller;312
9.5.4;4. Conclusion;317
9.5.5;Acknowledgements;317
9.5.6;References;318
9.6;MODELLING CONTROLLERS USING
FUZZY RELATIONS;319
9.6.1;INTRODUCTION;319
9.6.2;FORMATION OF THE FUZZY
RELATION;319
9.6.3;EXAMPLE;322
9.6.4;DISCUSSION;323
9.6.5;REFERENCES;325
9.6.6;HONORARY FELLOWSHIPS;326
9.7;THE USE OF FUZZY SETS FOR THE TREATMENT OF
FUZZY INFORMATION BY COMPUTER;327
9.7.1;1. Introduction;327
9.7.2;2. Fuzzy discretisation of a real space;327
9.7.3;3. Comparative properties of union and intersection of 2 fuzzy sets in X and
X;328
9.7.4;4. The principle of use;328
9.7.5;5. Conclusion;328
9.7.6;References;329
9.8;A Linguistic Self-Organizing
Process Controller;330
9.8.1;1. INTRODUCTION;330
9.8.2;2. THEORY OF THE SOC;331
9.8.3;3. EXPERIMENTS WITH THE SOC;337
9.8.4;4. CONCLUSION;344
9.8.5;REFERENCES;345
9.8.6;APPENDIX A:
RULE MODIFICATION PROCEDURE;345
9.9;Some Properties of Fuzzy Feedback Systems;346
9.9.1;I. INTRODUCTION;346
9.9.2;II. OPEN LOOP FUZZY SYSTEMS;346
9.9.3;III. THE CONCEPT OF STABILITY IN FUZZY SYSTEMS;347
9.9.4;IV. CLOSED LOOP FUZZY SYSTEMS;347
9.9.5;VI. A CONTROL PROBLEM;349
9.9.6;VI. CONCLUSION AND DISCUSSION;349
9.9.7;REFERENCES;349
9.10;CONTROL OF A CEMENT KILN BY FUZZY
LOGIC;350
9.10.1;1. INTRODUCTION;350
9.10.2;2. FUZZY CONTROL PRINCIPLES;351
9.10.3;3.
COMPUTERIZED FUZZY CONTROL;353
9.10.4;4. FUZZY CONTROL OF CEMENT KILN;357
9.10.5;5.
CONCLUSION;359
9.10.6;REFERENCES;360
9.10.7;APPENDIX: BASIC PCL PRINCIPLES;360
9.11;FUZZY CONTROL FOR AUTOMATIC TRAIN
OPERATION SYSTEM;361
9.11.1;INTRODUCTION;361
9.11.2;AUTOMATIC TRAIN OPERATION CONTROL
AND MODELING;361
9.11.3;TRAIN OPERATION BY A HUMAN
OPERATOR;362
9.11.4;FUZZY CONTROL;362
9.11.5;THE FUZZY CONTROLLED
ATO;363
9.11.6;SIMULATION;364
9.11.7;CONCLUSIONS;364
9.11.8;REFERENCES;365
9.12;FUZZY CONTROLLED ROBOT ARM PLAYING TWO
DIMENSIONALPING-PONG GAME;368
9.12.1;1. Introduction;368
9.12.2;2. The control algorithm of the two-dimensional ping-pong robot;368
9.12.3;3. Experimental result;372
9.12.4;4. Conclusions;372
9.12.5;References;373
9.13;A Fuzzy Logic Programming
Environment for Real-Time Control;374
9.13.1;ABSTRACT;374
9.13.2;INTRODUCTION;374
9.13.3;CONTROL RULE PROGRAMMING;376
9.13.4;SYSTEM DEVELOPMENT;378
9.13.5;SUMMARY;379
9.13.6;References;379
9.14;A Reinforcement
Learning-Based Architecture for Fuzzy Logic Control;381
9.14.1;ABSTRACT;381
9.14.2;INTRODUCTION;381
9.14.3;FUZZY LOGIC CONTROL;381
9.14.4;REINFORCEMENT LEARNING, CREDIT ASSIGNMENT, AND
TEMPORAL DIFFERENCE METHODS;383
9.14.5;THE ARIC ARCHITECTURE;384
9.14.6;APPLYING ARIC TO CART-POLE BALANCING;387
9.14.7;RESULTS;389
9.14.8;RELATION TO OTHER RESEARCH;391
9.14.9;CONCLUSIONS;392
9.14.10;ACKNOWLEDGMENTS;393
9.14.11;References;393
9.15;CHARACTERIZATION
OF ACLASS OF FUZZY OPTIMAL CONTROL PROBLEMS;394
9.15.1;SPECIAL PROBLEMS USING THE PESSIMISTIC CRITERION;395
9.15.2;METHODS OF SOLUTION;396
9.15.3;PROPERTIES OF THE OPTIMAL FUZZY CONTROL
IN SUBCASE (D);396
9.15.4;REFERENCES;398
9.16;Fuzzy Identification of Systems and Its
Applications to Modeling and Control;400
9.16.1;I. INTRODUCTION;400
9.16.2;II. FORMAT OF FUZZY IMPLICATION AND
REASONING ALGORITHM;400
9.16.3;III. ALGORITHM OF IDENTIFICATION;402
9.16.4;IV. APPLICATION TO FUZZY MODELING;410
9.16.5;V. CONCLUSION;415
9.16.6;REFERENCES;416
9.17;Fuzzy Modeling of
Complex Systems;417
9.17.1;ABSTRACT;417
9.17.2;INTRODUCTION;417
9.17.3;THE STATIC SUGENO'S FUZZY MODEL;417
9.17.4;QUASILINEAR FUZZY MODELS (QLFMs) OF NONLINEAR
DYNAMICAL SYSTEMS;417
9.17.5;IDENTIFICATION OF THE QLFM;419
9.17.6;TRANSFER FUNCTION AND STATE-SPACE DESCRIPTION OF
QLFM;420
9.17.7;CONCLUSION;421
9.17.8;References;421
10;Chapter 5.
Approximate Reasoning;422
10.1;Introduction;422
10.2;Further Readings;423
10.3;THE LOGIC OF INEXACT CONCEPTS;430
10.3.1;I. INTRODUCTION;430
10.3.2;II. A PARADOX;431
10.3.3;III. RTSOLUTION OF THE PARADOX;432
10.3.4;IV. REPRESENTING INEXACT CONCEPTS;435
10.3.5;V. THE AL.GEBRA OF INEXACT
PREDICATES;439
10.3.6;VI. OPTIMIZATION AND THE
TRUTHSET;442
10.3.7;VII. IMPLICATION AND NEGATION;445
10.3.8;VIII. THE LOGIC OF INEXACT CONCEPTS;448
10.3.9;ACKNOWLEDGMENT;453
10.3.10;BIBLIOGRAPHY;453
10.4;Fuzzy Logic and the Resolution Principle;455
10.4.1;1. Introduction;455
10.4.2;2. Fuzzy Logic;456
10.4.3;3. Satisfiability in Fuzzy Logic;457
10.4.4;4. The Concept of Logical Consequence in Fuzzy Logic;461
10.4.5;5. Conclusions;464
10.4.6;REFERENCES;465
10.5;Fundamentals of Fuzzy
Prolog;466
10.5.1;ABSTRACT;466
10.5.2;INTRODUCTION;466
10.5.3;CONFIDENCE, THE FUZZY RESOLUTION PRINCIPLE, AND
CONFIDENCE OF RESOLVENT;467
10.5.4;THE WEIGHT OF RULE;469
10.5.5;FUZZY RESOLUTION IN FUZZY FIRST-ORDER PREDICATE
LOGIC;471
10.5.6;CONCLUSION;472
10.6;Epistemic entrenchment and
possibilistic logic;474
10.6.1;1. Introduction;474
10.6.2;2. Necessity measures and their qualitative counterpart;475
10.6.3;3. Epistemic entrenchment and qualitative necessity measures;476
10.6.4;4. Dealing with uncertain knowledge bases;477
10.6.5;5. Possibilistic reasoning and belief revision;479
10.6.6;6. Possibility theory and Spohn's ordinal conditional functions;480
10.6.7;7. Conclusion;481
10.6.8;Acknowledgement;481
10.6.9;References;482
10.7;Fuzzy and Probability Uncertainty Logics;483
10.7.1;1. INTRODUCTION;483
10.7.2;2. MIN/MAX CONNECTIVES IN PROBABILITY LOGICS;483
10.7.3;3. A FORMAL BASIS FOR THE COMPARISON OF VARIOUS LOGICS OF
UNCERTAINTY;484
10.7.4;4. DERIVATION OF RESCHER'S
PROBABILITY LOGIC;487
10.7.5;5. DERIVATION OF LN1,
A FUZZY LOGIC;487
10.7.6;6. SEMANTICS FOR THE LOGICS IN TERMS OF POPULATION RESPONSES;488
10.7.7;7. SUMMARY AND CONCLUSIONS;489
10.7.8;ACKNOWLEDGMENTS;490
10.7.9;REFERENCES;490
10.8;Possibility Theory
and Soft Data Analysis;491
10.8.1;Abstract;491
10.8.2;1. Introduction;491
10.8.3;2. Basic Properties of Possibility Distributions;494
10.8.4;3. Translation Rules and Meaning Representation;499
10.8.5;4. Inference from Soft Data and Mathematical Programming;504
10.8.6;5. Examples of Inference from Soft Data;509
10.8.7;6. Evidence, Certainty and Possibility;514
10.8.8;7. Concluding Remark;519
10.8.9;8. References and Related Papers;519
10.9;AXIOMATIC APPROACH TO IMPLICATION FOR
APPROXIMATE REASONING WITH FUZZY LOGIC;522
10.9.1;1. Introduction;522
10.9.2;2. Some properties of implication rules found in the literature;522
10.9.3;3. Discussion of the properties desirable in an implication rule;528
10.9.4;4. Axioms for implication and derivation of classes of suitable functions;530
10.9.5;5. Conclusions;533
10.9.6;References;535
10.10;AN APPROACH TO FUZZY REASONING METHOD;536
10.10.1;1. INTRODUCTION;536
10.10.2;2 . PRELIMINARIES;536
10.10.3;3 . FUZZIFICATION OF
L;536
10.10.4;4 . RULES OF INFERENCE USING FUZZY ASSERTIONS;538
10.10.5;5 . COMPARISON WITH OTHER LOGICAL SYSTEMS;540
10.10.6;6. ILLUSTRATIVE EXAMPLES;540
10.10.7;7. CONCLUDING REMARKS;541
10.10.8;ACKNOWLEDGEMENTS;542
10.10.9;REFERENCES;542
10.11;SOME CONSIDERATIONS ON FUZZY CONDITIONAL INFERENCE;543
10.11.1;1. INTRODUCTION;543
10.11.2;2. Fuzzy sets—notation, terminology and basic operations;543
10.11.3;3. Fuzzy conditional inference;544
10.11.4;4. Formalization of improved methods;550
10.11.5;5. Concluding remarks;557
10.11.6;References;558
10.12;A FUZZY SYSTEM MODLE BASDE ON THE LOGICAL STRUCTEUR;559
10.12.1;ABSTRATC;559
10.12.2;KEYWORSD;559
10.12.3;INTRODUCTION;559
10.12.4;FUZZY
REASONING;559
10.12.5;THE MODEL OF FUZZY SYSTEM;561
10.12.6;CONCLUDING REMARKS;566
10.12.7;APPENDIX;567
10.13;ON MODE AND IMPLICATION IN APPROXIMATE REASONING;568
10.13.1;INTRODUCTION;568
10.13.2;MODUS PONENS IN BOOLEAN, PROBABILISTIC
AND FUZZY LOGIC;568
10.13.3;MODUS PONENS GENERATING
FUNCTIONS;569
10.13.4;MODUS TOLLENS GENERATING FUNCTIONS;571
10.13.5;CONCLUDING REMARKS;572
10.13.6;ACKNOWLEDGEMENTS;572
10.13.7;REFERENCES;572
10.14;FUZZY INFERENCES AND CONDITIONAL
POSSIBILITY DISTRIBUTIONS;573
10.14.1;1. Introduction;573
10.14.2;2. Conditional possibility distributions;573
10.14.3;3. Joint possibility distributions;574
10.14.4;4. Non-acting and non-interactivity;575
10.14.5;5. Conclusion;576
10.14.6;Acknowledgements;577
10.14.7;References;577
10.15;Fuzzy Modus Ponens: A New Model Suitable for Applications in Knowledge-Based
Systems;578
10.15.1;I. INTRODUCTION;578
10.15.2;II. GENERALIZED MODUS PONENS;578
10.15.3;III. CLASSICAL RULES OF IMPLICATION;579
10.15.4;IV. INTUITIVE PROPERTIES OF THE IMPLICATION;580
10.15.5;V. GMP WITH A SUP-TM COMPOSITION;581
10.15.6;VI. THE MODUS PONENS;581
10.15.7;VII. CONCLUSIONS;586
10.15.8;References;586
10.16;Using Approximate Reasoning to
Represent Default Knowledge;588
10.16.1;1. Introduction;588
10.16.2;2. A Theory of Approximate Reasoning;588
10.16.3;3. Default Variables and Possibility Qualification;591
10.16.4;4. Typical Default Reasoning Data;593
10.16.5;5. Conclusion;594
10.16.6;REFERENCES;594
10.17;FUZZY SET SIMULATION MODELS IN A SYSTEMS
DYNAMICS PERSPECTIVE;595
10.17.1;THE POINT OF DEPARTURE;595
10.17.2;CONTENTION;595
10.17.3;IMPLEMENTATION;596
10.17.4;VALIDATION OF A SEMANTICAL MODEL;599
10.17.5;EPILOGUE: VERBAL MODELS IN A SYSTEM DYNAMICS ENVIRONMENT;602
10.17.6;REFERENCES;603
11;Chapter
6. Information Processing;606
11.1;Introduction;606
11.2;Further Readings;606
11.3;Numerical Methods for Fuzzy Clustering;612
11.3.1;ABSTRACT;612
11.3.2;1. INTRODUCTION;612
11.3.3;2. DEFINITIONS AND NOMENCLATURE;612
11.3.4;3. SOME USEFUL RESULTS;613
11.3.5;4. FORMULAS BASED ON MEAN CLUSTER DENSITY (CLUSTERING I AND II);615
11.3.6;5. AN IMPROVED METHOD;623
11.3.7;6. DISCUSSION;627
11.3.8;ACKNOWLEDGMENT;627
11.3.9;REFERENCES;627
11.4;A Physical Interpretation of Fuzzy ISODATA;628
11.4.1;I. INTRODUCTION;628
11.4.2;II. AN ELECTRICAL ANALOG FOR FUZZY ISODATA;628
11.4.3;REFERENCES;629
11.5;Convex Decompositions of Fuzzy Partitions;630
11.5.1;I. INTRODUCTION AND CONCLUSIONS;630
11.5.2;II. PARTITION SPACES;630
11.5.3;III. THE DIMENSION OF FUZZY PARTITION SPACE;631
11.5.4;IV.MINIMAX
DECOMPOSITION;632
11.5.5;V. RECLASSIFICATION DECOMPOSITION;635
11.5.6;VI. CONVEX DECOMPOSITION AND THE PARTITION COEFFICIENT;637
11.5.7;VII. SUMMARY;640
11.5.8;ACKNOWLEDGEMENT;640
11.5.9;REFERENCES;641
11.6;New Results in Fuzzy Clustering Based on the
Concept of Indistinguishability Relation;642
11.6.1;I. INTRODUCTION;642
11.6.2;II. ON INDISTINGUISHABILITY RELATIONS;642
11.6.3;IV. CONCLUDING REMARKS;645
11.6.4;REFERENCES;645
11.7;The fuzzy geometry of image subsets;646
11.7.1;1. Introduction;646
11.7.2;2. Fuzzy subsets [2];646
11.7.3;3. Connectedness and surroundedness
[2,3];647
11.7.4;4. Adjacency;648
11.7.5;5. Convexity [4-6] and starshapedness;648
11.7.6;6. Area, perimeter, and compactness [7];649
11.7.7;7. Extent and diameter [8];649
11.7.8;8. Shrinking and expanding, medial axes,
elongatedness, and thinning;650
11.7.9;9. Gray-level-dependent properties; splitting and
merging;651
11.7.10;10. Representation of fuzzy subsets;651
11.7.11;11. Concluding remarks;652
11.7.12;References;652
11.8;Fuzzy Confidence Measures in Midlevel Vision;653
11.8.1;I. INTRODUCTION;653
11.8.2;II. THE LINGUISTIC CONFIDENCE STRUCTURE
IN IMAGE ANALYSIS;653
11.8.3;III. APPLICATIONS;656
11.8.4;IV. CONCLUSION;658
11.8.5;REFERENCES;659
11.9;Fuzzy sets and generalized Boolean retrieval systems;661
11.9.1;1. Introduction;661
11.9.2;2. Fuzzy retrieval;663
11.9.3;3. Generalized queries;664
11.9.4;4. Thresholds;665
11.9.5;5. Other approaches;666
11.9.6;6. Performance measures;668
11.9.7;7. Summary and conclusions;668
11.9.8;References;669
11.9.9;Appendix A:
WALLER-KRAFT CRITERIA FOR WEIGHTED RETRIEVAL;671
11.10;A FUZZY REPRESENTATION OF DATA FOR
RELATIONAL DATABASES;673
11.10.1;1. Introduction;673
11.10.2;2. Organization of a fuzzy database;673
11.10.3;3. Redundancy and determinancy properties;675
11.10.4;4. Application;676
11.10.5;5. Conclusions;679
11.10.6;References;679
11.11;FREEDOM-O: A FUZZY DATABASE SYSTEM;680
11.11.1;1. INTRODUCTION;680
11.11.2;2. POSSIBILITY DISTRIBUTIONS;680
11.11.3;3. FUZZY DATABASES;682
11.11.4;4. DATA MANIPULATION LANGUAGE;683
11.11.5;5. EXAMPLES OF QUERY STATEMENTS;686
11.11.6;6. CONCLUSIONS;688
11.11.7;REFERENCES;688
11.12;WEIGHTED FUZZY PATTERN MATCHING;689
11.12.1;1. Introduction;689
11.12.2;2. Fuzzy pattern matching;689
11.12.3;3. Importance assignment;693
11.12.4;4. Conclusion;697
11.12.5;References;697
11.13;FUZZY QUERYING WITH SQL: EXTENSIONS AND
IMPLEMENTATION ASPECTS;699
11.13.1;1. Introduction;699
11.13.2;2. Non-fuzzy sets based interpretations;699
11.13.3;3. Extending the SQL language;701
11.13.4;4. Some implementation aspects;704
11.13.5;5. Conclusions and future works;706
11.13.6;References;707
11.14;FILIP: A FUZZY INTELLIGENT INFORMATION
SYSTEM WITH LEARNING CAPABILITIES;708
11.14.1;1. INTRODUCTION;708
11.14.2;2. INTELLIGENT INFORMATION SYSTEM
WITH LEARNING CAPABILITIES;709
11.14.3;3. CONCEPT LEARNING IN FILIP;714
11.14.4;4. CONCLUSION;720
11.14.5;REFERENCES;721
12;Chapter 7.
Decision and Management Sciences;722
12.1;Introduction;722
12.2;Further Readings;723
12.3;DECISION-MAKING WITH A
FUZZY PREFERENCE RELATION;730
12.3.1;1. Introduction;730
12.3.2;2. Preliminary definitions;730
12.3.3;3. Fuzzy preference relations and nondominated alternatives;731
12.3.4;4 . Transitive fuzzy preference relations;732
12.3.5;5. Properties of unfuzzy nondominated elements;734
12.3.6;6. Existence of unfuzzy nondominated elements;735
12.3.7;7. Summary;736
12.3.8;Acknowledgement;736
12.3.9;References;736
12.4;STRUCTURE OF FUZZY BINARY RELATIONS;737
12.4.1;Introduction;737
12.4.2;1. Fuzzy sets;737
12.4.3;2. V-correspondences, V-relations, and V-mappings;738
12.4.4;3. Structure of V-equivalences;739
12.4.5;4. V-preferences;741
12.4.6;5. Relations between transitivity properties;742
12.4.7;6. V-quasi-orders;744
12.4.8;7. V-choice functions;745
12.4.9;8. Scalar V-criteria and quasi-transitive V-preferences;746
12.4.10;9. Stability;747
12.4.11;10. Indices of fuzziness;747
12.4.12;11. Stability of the transitivity property;748
12.4.13;References;749
12.5;Some properties of choice functions based
on valued binary relations;751
12.5.1;1. Introduction;751
12.5.2;2. The operators
T and T;752
12.5.3;3. Choice functions for valued pairwise comparisons;753
12.5.4;4. Some properties of DR(y), NR(y), SDR( y) and
SNR(Y);756
12.5.5;5. Rationality properties of the choice functions;759
12.5.6;6. Some particular cases;761
12.5.7;7. Conclusions;762
12.5.8;Acknowledgment;762
12.5.9;References;762
12.6;A NEW METHODOLOGY FOR ORDINAL MULTIOBJECTIVE DECISIONS BASED ON FUZZY
SETS;764
12.6.1;INTRODUCTION;764
12.6.2;MULTIOBJECTIVE DECISION MAKING;764
12.6.3;FUZZY SET APPROACH;765
12.6.4;SET THEORY AND LOGIC;766
12.6.5;A MODEL USING ORDINAL INFORMATION;766
12.6.6;EXAMPLE;767
12.6.7;DISCUSSION OF PROPERTIES OF THE MODEL;768
12.6.8;REFERENCES;769
12.7;Decision Making Under Uncertainty
with Various Assumptions About Available Information;770
12.7.1;PART I: SURVEY
OF METHODOLOGIES;770
12.7.2;PART II: EXAMPLE;777
12.7.3;PART III: CONCLUSION;787
12.7.4;REFERENCES;789
12.8;A Linguistic Approach to Decisionmaking
with Fuzzy Sets;790
12.8.1;I. INTRODUCTION;790
12.8.2;II. A MULTICHOICE DECISION PROBLEM;790
12.8.3;III. TRUTH QUALIFICATION AND LINGUISTIC APPROXIMATION;792
12.8.4;IV. AN INVESTMENT DECISION PROBLEM;793
12.8.5;V. DISCUSSION;795
12.8.6;VI. CONCLUSION;796
12.8.7;ACKNOWLEDGMENT;796
12.8.8;REFERENCES;797
12.9;GROUP DECISION MAKING WITH FUZZY MAJORITIES REPRESENTED
BY LINGUISTIC QUANTIFIERS;798
12.9.1;1. INTRODUCTION;798
12.9.2;2. ZADEH'S DISPOSITION - BASED REPRESENTATION OF COMMONSENSE KNOWLEDGE, AND ITS
UNDERLYING CALCULI OF LINGUISTICALLY QUANTIFIED PROPOSITION;799
12.9.3;3. GROUP DECISION MAKING WITH A FUZZY MAJORITY;800
12.9.4;4. REMARKS ON "SOFT" DEGREES OF CONSENSUS;806
12.9.5;5. CONCLUDING REMARKS;806
12.9.6;LITERATURE;806
12.10;Applications of Fuzzy Set Theory to Mathematical Programming;808
12.10.1;1. INTRODUCTION;808
12.10.2;2. SYMMETRICAL MODELS;808
12.10.3;3. NONSYMMETRICAL MODELS;811
12.10.4;4. DUALITY AND SENSITIVITY ANALYSIS;813
12.10.5;5. EXTENSIONS;816
12.10.6;6. MULTIPLE OBJECTIVE PROGRAMMING;819
12.10.7;7. FUTURE PERSPECTIVES;821
12.10.8;REFERENCES;822
12.11;Fuzzy Versus Stochastic Approaches to Multicriteria
Linear Programming under Uncertainty;823
12.11.1;1. INTRODUCTION;823
12.11.2;2. SHORT DESCRIPTION OF STRANGE AND FLIP;824
12.11.3;3. DIDACTIC EXAMPLE;826
12.11.4;4. COMPARISON OF STRANGE AND FLIP;832
12.11.5;5. CONCLUSIONS;833
12.11.6;REFERENCES;834
12.12;FUZZY SETS IN FEW CLASSICAL OPERATIONAL RESEARCH PROBLEMS;835
12.12.1;1. INTRODUCTION;835
12.12.2;2. FLOWS IN NETWORKS WITH FUZZY CAPACITY CONSTRAINTS;836
12.12.3;3. FUZZY APPROACH TO PROJECT ANALYSIS;839
12.12.4;4. FUZZY ASSIGNMENT PROBLEM;843
12.12.5;5. FINAL REMARKS;845
12.12.6;FOOTNOTES;847
12.12.7;REFERENCES;847
12.13;FUZZY ZERO-BASE BUDGETING;848
12.13.1;References;852
12.14;THE FUZZY MATHEMATICS OF FINANCE;853
12.14.1;1. Introduction;853
12.14.2;2. Future and present value;854
12.14.3;3 . Fuzzy annuities;857
12.14.4;4. Fuzzy cash flows;858
12.14.5;5. Summary and conclusions;861
12.14.6;References;861
13;Chapter 8.
Membership, Cognition, Neural Networks, and Learning;862
13.1;Introduction;862
13.2;Further Readings;862
13.3;RECONCILIATION OF THE YES-NO VERSUS GRADE OF MEMBERSHIP DUALISM IN HUMAN
THINKING;867
13.3.1;1 . INTRODUCTION;867
13.3.2;2 . THE GRADE OF MEMBERSHIP CONCEPT;867
13.3.3;3 . THE MAX MIN OPERATIONS;869
13.3.4;4 . FORMER POSTULATES AND DIFFUCULTIES;870
13.3.5;5 . RESOLUTION OF THE DUALISM OF THE LAW OF THE EXCLUDED MIDDLE (LEM) VERSUS GRADABLE CONCEPTS AND OF AFFIRMATION-NEGATION VERSUS
ANTONYMS;871
13.3.6;6 . CONCLUDING REMARKS;872
13.3.7;REFERENCES;873
13.4;A MODEL FOR THE MEASUREMENT OF MEMBERSHIP
AND THE CONSEQUENCES OF ITS EMPIRICAL IMPLEMENTATION;874
13.4.1;1. Introduction;874
13.4.2;2. The fundamental measurement of fuzziness;874
13.4.3;3. The construction of membership functions;878
13.4.4;4. The meaningfulness of operations on membership;883
13.4.5;5. Conclusions;885
13.4.6;Acknowledgement;885
13.4.7;References;885
13.5;THE CONCEPT OF GRADE OF MEMBERSHIP;887
13.5.1;1. Nature of the problem;887
13.5.2;2. Two approaches to the problem;887
13.5.3;3. The semantic approach to fuzzy reasoning;888
13.5.4;4. Decision theory;889
13.5.5;5. Application to assertions;890
13.5.6;6. Mathematical foundations;892
13.5.7;7. Simple assertions;893
13.5.8;8. The concept of grade of membership;895
13.5.9;9. The definition of connectives;897
13.5.10;10. The logic of assertions;898
13.5.11;References;900
13.6;ADAPTIVE INFERENCE IN FUZZY KNOWLEDGE NETWORKS;901
13.6.1;I. Knowledge Nets vs.
Expert Systems: Digraphs vs. Trees;901
13.6.2;II. Combining Fuzzv Knowledge Networks;902
13.6.3;III. ADAPTIVE INFERENCE THROUGH CONCOMITANT VARIATION;904
13.7;A Formulation of Fuzzy Automata and its Application
as a Model of Learning Systems;905
13.7.1;I. INTRODUCTION;905
13.7.2;II. FOR MULATION OF FUZZY
AUTOMATA;905
13.7.3;III. SPECIAL CASES OF FUZZY AUTOMATA;907
13.7.4;IV. FUZZY AUTOMATA AS MODELS
OF LEARNING SYSTEMS;908
13.7.5;VI.
CONCLUSION;911
13.7.6;APPENDIX I;911
13.7.7;APPENDIX II:
PROPERTIES OF Fuzzy AUTOMATA;912
13.7.8;REFERENCES;913
13.8;LEARNING OF FUZZY PRODUCTION RULES FOR MEDICAL DIAGNOSIS;914
13.8.1;1. INTRODUCTION;914
13.8.2;2. OVERVIEW OF THE SYSTEM;915
13.8.3;3. FUZZY PRODUCTION RULES;917
13.8.4;4. EXPERIMENTAL RESULTS;920
13.8.5;5. CONCLUDING REMARKS;922
13.8.6;ACKNOWLEDGEMENTS;924
13.8.7;REFERENCES;924
14;Index;926