Fuzzy Sets in Approximate Reasoning and Information Systems

514 Pages·1999·13.42 MB·English

by  Didier Dubois

FUZZY SETS IN
APPROXIMATE REASONING
AND INFORMATION
SYSTEMS
THE HANDBOOKS
OF FUZZY SETS SERIES
Series Editors
DidierDubois and HenriPrade
IRIT, UniversitePaulSabatier, Toulouse, France
FUNDAMENTALSOFFUZZYSETS,editedbyDidierDuboisandHenriPrade
MATHEMATICSOFFUZZYSETS:Logic, Topology,andMeasureTheory,edited
byUlrichHohleandStephenErnestRodabaugh
FUZZY SETS IN APPROXIMATE REASONING AND INFORMATION
SYSTEMS,editedbyJamesC. Bezdek,DidierDuboisandHenriPrade
FUZZYMODELSANDALGORITHMSFORPATTERNRECOGNITIONAND
IMAGEPROCESSING,byJamesC. Bezdek, JamesKeller, Raghu Krisnapuram
andNikhilR. Pal
FUZZY SETSINDECISIONANALYSIS,OPERATIONSRESEARCHAND
STATISTICS,editedbyRoman Slowinski
FUZZYSYSTEMS: ModelingandControl, editedbyHungT. NguyenandMichio
Sugeno
PRACTICALAPPLICATIONSOFFUZZYTECHNOLOGIES,editedbyHans
JUrgenZimmermann
FUZZY SETS IN
APPROXIMATE REASONING
AND INFORMATION
SYSTEMS
edited by
c.
James Bezdek
University o/West Florida
Didier Dubois
lRIT, CNRS & University o/Toulouse III
and
Henri Prade
IRIT, CNRS & University o/Toulouse III
.
. ,
~
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging-in-Publication Data
Fuzzy sets in approximate reasoning and information systems / edited
by James C. Bezdek, Didier Dubois and Renri Prade.
p. cm. -- (The Randbooks of fuzzy sets series ; FSRS 5)
Includes bibliographical references.
ISBN 978-1-4613-7390-2 ISBN 978-1-4615-5243-7 (eBook)
DOI 10.1007/978-1-4615-5243-7
1. Expert systems (Computer science) 2. Reasoning. 3. Fuzzy
sets. 1. Bezdek, James C., 1939-. II. Dubois, Didier.
III. Prade, Renri M. IV. Series.
QA76.76.E95F8866 1999
006.3 3--dc21 99-16664
I
CIP
Copyright © 1999 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1999
Softcover reprint of the hardcover 1s t edition 1999
AlI rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer Science+Business Media, LLC
Printed an acid-free paper.
Contents
Series Foreword xi
ContributingAuthors xiii
Introduction
DidierDubois, HenriPrade, JamesBezdek
PARTI REASONING
1
Fuzzy Sets and Possibility Theory in Approximate and Plausible
Reasoning 15
BernadetteBouchon-Meunier, DidierDubois, L1uisGodoandHenriPrade
1.1 Introduction 15
1.1.1 The Emergence of Fuzzy Set-Based Approximate
Reasoning 16
1.1.2 Organization oftheChapter 17
1.2 ARefresheron PossibilityTheoryar:ld FuzzyConnectives 18
1.2.1 AShortCourseon PossibilityTheory 18
1.2.2 Fuzzy Connectives: Negations, Conjunctions and
Disjunctions 25
1.2.3 FuzzyImplications 27
1.3 Representationof FuzzyStatements 31
1.3.1 Linguistic Variables 32
1.3.2 LinguisticModifiers 33
1.3.3 Modelling ElementaryFuzzyStatements 35
1.3.4 Compound FuzzyStatements 39
1.4 RepresentationofQualifiedandQuantified FuzzyStatements 43
1.4.1 FuzzyTruth-Values 43
1.4.2 Truth QualifiedStatements 46
1.4.3 Graded Truth Versus Degrees of Uncertainty: The
Compositionality Problem 48
1.4.4 Certaintyand PossibilityQualification 51
1.4.5 ProbabilityQualification 54
1.4.6 FuzzyQuantifiedStatements 55
vi APPROXIMATE REASONING AND INFORMATIONSYSTEMS
1.5 The RepresentationofFuzzyRules 59
1.5.1 TheNon FuzzyCase: RelationalandMetaLevelViews 59
1.5.2 Fuzzy Rules: Implication-BasedModels 61
1.5.3 Fuzzy Rules: Conjunction-Based Models 64
1.5.4 MetalevelModelsofFuzzyRules 66
1.5.5 Comparison BetweenFuzzy Rules 66
1.5.6 OtherRepresentational Issueson Fuzzy Rules 71
1.6 Basic Principlesof FuzzyInference 74
1.6.1 CombinationandProjection Principles 74
1.6.2 GenericFuzzyReasoningTechniques 77
1.7 ElementaryFuzzyInference:TheGeneralizedModusPonens 84
1.7.1 BasicFormsoftheGeneralizedModusPonens 84
1.7.2 PropertiesoftheGeneralizedModusPonens 86
1.7.3 RetrievingModusPonensasaParticularCase 88
1.7.4 GeneralizedModusPonensand FuzzyTruth-Values 91
1.7.5 Inferencewith FuzzyMeta-Rules 93
1.8 Systemsof FuzzyRules 96
1.8.1 CompoundCondition Parts 96
1.8.2 Systemsof Parallel Rules 97
1.8.3 Chaining 105
1.8.4 ConsistencyCheckingand Redundancy Elimination 107
1.9 InferencewithCertainty FactorsandApproximate Reasoning 110
1.9.1 TheCertaintyFactorApproach 110
1.9.2 CertaintyFactorsandMultiple-ValuedLogic 112
1.9.3 APossibility-TheoreticViewofCertainty Factors 116
1.10 Specialized Formsof FuzzyDeductiveReasoning 118
1.10.1 ADeductiveApproachtoInterpolativeReasoning 118
1.10.2 Deductive SimilarityReasoning 122
1.10.3 PossibilisticLogic 125
1.10.4 TheSemanticAgreementBetween Many-Valued
Logics, Uncertain, andApproximate Reasoning 127
1.10.5 ReasoningaboutPreferences 129
1.10.6 FuzzyTemporal Reasoning 133
1.10.7 FuzzyQualitative Reasoning 136
1.11 FuzzySet-BasedPlausible Reasoning 139
1.11.1 Plausible Similarity-BasedReasoning 139
1.11.2 Reasoningwith FuzzyQuantifiers 143
1.11.3 NonmonotonicReasoning 146
1.11.4 FuzzyCausal Reasoning 154
1.12 Conclusion 160
References 162
2
Weighted InferenceSystems 191
Vi/em Novak
2.1 Introduction 191
Contents vii
2.2 Formal Inferenceand its Interpretation 193
2.3 Truth ValuesinMany-ValuedLogic 196
2.4 Many-Valued, Fuzzyand Possibilistic Logic 199
2.4.1 Historical Remarks 199
2.4.2 Lukasiewicz Logic 201
2.4.3 FuzzyLogicin NarrowSense 204
2.4.4 MonoidalLogicasAggregatingView 212
2.4.5 PossibilisticLogic 215
2.4.6 Some Other Logic Kinds of Weighted Reasoning
Systems 223
2.5 Rule-Based InferenceSystems 225
2.6 FuzzyLogic Programming 227
2.6.1 TheoreticalAspectsof FuzzyLogic Programming 227
2.6.2 Attemptsat Fuzzy PROLOG 233
2.7 SummaryandConclusion 234
References 235
3
ClosureOperatorsinFuzzysetTheory 243
LoredanaBiacinoandGiangiacomoGerla
3.1 Introduction 243
3.2 ClosureOperatorsinaLattice 244
3.3 AbstractLogics 246
3.4 ContinuityforAbstractLogics 247
3.5 Step-by-stepDeduction Systems 247
3.6 LogicalCompactness 249
3.7 Basic Notionsin FuzzySetsTheory 250
3.8 Abstract FuzzyLogic 253
3.9 Pavelka'sLogic 256
3.10 LogicalCompactnessand Ultraproducts 259
3.11 An extension PrincipleforDeductionOperators 261
3.12 ExtendingCompactDeductionSystems 264
3.13 SimilarityLogic 266
3.14 Ying'sSimilarityLOgic 267
3.15 StratifiedFuzzylogic 269
3.16 GradedConsequence Relations 272
References 277
viii APPROXIMATE REASONING AND INFORMATION SYSTEMS
PARTII LEARNINGAND FUSION
4
Learning FuzzyDecision Rules 279
BernadetteBouchon-MeunierandChristopheMarsala
4.1 Introduction 279
4.2 FuzzyDecision Rules 280
4.3 InductiveLearningofFuzzyRules 281
4.3.1 GeneralInductiveLearning Method 282
4.3.2 Fuzzy DecisionTrees 283
4.3.3 Fuzzy Prototypes 290
4.3.4 Determinationof Fuzzy Modalities 291
4.3.5 OtherMethodsofInductiveRuleConstruction 292
4.3.6 Applications of FuzzyInductiveLearning 292
4.4 Tuning FuzzyRules 293
4.4.1 SelfLeaming 294
4.4.2 UtilizationofaFuzzyModelofthe Process 294
4.4.3 Fuzzy Relational Equations 296
4.4.4 ClusteringMethods 297
4.4.5 Optimization 297
4.4.6 GeneticAlgorithms 298
4.5 Conclusion 299
References 299
5
Neuro-FuzzyMethodsin FuzzyRuleGeneration 305
DetlefNauckandRudolfKruse
5.1 Introduction 305
5.2 StructureLearning- Creating Fuzzy Rules 307
5.2.1 Cluster-OrientedApproachestoLearn Fuzzy Rules 308
5.2.2 Hyperbox-OrientedApproachestoLearn FuzzyRules 311
5.2.3 Structure-OrientedApproachesto Learn FuzzyRules 312
5.3 ParameterLearing- Adapting FuzzySets 315
5.3.1 Adapting FuzzySetsbyGradientDescent 316
5.3.2 Adapting FuzzySetswith ANFIS 318
5.3.3 Adapting FuzzySetsbySimpleHeuristics 320
5.4 Tuning Fuzzy RuleswithAdaptive Weights 327
5.5 Conclusions 331
References 332
6
MergingFuzzy Information 335
DidierDubois, HenriPradeand RonaldYager
6.1 Introduction 335
Contents ix
6.2 Information Fusion 337
6.2.1 Characteristicsofthe Information Fusion Problem 337
6.2.2 Some Limitations of a Pure Probabilistic Approach to
Fusion 339
6.3 The Possibilistic RepresentationofIncompleteData 343
6.3.1 PossibilityTheory: ARefresher 343
6.3.2 BuildingPossibilityDistributions 345
6.4 The Fusionof PossibilityDistributions 350
6.4.1 BasicSymmetricCombination Modesinthe Possibilistic
Setting 351
6.4.2 ExpectedFormal PropertiesofCombination Rules 356
6.5. Refined Fusion Modes 361
6.5.1 Assessmentofthe ReliabilityofSources 361
6.5.2 Fusionwith UnequallyReliable Sources
6.5.3 Adaptive Fusion 369
6.6 Fuzzy Estimation 373
6.6.1 Fuzzy Extension of EstimationTechniques 373
6.6.2 Constrained Merging 375
6.7 Possibilistic FusionUnderAPriori Knowledge 380
6.7.1 Revisionwith FuzzyInputs 381
6.7.2 MultisourceFusion underaPriori Knowledge 385
6.8 SyntacticCombinationof Logical Databasesin Possibilistic Logic 386
6.8.1 From Prioritized Logical Databases to Possibility
Distributions 387
6.8.2 SyntacticCombination Modes 388
6.9 Conclusion 391
References 392
PARTIII FUZZYINFORMATION SYSTEMS
7
Fuzzy Databases 403
PatrickBose, BillB. Buckles, FrederickE. Petry, and 0. Pivert
7.1 Introduction 403
7.2 FuzzyQueryingin Relational Databases 404
7.2.1 Overviewof Relational Databases 405
7.2.2 RelationswithWeightedTuples 409
7.2.3 ModelingFuzzyQueries 409
7.2.4 FlexibleQueriesAgainstCrisp Databases 414
7.2.5 ARelational AlgebraforFuzzy Relations 416
7.2.6 SQU, anSQL-LikeFuzzyQueryLanguage 419
7.2.7 ImplementationAspectsandQuery Processing 422
7.2.8 Cooperative Answering 427
7.3 UncertaintyandIncompletenessin Relational Databases 428
7.3.1 Non PossibilisticApproachesto ImpreciseValues 429
x APPROXIMATE REASONING AND INFORMATION SYSTEMS
7.3.2 The PossibilisticModelforRelational Databases 432
7.3.3 Queriesinthe PossibilisticFramework 435
7.4 Similarityin Relational Databases 440
7.4.1 Introduction 440
7.4.2 Similarity Thresholds and Relational Algebra
Operations 442
7.4.3 Redundancyand UniquenessProperties 443
7.4.4 Query Evaluation intheSimilarityBased Framework 446
7.4.5 Proximity-Based Models 447
7.4.6 ApplicationtoSecurity 448
7.5 DataIntegrityand Dependencies 452
7.6 OtherData Models 454
7.6.1 Fuzzy ExtensionsforEntity-RelationshipModels 454
7.6.2 ExtendedSemanticModel 455
7.6.3 NetworkDataModels 457
7.6.4 Object-Oriented Model 457
7.7 Conclusion 460
References 462
8
FuzzySetTechniquesinInformationRetrieval 469
DonaldH. Kraft, Gloria BordognaandGabriellaPas;
8.1 Introduction 469
8.2 Information Retrieval Models 472
8.3 FuzzyInformationRetrieval Models 474
8.3.1 Fuzzy ExtendedBoolean Models 475
8.3.2 Fuzzy KnowledgeBased IR Models 476
8.3.3 FuzzyAssociative Mechanisms 477
8.4 The FuzzyIndexingProcedure 477
8.4.1 Issuesin Indexingand Retrieval 477
8.4.2 Fuzzy RepresentationofStructuredDocuments 479
8.5 FuzzyQuerying 482
8.5.1 QueryWeights 483
8.5.2 AggregationOperators 487
8.6 Enhancing RetrievalThrough FuzzyAssociations 489
8.6.1 FuzzyThesauriforTerms 489
8.6.2 FuzzyClusteringforDocuments 492
8.7 Relevance Feedback 493
8.7.1 PreviousWorkson Relevance Feedback 494
8.7.2 Relevance FeedbackBasedon NaiveLearning 496
8.8 EvaluationofInformation RetrievalSystems 500
Summary 501
References 502