The manipulation of databases is an integral part of a world which is becoming increasingly and pervasively information-focused. This book puts forward a suggestion to advocate preference queries and fuzzy sets as a central concern in database queries and offers an important contribution to the design of intelligent information systems. It provides a comprehensive study on fuzzy preference queries in the context of relational databases. Preference queries, a recent hot topic in database research, provide a basis for rank-ordering the items retrieved, which is especially valuable for large sets of answers.
This book aims to show that fuzzy set theory constitutes a highly expressive framework for modeling preference queries. It presents a study of the algorithmic aspects related to the evaluation of such queries in order to demonstrate that this framework offers a good trade-off between expressivity and efficiency. Numerous examples and proofs are liberally and lucidly demonstrated throughout, and greatly enhance the detailed theoretical aspects explored in the book.
Researchers working in databases will greatly benefit from this comprehensive and up-to-date study of fuzzy preference queries, and it will also become an invaluable reference point for postgraduate students interested in advanced database techniques." Reminders on Relational Databases; Basic Notions on Fuzzy Sets; An Overview of Non-Fuzzy Approaches to Preference Queries; Simple Fuzzy Queries; Fuzzy Queries Involving Quantified Statements and Aggregates; Division and Antidivision of Fuzzy Relations; Bipolar Fuzzy Queries; Fuzzy Group-By; Empty and Plethoric Answers.
Foreword vii
Acknowledgments ix
1 Introduction 1
1.1 Databases and their Evolution 1
1.2 Preferences and Fuzzy Sets 2
1.3 Overview of the Book 3
2 Reminders on Relational Databases 5
2.1 Basic Notions and Vocabulary 5
2.2 Algebraic Operations 8
2.2.1 Set operations 8
2.2.2 Relational operations 11
2.2.3 Properties 20
2.3 An Overview of SQL 21
2.3.1 The base block 22
2.3.2 Combining base blocks 25
2.3.3 Partitioning 27
2.3.4 Expressing division and antidivision 29
3 Basic Notions on Fuzzy Sets 31
3.1 Introduction 31
3.2 Definitions and Notations 33
3.3 Composition of Fuzzy Sets 36
3.3.1 Intersection and union of fuzzy sets 36
3.3.2 Difference between fuzzy sets 41
3.3.3 Cartesian product of fuzzy sets 43
3.3.4 Trade-off operators 44
3.3.5 Nonsymmetric operators 45
3.4 Inclusions and Implications 50
3.4.1 Fuzzy implications 50
3.4.2 Inclusions 58
3.5 Fuzzy Measures and Integrals 68
3.5.1 Introduction 68
3.5.2 Fuzzy measures 68
3.5.3 Fuzzy integrals 70
3.6 The Extension Principle 71
3.7 Fuzzy Quantified Propositions 73
3.7.1 Fuzzy linguistic quantifiers 73
3.7.2 Quantified propositions 73
4 Non-Fuzzy Approaches to Preference Queries: A Brief Overview 77
4.1 Introduction 77
4.2 Quantitative Approaches 78
4.2.1 Distances and similarity 78
4.2.2 Linguistic preferences 79
4.2.3 Explicit scores attached to entities 80
4.2.4 Top-k queries 81
4.2.5 Outranking 81
4.3 Qualitative Approaches 82
4.3.1 Secondary preference criterion 82
4.3.2 Pareto-order-based approaches 83
4.3.3 CP-nets 85
4.3.4 Domain linearization 87
4.3.5 Possibilistic-logic-based approach 88
4.4 Conclusion 89
5 Simple Fuzzy Queries 91
5.1 Introduction 91
5.2 An Extended Relational Algebra 93
5.3 An Overview of a Basic Version of SQLf 94
5.3.1 Introduction 94
5.3.2 The multiple relation base block 95
5.3.3 Subqueries 96
5.3.4 Set-oriented operators 102
5.3.5 Relation partitioning 104
5.4 Interface for User-Defined Terms and Operators 106
5.5 Contextual Queries 108
5.5.1 Queries with one level of context 110
5.5.2 Queries with several levels of context 113
5.6 Evaluation of Simple Fuzzy Queries 114
5.6.1 Derivation principle 114
5.6.2 Derivation-based processing of SQLf queries 121
5.7 Conclusion 128
6 Fuzzy Queries Involving Quantified Statements or Aggregates 131
6.1 Introduction 131
6.2 Quantified Statements 132
6.2.1 Introduction 132
6.2.2 Quantified statements and fuzzy integral theory 133
6.2.3 Interpretation of statements of the type "Q X are A" 138
6.2.4 Integration into SQLf 149
6.2.5 Evaluation of SQLf queries involving quantified statements 152
6.3 Aggregates 160
6.3.1 Introduction 160
6.3.2 The case of monotonic predicates and aggregates 161
6.3.3 Dealing with the general case 164
6.3.4 SQLf queries involving aggregates 169
6.3.5 Evaluation of SQLf queries involving aggregates 176
6.4 Conclusion 182
7 Division and Antidivision of Fuzzy Relations 185
7.1 Introduction 185
7.2 Division of Fuzzy Relations 186
7.2.1 Principles 186
7.2.2 On the choice of implication 189
7.2.3 Primitivity of the extended division operator 190
7.2.4 Expressing extended division in SQLf 192
7.3 Tolerant Division 193
7.3.1 Exception-based tolerant division 193
7.3.2 Resemblance-based tolerant division 196
7.4 Stratified Division 198
7.4.1 Introduction 198
7.4.2 The queries 200
7.4.3 Quotient property of the result delivered 205
7.5 Queries Mixing Division and Antidivision 207
7.5.1 Motivation 207
7.5.2 Mixed stratified queries 208
7.6 Evaluation of Division Queries 211
7.6.1 Processing the division of fuzzy relations 211
7.6.2 Processing the tolerant divisions of fuzzy relations 213
7.6.3 Processing the conjunctive stratified division 215
7.7 Conclusion 219
8 Bipolar Fuzzy Queries 221
8.1 Introduction 221
8.2 Preliminaries 222
8.2.1 About bipolarity 222
8.3 Extended Algebraic Operators 224
8.3.1 Intersection 224
8.3.2 Union 225
8.3.3 Cartesian product 226
8.3.4 Negation 226
8.3.5 Difference 232
8.3.6 Selection 235
8.3.7 Projection 237
8.3.8 Join 238
8.3.9 Division 239
8.4 Implementation Aspects 246
8.5 Conclusion 248
9 Fuzzy Group By 251
9.1 Introduction 251
9.2 An Extended Group By Clause 252
9.2.1 Use of a crisp partition 252
9.2.2 Use of a fuzzy partition 253
9.3 Having Clause 255
9.3.1 Inclusion constraint 256
9.3.2 Aggregate1 θ aggregate2 256
9.3.3 Aggregate is ψ 257
9.4 Application to Association Rule Mining 258
9.4.1 Rules of the type A is Li -> B is L 259
9.4.2 Rules of the type A is L -> B is Li 261
9.5 Evaluation of a Fuzzy Group By 262
9.6 Related Work 262
9.6.1 Extended group by 262
9.6.2 Fuzzy OLAP 263
9.6.3 Fuzzy database summarization techniques 263
9.6.4 Mining association rules with SQL 264
9.7 Conclusion 264
10 Empty and Plethoric Answers 267
10.1 Introduction 267
10.2 Empty Answer Problem 268
10.2.1 Query relaxation 268
10.2.2 Relaxation by predicate weakening 269
10.2.3 Case-based reasoning approach 279
10.3 Plethoric Answer Problem 287
10.3.1 Introduction 287
10.3.2 Approach based on predicate strengthening 288
10.3.3 Approach based on query expansion 294
10.4 Conclusion 306
11 Conclusion 309
Bibliography 313
Index 327
Overview
The manipulation of databases is an integral part of a world which is becoming increasingly and pervasively information-focused. This book puts forward a suggestion to advocate preference queries and fuzzy sets as a central concern in database queries and offers an important contribution to the design of intelligent information systems. It provides a comprehensive study on fuzzy preference queries in the context of relational databases. Preference queries, a recent hot topic in database research, provide a basis ...