Fuzzy Logic in Data Modeling: Semantics, Constraints, and Database Design

222 Pages·1998·11.518 MB·English

by  Guoqing Chen (auth.)

FUZZY LOGIC IN
DATA MODELING
Semantics, Constraints, and
Database Design
The Kluwer International Series on
ADVANCES IN DATABASE SYSTEMS
Series Editor
Ahmed K. Elmagarmid
Purdue University
West Lafayette, IN 47907
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FUZZY LOGIC IN
DATA MODELING
Semantics, Constraints, and
Database Design
Guoqing Chen
School of Economics and Management
Tsinghua University
Beijing, China
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Electronic Services <http://www.wkap.nl>
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is available
from the Library of Congress.
ISBN 978-1-4613-6822-9 ISBN 978-1-4615-4068-7 (eBook)
DOI 10.1007/978-1-4615-4068-7
© Springer Science+Business Media New York 1998
Originally published by Kluwer Academic Publishers 1998
Softcover reprint of the hardcover 1st edition 1998
All 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 on acid-free paper.
Table o/Contents v
TABLE OF CONTENTS
PREFACE ...........................................•.••..........•........••................................. Dii
ACKN'OWLEDGEmNTs ................................•••.........•............•..•••••....•..... xv
PART I BASIC CONCEPTS .......................................................................... 1
CHAPTER 1 THE RELATIONAL DATA MODEL ..................................... 3
1.1. The Relational Model Concepts .......................................................... 3
1.1.1. Relations and the Underlying Asswnptions ............................................. 3
1.1.2. Data Constraints ..................................................................................... 6
1.2. The Relational Algebra ...................................................................... 9
1.3. Relational Database Design .............................................................. 12
References ............................................................................................... 17
CHAPTER 2 CONCEPTUAL MODELING WITH THE
ENTITY-RELATIONSIDP MODEL ..................................... 19
2.1. ER Diagrammatic Notations ............................................................. 19
2.2. The ER Model Concepts ................................................................... 22
2.2.1. Entities ................................................................................................. 22
2.2.2. Attributes ............................................................................................. 22
2.2.3. Relationships ........................................................................................ 25
2.3. Enhanced ER (EER) Model Concepts ............................................... 27
References ............................................................................................... 33
CHAPTER 3 FUZZY LOGIC ..................................................................... 35
3.1. Uncertainty and Imprecision ............................................................ 35
3.2. Fuzzy Sets and Possibility Distributions ........................................... 37
3.2.1. Support, Kernel, «-Cut, Height and Plinth of a Fuzzy Set ...................... 40
3.2.2. Some Max-Min Operations on Fuzzy Sets ............................................. 40
3.2.3. Zadeh's Extension Principle .................................................................. 42
3.2.4. Fuzzy IInplication Operators .................................................................. 43
3.3. Linguistic Variable ........................................................................... 45
3.4. Closeness Measures Between Fuzzy Sets .......................................... 49
3.5. Fuzzy Relations ................................................................................ 53
References ................................................................................................ 57
PART n FUZZY CONCEPTUAL MODELING ......................................... 59
vi Table o/Contents
CHAPTER 4 FUZZY ER CONCEPTS ....................................................... 61
4.1. Levels of Concepts ........................................................................... 61
4.2. Fuzzy Entities, Relationships and Attributes ..................................... 64
4.3. Relationships and Constraints .......................................................... 69
4.4. Fuzzy ER Manipulation ................................................................... 75
References ............................................................................................... 76
CHAPTER 5 FUZZY EER CONCEPTS ..................................................... 79
5.1. Fuzzy Subclass and Superclass ......................................................... 79
5.2. Specialization and Generalization with Fuzziness ............................ 81
5.3. Fuzzy Shared Subclass and Category ................................................ 87
5.4 Inheritance of Relationships and Attributes ...................................... 90
References ............................................................................................... 92
PART m REPRESENTATION OF FUZZY DATA
AND CONSTRAIN'TS •••••••••••••••••••••••••••••.•.••••••••••..••••.•••••••••••••••• 9S
CHAPTER 6 FUZZY DATA REPRESENTATION .................................... 97
6.1. Data Representation Frameworks ..................................................... 98
6.1.1. The Fuzzy-relation-based Framework .................................................... 98
6.1.2. The Similarity-based Framework .......................................................... 99
6.1.3. The Possibility-based Framework ........................................................ 100
6.1.4. The Extended Possibility-based Framework ........................................ 100
6.2. Fuzzy Data Closeness and Redundancy .......................................... 102
6.2.1. The ProbleJJl ....................................................................................... 102
6.2.2. Some Existing Treabnents .................................................................. 103
6.3. The CVK Treatment ...................................................................... 108
6.3.1. The KS Treabnent ofTuple Equality ................................................... 109
6.3.2. The Extension to the KS Treabnent.. ................................................... 110
6.3.3. More Discussions On the CVK Treabnent ........................................... 113
References ............................................................................................. 117
CHAPTER 7 FUZZY FUNCTIONAL DEPENDENCIES (FFDs)
AS INTEGRITY CONSTRAINTS ....................................... 119
7.1. A General Fonn ofFFDs ................................................................ 120
7.2. FFD Inference Rules ...................................................................... 122
7.3. Fuzzy Implication Operators versus the Properties Ct,~, C3 ......... 124
7.4. Semantics Represented by Specific Forms ofFFDs ......................... 127
7.5. Extended Keys and Integrity Rules ................................................. 130
References ............................................................................................. 134
Table o/Contents vii
CHAPTER 8 A FFD INFERENCE SYSTEM ........................................... 135
8.1. Inference Rules in the FFD Axiomatic System ................................ 136
8.2. Transitive Closure and a Computational Algorithm ........................ 138
8.3. Soundness and Completeness of the Axiomatic System .................. 148
8.4. Equivalence of the Dependency Sets ............................................... 150
References ............................................................................................. 154
PART IV FUZZY DATABASE DESIGN AND
INFORMATION MAIN'TENANCE ....••.....•...•••........................ 15S
CHAPTER 9 SCHEME DECOMPOSITION AND INFORMATION
MAINTENANCE ................................................................. 157
9.1. Fuzzy Data Manipulation ............................................................... 158
9.2. loin and Projection on Base Relations ............................................ 160
9.3. Lossless-loin Decomposition .......................................................... 162
9.4. Dependency-Preserving Decomposition .......................................... 167
References ............................................................................................. 176
CHAPTER 10 DESIGN OF FUZZY DATABASES TO AVOID
UPDATE ANOMALIES ................................................... 179
10.1. The Update Anomaly Problems .................................................... ISO
10.2. Use of Fuzzy Normal Forms to Deal with Update Anomalies ........ 182
10.2.1. Fuzzy First Nonnal Fonn (FINF). ..................................................... 183
10.2.2. 9-Fuzzy Nonnal Fonns ...................................................................... 185
10.3. Design Algorithm and Information Maintenance .......................... 189
10.3.1. Dependency-Preserving Decomposition into Fuzzy
Third Nonnal Fonns ........................................................................ 190
10.3.2. Dependency-Preserving and Lossless-loin Decomposition
into Fuzzy Third Nonnal Fonns ........................................................ 194
10.3.3. Lossless-loin Decomposition into Fuzzy Boyce-Codd
Nonnal Fonns .................................................................................. 196
References ............................................................................................. 199
BmLIOGRAPBY ....................................................................................... 201
APPENDIX .......•.....•.•.•.•.•...•..............••.•.....•••..•..•.............................•...•...... 207
A. List of Examples .............................................................................. 209
B. List of Definitions ............................................................................. 211
C. List of Theorems .............................................................................. 213
D. List of Lemmas ................................................................................ 215
E. List of Algorithms ............................................................................ 217
IN'DEX •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••.••••••••••••••••••••••••••••••••••••••••••• 119
LIST OF FIGURES
Figure 1.1 Nonna! fonns based FDs ............................................................ 16
Figure 2.1 ER diagram notations ................................................................ 20
Figure 2.2 A university ER diagram ............................................................ 21
Figure 2.3 A hierarchy of composite attribute address ................................. 24
Figure 2.4 Subclass and superclass .............................................................. 28
Figure 2.5 Shared subclass and category ..................................................... 30
Figure 2.6 A university EER diagram additional component. ...................... 31
Figure 2.7 Subclasses and the attributes of their own ................................... 32
Figure 3.1 The membership function for "Large" ........................................ 38
Figure 3.2 The membership function for "Young" ....................................... 38
Figure 3.3 The linguistic variable Age with values ...................................... 46
Figure 3.4 Linguistic hedges ....................................................................... 48
Figure 3.5 The inclusion-based closeness measure ...................................... 50
Figure 3.6 ~(A, B)=d and 1tIJf(A, B) = e ................................................. SO
Figure 4.1 Fuzzy ER diagrammatic notations .............................................. 65
Figure 4.2 An entity type Company with a partial degree 0.9 ...................... 66
Figure 4.3 Diagrammatic notations for fuzzy participation constraints ........ 70
Figure 4.4 An example with a cardinality ratio n:M .................................... 70
Figure 4.5 Cardinality ratios with fuzziness ................................................ 71
Figure 5.1 The attribute-defined specialization with ai e Dom(A) ................ 82
Figure 5.2 The attribute-defined specialization with FSi e F(Dom(A» ......... 83
Figure 5.3 The attribute-defined generalization upon fuzzy values of age .... 84
Figure 5.4 Membership functions for "young", "mid-aged" and "old" .......... 85
Figure 5.5 A membership function for "about 55" ....................................... 85
Figure 5.6 The attribute-defined specialization/generalization at the
L1(1.f) level. ............................................................................... 87
Figure 5.7 A fuzzy shared subclass E .......................................................... 88
Figure 5.8 A fuzzy category E ..................................................................... 89
Figure 6.1 Relationships among X, y and z ................................................ 101
Figure 7.l Me mb e rs hi p functi·o ns 0 f"y oung"," high" and "a verage ". ........ . 129
Figure 7.2 The spectrum of possible a-key values ...................................... 133
Figure 8.1 Two dependency paths from X to Y ......................................... 140
Figure 8.2 A dependency diagram (Example 8.1) ...................................... 142
Figure 8.3 A dependency diagram (Example 8.2) ...................................... 146
Figure 8.4 A dependency diagram (Example 8.3) ...................................... 147
Figure 8.5 A dependency diagram (Example 8.4) ...................................... 148
x List ofF igures
Figure 9.1 A dependency diagram (Example 9.6 with F\) ......................... 171
Figure 9.2 A dependency diagram (Example 9.6 with F2) ••••••••••••••••••••••••• 173
Figure 9.3 Dependency diagrams (Example 9.7) ....................................... 174
LIST OF TABLES
Table 1.1 A relation (table) R. ....................................................................... 4
Table 1.2 Relation Rl (customers' physical characteristics) ............................ 4
Table 1.3 . Relation R2 (finished products) ...................................................... 5
Table 1.4 Relation R3 (customers and products) ............................................ 5
Table 3.1 The truth table for -+ ................................................................... 43
Table 3.2 Fuzzy implication operators (FIOs) ............................................. 44
Table 3.3 A fuzzy relation R (company's ordering information) .................. 53
Table 4.1 A relationship matrix R on ExF .................................................. 73
Table 4.2 A relationship matrix ~(2) on Ex2F ........................................... 74
Table 4.3 A relationship matrix R( ) of 1: 1 cardinality ................................ 74
Table 6.1 A closeness relation CHealth ........................................................... 98
Table 6.2 A resemblance relation ReSj ....................................................... 107
Table 6.3 A closeness relation Cj. .............................................................. 112
Table 6.4 A relation Itt (employees' performance) ..................................... 115
Table 6.5 The tuple closeness Fe (Case 1) .................................................. 116
Table 6.6 The tuple closeness Fc (Case 2) ................................................. 116
Table 6.7 Closeness classes and equivalence classes .................................. 116
Table 6.8 A closeness relation Cp. .............................................................. 117
Table 6.9 The tuple closeness Fe (Case 3) .................................................. 117
Table 3.2 Fuzzy implication operators (FIOs) ........................................... 124
Table 7.1 FIOs versus C), C2, C3. .............................................................. 126
Table 7.2 A relation C (customers) ........................................................... 129
Table 8.1 A relation R with DomOC"F) and U-DomOr-F) ........................... 149
Table 9.1 Relation R and two ofits projections Rl and R2 ......................... 161
Table 9.2 Relation R reconstructed via join on close elements ................... 162
Table 9.3 Relation R' and two of its projections R'l and R'2 ....................... 163
Table 9.4 Relation R'l * R'2 that is not equal to the original R' .................. 163
Table 9.5 Relation R" and two of its projections R"l and R"2 ..................... 163
Table 9.6 Rill * R"2 = R" when B-+~ ...................................................... 164
Table 9.7 Testing the lossless-join property (Example 9.3 with Fl). ........... 166
Table 9.8 Testing the lossless-join property (Example 9.3 with F2). ........... 166
Table 10.1 A non-FINF relation ................................................................. 184
Table 10.2 A FINF relation ........................................................................ 184
Table 10.3 Testing for lossless-join (Example 10.9) .................................... 196
Table 10.4 Testing for 10ssless-join (Example 10.10) .................................. 198