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.

I S B N 978-1-4613-6822-9 I S B N 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

A l l 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, photo­copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, L L C

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

Figure 1.1 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6

Figure 5.7 Figure 5.8 Figure 6.1 Figure 7.l Figure 7.2 Figure 8.1 Figure 8.2 Figure 8.3 Figure 8.4 Figure 8.5

LIST OF FIGURES

Nonna! fonns based FDs ............................................................ 16 ER diagram notations ................................................................ 20 A university ER diagram ............................................................ 21 A hierarchy of composite attribute address ................................. 24 Subclass and superclass .............................................................. 28 Shared subclass and category ..................................................... 30 A university EER diagram additional component. ...................... 31 Subclasses and the attributes of their own ................................... 32 The membership function for "Large" ........................................ 38 The membership function for "Young" ....................................... 38 The linguistic variable Age with values ...................................... 46 Linguistic hedges ....................................................................... 48 The inclusion-based closeness measure ...................................... 50 ~(A, B)=d and 1tIJf(A, B) = e ................................................. SO Fuzzy ER diagrammatic notations .............................................. 65 An entity type Company with a partial degree 0.9 ...................... 66 Diagrammatic notations for fuzzy participation constraints ........ 70 An example with a cardinality ratio n:M .................................... 70 Cardinality ratios with fuzziness ................................................ 71 The attribute-defined specialization with ai e Dom(A) ................ 82 The attribute-defined specialization with FSi e F(Dom(A» ......... 83 The attribute-defined generalization upon fuzzy values of age .... 84 Membership functions for "young", "mid-aged" and "old" .......... 85 A membership function for "about 55" ....................................... 85 The attribute-defined specialization/generalization at the L1(1.f) level. ............................................................................... 87 A fuzzy shared subclass E .......................................................... 88 A fuzzy category E ..................................................................... 89 Relationships among X, y and z ................................................ 101 M be hi fun · f" ""high" d " " 129 em rs p ctions 0 young, an average ......... . The spectrum of possible a-key values ...................................... 133 Two dependency paths from X to Y ......................................... 140 A dependency diagram (Example 8.1) ...................................... 142 A dependency diagram (Example 8.2) ...................................... 146 A dependency diagram (Example 8.3) ...................................... 147 A dependency diagram (Example 8.4) ...................................... 148

x List of Figures

Figure 9.1 Figure 9.2 Figure 9.3

A dependency diagram (Example 9.6 with F\) ......................... 171 A dependency diagram (Example 9.6 with F2) ••••••••••••••••••••••••• 173 Dependency diagrams (Example 9.7) ....................................... 174

Table 1.1 Table 1.2 Table 1.3 Table 1.4 Table 3.1 Table 3.2 Table 3.3 Table 4.1 Table 4.2 Table 4.3 Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 6.8 Table 6.9 Table 3.2 Table 7.1 Table 7.2 Table 8.1 Table 9.1 Table 9.2 Table 9.3 Table 9.4 Table 9.5 Table 9.6 Table 9.7 Table 9.8 Table 10.1 Table 10.2 Table 10.3 Table 10.4

LIST OF TABLES

A relation (table) R ........................................................................ 4 Relation Rl (customers' physical characteristics) ............................ 4

. Relation R2 (finished products) ...................................................... 5 Relation R3 (customers and products) ............................................ 5 The truth table for -+ ................................................................... 43 Fuzzy implication operators (FIOs) ............................................. 44 A fuzzy relation R (company's ordering information) .................. 53 A relationship matrix R on ExF .................................................. 73 A relationship matrix ~(2) on Ex2F ........................................... 74 A relationship matrix R( ) of 1: 1 cardinality ................................ 74 A closeness relation CHealth ........................................................... 98 A resemblance relation ReSj ....................................................... 107 A closeness relation Cj ............................................................... 112 A relation Itt (employees' performance) ..................................... 115 The tuple closeness Fe (Case 1) .................................................. 116 The tuple closeness Fc (Case 2) ................................................. 116 Closeness classes and equivalence classes .................................. 116 A closeness relation Cp. .............................................................. 117 The tuple closeness Fe (Case 3) .................................................. 117 Fuzzy implication operators (FIOs) ........................................... 124 FIOs versus C), C2, C3 ............................................................... 126 A relation C (customers) ........................................................... 129 A relation R with DomOC"F) and U-DomOr-F) ........................... 149 Relation R and two ofits projections Rl and R2 ......................... 161 Relation R reconstructed via join on close elements ................... 162 Relation R' and two of its projections R'l and R'2 ....................... 163 Relation R'l * R'2 that is not equal to the original R' .................. 163 Relation R" and two of its projections R"l and R"2 ..................... 163 Rill * R"2 = R" when B-+~ ...................................................... 164 Testing the lossless-join property (Example 9.3 with Fl) ............ 166 Testing the lossless-join property (Example 9.3 with F2) ............ 166 A non-FINF relation ................................................................. 184 A FINF relation ........................................................................ 184 Testing for lossless-join (Example 10.9) .................................... 196 Testing for 10ssless-join (Example 10.10) .................................. 198

PREFACE

Classical data models often suffer from their incapability of representing and manipulating uncertain and imprecise information. Since the early 1980's, Zadeh's fuzzy logic has been used to extend various data models, aimed at being able to deal with uncertainty and imprecision of a particular kind - fuzziness in concept, with which people usually think and reason in their decision-making and problem-solving processes. Primary attention has been paid to Codd's relational model, resulting in a number of fuzzy relational data models. Consequently, two important directions have emerged, namely, fuzzy queries and fuzzy data modeling. Fuzzy queries deal with the problems related to retrieving databases using linguistic variables and fuzzy predicates, while fuzzy data modeling focuses on the issues such as fuzzy data representation, fuzzy integrity 'constraints, fuzzy conceptual modeling, and fuzzy database design/

Like its classical counterpart, fuzzy data modeling addresses fundamental and important issues of fuzzy databases. Fuzzy data representation reflects how, where, and to what extent fuzziness is incorporated into classical models. Fuzzy integrity constraints are a sort of fuzziness-involved business rules and semantic restrictions that need to be specified and enforced. Fuzzy conceptual modeling is to describe and treat high-level data concepts and related seniantics in a fuzzy context, allowing the model to tolerate imprecision at different degrees. Fuzzy database design provides guidelines for how relation schemes of fuzzy databases should be fonned, and develops remedies to possible problems of data redundancy and update anomaly.

With in-depth discussions, this book, Fuzzy Logic in Data Modeling --­semantics, constraints, and database deSign, is exclusively devoted to fuzzy data modeling. The issues the book addresses are highly relevant to many fundamental concerns of both researchers and practitioners of (fuzzy/conventional) databases. The material in the book is the outgrowth of research the author has conducted in recent years while at Catholic University of Leuven (K.U.Leuven) in Belgium, and at Tsinghua University in China.

Organization of tbe Book

This book is organized into four parts. Part I descries the basic concepts necessary for a good understanding of data models and fuzzy logic. Chapter 1

xiv Preface

introduces Codd's relational data model, the relational algebra, and database design concepts. In Chapter 2, the Entity-Relationship (ER) model and the Enhanced Entity-Relationship (EER) model are presented, and used to illustrate the conceptual modeling concepts. Chapter 3 describes basic ideas of fuzzy logic and introduces some of the aspects relevant to fuzzy extension of data models.

Part II covers the topics of fuzzy conceptual modeling. Chapter 4 describes how fuzzy logic can be used to model ER concepts. A fuzzy ER model is presented at three levels in accordance with fuzzy types and occurrences of entities, attributes and relationships. It also provides an extensive look at relationships between entities in terms of participation constraints, cardinality constraints and relationship metrics. Chapter 5 presents fuzzy extensions to the enhanced ER concepts, such as superclass, subclass, generalization, specialization, shared subclass, and category, as well as the inheritance of attributes and relationships.

Part III contains discussions on the representation of fuzzy data and certain business rules. In Chapter 6, different frameworks of fuzzy data representation are described. Chapters 7 and 8 deal with fuzzy integrity constraints. Chapter 7 discusses semantic restrictions between attributes in terms of fuzzy functional dependencies (FFDs). A number of related notions and properties are covered, including fuzzy implication operators, extended Armstrong's axioms. a-keys concepts, and fuzzy extensions to entity and referential integrity rules. Chapter 8 formulates the FFD axiomatic system, develops a computational algorithm for FFD transitive closure, and provides a complete axiomatization of FFDs, which serves as a fundamental step towards the theory of fuzzy database design.

Part IV discusses the issues of fuzzy database design and information maintenance. Chapter 9 describes some extended relational algebraic operations in the light of fuzzy data manipulation, and shows that some update anomaly problems can be remedied by scheme decomposition. It further discusses the desirable properties of lossiess-join and dependency-preservation, as well as the corresponding testing algorithms. Finally, Chapter 10 introduces a number of fuzzy normal forms (FINF, F2NF, F3NF, FBCNF) along with some possible ways (algorithms) to obtain the normal forms. Fuzzy normal forms impose restrictions on the presence of partial and transitive FFDs in a scheme, such that the problems of data redundancy and update anomaly can be avoided.

Acknowledgements

It is a great pleasure for me to acknowledge a number of individuals and institutions who provided valuable supports for my research and for the completion of this book in many ways.

First of all, I would like to acknowledge Professor Etienne E. Kerre (University of Gent, Belgium) for his encouragement, expertise, and coorporation. It could be dated back to the late 1980's that I was encouraged and guided by Professor Kerre to pursue a deep exploration of fuzzy databases. I very much appreciated many discussions and exchanges of views with him during and thenafter my Ph.D. studies at KU. Leuven, Belgium, and have enjoyed close and fruitful collaborations with him for more than ten years, while in Belgium and U.S.A., as well as in China.

I am grateful for the supports and help that I received from my former colleagues at KU. Leuven. Special thanks should go to Professors Jacques Vandenbulcke, Maurice Verhelst and Jan Vanthienen. I am also indebted to Professor Philippe Smets (Universite Libre de Bruxelles, Belgium) for his comments and help at the early stage of my research on fuzzy databases. Many thanks go to Professor Janusz Kacprzyk (polish Academy of Sciences, Poland), and Professor Don Kraft (Louisiana State University, USA) for their kind help during the past years and their reviews of the outlines of this book.

In particular, I would like to express my sincere thanks to the A.B.O.S. of the Belgium government, the China's National Science Foundation, and the Flemish government for their financial supports in several research projects. Additionally, the assistance and facilities of the School of Economics and Management, Tsinghua University, are deemed important, and are highly appreciated.

Special thanks are also due to Scott Delman and his assistant Sharon Fletcher (Kluwer Academic Publishers) and Kai-Yuan Cai for their advice and help to prepare and publish this book, and to Qiang Wei (Tsinghua University) for his careful typesetting of this book.

Finally, this book will not be completed without the enduring support and understanding from my family.

Tsinghua University March 1998. Beijing

Guoqing Chen