FUZZY RELATION EQUATIONS AND THEIR APPLICATIONS TO KNOWLEDGE ENGINEERING

THEORY AND DECISION LIBRARY

General Editors: W. Leinfellner and G. Ebedein

Series A: Philosophy and Methodology of the Social Sciences Editors: W. Leinfellner (Technical Universtiy of Vienna) G. Eherlein (Technical University of Munich)

Series B: Mathematical and Statistical Methods Editor: H. Skala (University of Paderborn)

Series C: Game Theory, Mathematical Programming and Operations Research Editor: S. H. Tijs (University ofNijmegen)

Series D: System Theory, Knowledge Engineering and Problem Solving Editor: W. Janko (University of Economics, Vienna)

SERIES D: SYSTEM THEORY, KNOWLEDGE ENGINEERING AND PROBLEM SOLVING

Volume 3

Editor: W. Janko (Vienna)

Editorial Board

G. Feichtinger (Vienna), H. T. Nguyen (Las Cruces), N. B. Nicolau (Palma de Mallorca), O. Opitz (Augsburg), H. J. Skala (Paderborn), M. Sugeno (Yokohama).

Scope

This series focuses on the design and description of organisations and systems with application to the social sciences. Formal treatment of the subjects is encouraged. Systems theory, information systems, system analysis, interrelated structures, program systems and expert systems are considered to be a theme within the series. The fundamental basics of such concepts inc1uding computational and algorithmic aspects and the investigation of the empirical behaviour of systems and organisations will be an essential part of thls library. The study of problems related to the interface of systems and organisations to their environment is supported. Interdisciplinary considerations are welcome. The publication of recent and original results will be favoured.

For a list of titles published in this series, see final page.

FUZZY RELATION EQUATIONS AND

THEIR APPLICATIONS TO KNOWLEDGE ENGINEERING

by

ANTONIO DI NOLA and SALVATORE SESSA Universita di Napoli, Facolta di Architettura, Napoli, 1taly

WITOLD PEDRYCZ University 01 Manitoba,

Department 01 Electrical Engineering, Winnipeg, Canada

and

ELlE SANCHEZ Universite Aix-Marseille 11,

Faculte de Medecine, Marseille, France

Forewordby

LOTFI A. ZADEH University 01 California, Berkeley, U.S.A.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging in Publication Data

Fuzzy relation equations and their appl ications to knowledge engineering ; Antonio DI Nola ... let al.].

p. cm. -- (Theory and decision library. Serles D. System theory, knowledge engineering, and problem solvlng)

Inc 1 udes b i b 1 i ograph 1 es and i ndexes.

1. Expert systems (Computer sclence) 2. Fuzzy systems. Nola, Antonlo. II. Serles. QA76.76.E95F886 1989 006.3'3'01511322--dc20

ISBN 978-90-481-4050-3

printed on acid free paper

AII Rights Reserved

1. DI

89-34246

© 1989 Springer Science+Business Media Dordrecht Originally pub1ished by Kluwer Academic Publishers in 1989

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical

including photocopying, recording or by any information storage and retrieval system, without written permis sion from the copyright owner.

ISBN 978-90-481-4050-3 ISBN 978-94-017-1650-5 (eBook) DOI 10.1007/978-94-017-1650-5

Foreword

It took many decades for Peirce's coneept of a relation to find its way into the microelectronic innards of control systems of eement kilns, subway trains, and tunnel-digging machinery. But what is amazing is that the more we leam about the basically simple coneept of a relation, the more aware we become of its fundamental importanee and wide ranging ramifications. The work by Di Nola, Pedrycz, Sanchez, and Sessa takes us a long distanee in this direction by opening new vistas on both the theory and applications of fuzzy relations - relations which serve to model the imprecise coneepts which pervade the real world.

Di Nola, Pedrycz, Sanchez, and Sessa focus their attention on a eentral problem in the theory of fuzzy relations, namely the solution of fuzzy relational equations. The theory of such equations was initiated by Sanchez in 1976, ina seminal paper dealing with the resolution of composite fuzzy relational equations. Sinee then, hundreds of papers have been written on this and related topics, with major contributions originating in France, Italy, Spain, Germany, Poland, Japan, China, the Soviet Union, India, and other countries. The bibliography included in this volume highlights the widespread interest in the theory of fuzzy relational equations and the broad spectrum of its applications.

In the context of applications, the importance of the theory of fuzzy relational equations derives from the fact that human knowledge may be viewed as a collection of facts and rules, each of which may be represented as the assignment of a fuzzy relation to the unconditional or conditional possibility distribution of a variable. What this implies is that knowledge may be viewed as a system of fuzzy relational equations. In this perspective, then, inference from a body of knowledge reduces to the solution of a system of fuzzy relational equations. This basic idea underlies the theory of approximate reasoning based on fuzzy logic as well as various versions of fuzzy Prolog and, in particular, Professor Baldwin's language FRIL, which is a Prolog-based language for inferenee from fuzzy relations.

The work of Di Nola, Pedrycz, Sanchez, and Sessa has a dual pUl' pose: first, to present an authoritative and up-to-date account of the theory in a rigorous, thorough, and complete fashion; and second, to describe its applications, especially in the realm of knowledge-based systems.

The theoretical part addresses the major issues, among them: fuzzy relational equations in residuated lattiees, the lower solutions of max-min equations, the measures of fuzziness of solutions, the max-min decomposition, and fuzzy relational equations with triangular norms. In a transition to applications, the authors consider an issue which is of high intrinsic importanee, namely, the approximate solution of fuzzy relational equations. In this and other chapters, the authors make the reading easier for the non-mathematician by describing solution algorithms and applying them to well-chosen examples.

The second part, which deals with applications, develops a systematic approach to knowledge representation and inference based on the theory developed in the earlier chapters. In addition to the applications to knowledge-based systems, the authors present a lucid account of the basic ideas underlying the analysis and design of fuzzy logic controllers. Such controllers have proved

v

vi

to be highly successful in a variety of applications ranging from industrial process control and robotics to medical diagnosis and trafik control.

An important issue which is addressed in the chapters dealing with knowledge-based systems is that of the validation of production rules and the related problems of reduction and reconstruc­tion. In these chapters, there is a great deal that is new in the application of the theory of fuzzy relational equations to the problem of inference.

The authors of this volume have played a leading role in the development of the theory of fuzzy relational equations and its applications. Not surprisingly, the book reflects their high expertise and expository skills. Much of the material is new; the writing is lucid and well­motivated; and the references are a model of thoroughness and organization. Di Nola, Pedrycz, Sanchez, and Sessa deserve our thanks and congratulations for authoring an outstanding text which is certain to become an important landmark in the development of the theory of fuzzy sets and its applications.

Lotfi A. Zadeh Berkeley, California

Table of Contents

Preface

Acknow1edgements

List of Abbreviations

CHAPTER 1: INTRODUCTORY REMARKS ON FUZZY SETS

1. 1. Remarks on Fuzzy Sets. 1.2. Some Usefu1 Bib1iographies.

References.

CHAPTER 2: FUZZY RELATION EQUATIONS IN RESIDUATED LATTICES

2.1. Basic Definitions. 2.2. Max-0 Composite Fuzzy Equations. 2.3. Another Characterization Theorem. 2.4. Some Theoretical Results. 2.5. Dual Composite Fuzzy Equations.

References.

xi

xii

xiii

1

1 4 5

7

7 10 14 17 21 22

CHAPTER 3: LOWER SOLUTIONS OF MAX-MIN FUZZY EQUATIONS 23

3.1. Lower Solutions of Eq.(2.5) in Linear Lattices. 3.2. Lower So1utions of Eq.(2.6) in Linear Lattices. 3.3. Further Lattice Results in 8. 3.4. Further Properties of the Fuzzy Relation L. 3.5. On Lower Solutions ofEq.(3.3) in CompIete Brouwerian Lattices. 3.6. On Lower Solutions of Eq.(3.3) in Complete CompIetely

Distributive Lattices. 3.7. Concluding Comments.

References.

CHAPTER 4: MEASURES OF FUZZINESS OF SOLUTIONS OF MAX-MIN

23 29 33 35 40

42 46 46

FUZZY RELATION EQUATIONS ON LINEAR LATTICES 49

4. 1. Basic Preliminaries. 4.2. Two Optimization Problems. 4.3. Some Measures of Fuzziness. 4.4. Further Lattice Results. 4.5. Concluding Remarks.

References.

49 51 52 54 54 55

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CHAPTER 5: BOOLEAN SOLUTIONS OF MAX-MIN FUZZY EQUATIONS 59

5.1. A Fundamental Theorem. 5.2. Minimal Boo1ean Solutions. 5.3. Solutions with Maximum Boolean Degree.

References.

CHAPTER 6: a-FUZZY RELATION EQUATIONS AND DECOMPOSABLE

59 61 64 69

FUZZY RELATIONS 71

6.1. a-Fuzzy Relation Equations on Complete Brouwerian Lattices. 6.2. Re1ated Results on Linear Lattices. 6.3. Composite a-Fuzzy Relation Equations. 6.4. Fuzzy Relations Decomposable into Two Fuzzy Sets. 6.5. Decomposable Fuzzy Relations and the Transitivity Property. 6.6. Convergence of Powers of Decomposable Fuzzy Relations.

References.

CHAPTER 7: MAX-MIN DECOMPOSITION PROBLEM OF A FUZZY RELATION IN LINEAR LATTICES

7. 1. Preliminaries. 7.2. Solution of the Decomposition Problem. 7.3. Two N umerica1 Examples. 7.4. Max-min Decomposition of Transitive Fuzzy Relations.

References.

CHAPTER 8: FUZZY RELATION EQUATIONS WITH LOWER AND UPPER SEMICONTINUOUS TRIANGULAR NORMS

71 72 73 74 77 79 83

87

87 92 92 99

103

105

8.1. Triangular Norms as Logica1 Connectives in Fuzzy Set Theory. 105 8.2. Fuzzy Relation Equations with Lower Semicontinuous Triangular Norms

and Upper Semicontinuous Conorms. 108 8.3. Fuzzy Relation Equations of Comp1ex Structure. 117 8.4. A Re1ated Adjoint Fuzzy Relation Equation. 119 8.5. Fuzzy Relation Equations with Upper Semicontinuous Triangular Norms. 121

References. 126

CHAPTER 9: FUZZY RELATION EQUATIONS WITH EQUALITY AND DIFFERENCE COMPOSITION OPERATORS

9.1. Equality and Difference Operators. 9.2. Basic Lemmas. 9.3. Resolution of the Fuzzy Equation (9.4). 9.4. Resolution of the Fuzzy Equation (9.4'). 9.5. Resolution of Fuzzy Equations with Difference Operator. 9.6. Concluding Remarks.

References.

129

129 131 133 138 142 144 144

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CHAPTER 10: APPROXIMATE SOLUTIONSOF FUZZY RELATION EQUATIONS 145

10.1. Preliminaries. 145 10.2. On Expressing a Property of Solvability ofFuzzy Relation Equations. 146 10.3. On Solving Fuzzr Relation Equations via Modification ofFuzzy Sets:

a Role ofy-Leve Fuzzy Sets. 150 10.4. Hierarchy in the Data Set 153

1. Preliminaries. Determination of the Hierarchy in the Families of Fuzzy Sets. 154

2. On Measuring the Representation Power of the Fuzzy Data. 162 10.5. On Solving a System ofMax-min Fuzzy Relation Equations by

Gathering Statistics of Partial Solutions. 165 10.6. On Solving a Max-min Fuzzy Relation Equation. 170 10.7. Concluding Remarks. 173

References. 174

CHAPTER 11: HANDLING FUZZINESS IN KNOWLEDGE-BASED SYSTEMS 175

11.1. Handling of Factors of Uncertainty in Knowledge-Based Systems: Approaches and Requirements. 176

11.2. Selected Aspects of Reasoning with Fuzziness. 178 11.3. Principle of Consistency of Aggregation and Inference Mechanisms. 179 11.4. Information Granularity and Fuzzy Sets. 181 11.5. Conclusions. 183

References. 184

CHAPTER 12: CONSTRUCTION OF KNOWLEDGE BASE, ITS VALIDATION AND OPTIMIZATION 187

12.1. Preliminaries. 187 12.2. Validation of Production Rules. 189 12.3. Distributed Knowledge Bases. 193 12.4. Reduction Problem in Knowledge Bases. 194 12.5. Reconstruction Problem of Knowledge Bases. 200 12.6. Concluding Remarks. 202

References. 203

CHAPTER 13: INFERENCE ALGORITHMS IN KNOWLEDGE-BASED SYSTEMS 205

13.1. Preliminaries. 205 13.2. Representation ofModus Ponens with Fuzzy Premises. 208 13.3. Expressing the Scheme of Modus Ponens in Terms of Fuzzy Relation

Equations. 212 13.4. Backward Chaining. 214 13.5. Modus Ponens Performed in the Framework ofFuzzy Logic. 216 13.6. Detachment Principle and Its Implementation with the Aid ofFuzzy

x

Equations. References.

CHAPTER 14: A FUZZY CONTROLLER AND ITS REALIZATION

14.1. Theoretical Preliminaries. 14.2. Schemes ofInference with Fuzzy Information. 14.3. Realization of the Fuzzy Controller and an Applicational Example. 14.4. Concluding Remarks.

References.

CHAPTER 15: BIBLIOGRAPHIES

15.1. A List of Papers on Fuzzy Relation Equations and Related Topics. 15.2. A List of Papers on Fuzzy Relations and Related Topics.

Author Index

Subject Index

219 222

225

225 227 230 233 240

241

241 253

263

270

Preface

The growing interest to knowledge engineering has forced the buming need to search novel tools which mimic human processes of perception, decision-making, object recognition, etc.

Fuzzy set theory is an approach studied extensively in this area. On the other hand, the notion of fuzzy set, i.e. a set with not sharply defined boundaries, is very natural for a human being. Moreover, this approach is very convenient since the user-friendly man­system communication is performed in a linguistic (not numerical) fashion.

The aim of this monograph is to provide the reader with fundamentals of fuzzy relations indicating clearly their applications to knowledge engineering as, e.g., verification of a knowledge base (in the sense of its consistency and re1evancy), designing of inference mechanisms, reduction of a knowledge base, propagation of uncertainty in different reasoning schemes.

This book is organized in 15 Chapters. Chapters 1+9 contain theoretical backgrounds of the theory of fuzzy relation equations. In these Chapters, the fuzzy sets are defined and studied on several types of lattice: of course, this in the spirit of the symbolic computations, characteristic for Artificial Intelligence. In Chapters 1ü+ 14, containing the above mentioned applications to know1edge engineering, the fuzzy sets are expressed in the real unit interval in order to process the vague information coming from the expert.

Chapter 15 contains two useful and extensive bibliographies of papers on fuzzy equations, fuzzy relations and related topics. Each Chapter has its appropriate references.

The following diagram illustrates some main routes for studying the Chapters of this book. Of course, the reader should consider this diagram only as indicative.

7

6'\t/9 t-t_3 _ 4 _ 5

8

~ 10

~ 124--11----..14

~ 13

xi

Acknowledgements

Thanks are due to N. Blanchard, Z.Q. Cao, D. Dubois, S. Gottwald, H. Hashimoto, M. Higashi, G. Hirsch, K.H. Kim, G.J. Klir, W. Kolodziejczyk, V. Novak, A. Ohsato, H. Prade, R.M. Tong, P.Z. Wang, D. Willaeys, R.R. Yager jor providing preprints and reprints ojtheir works and S.Z.Guojor providing a list ojChinese papers onjuzzy relation equations, partially contained in the bibliographies oj Ch.I5.

xii

List of Abbreviations

Ch(s). Chapter(s)

Corol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corollary

Def(s). .............. ... .............. ... ... ....... ..... ...... ...... .... Defmition(s)

Eq(s). .................................................................. Equation(s)

Ex(s). .................................................................. Exarnple(s)

Fig(s). .................................................................. Figure(s)

KB . ... . .. . .. . . . . .. . .. ... . .. . . . . .. . .. . .. . .. ... ... ....... ... .. . ... .. .. Knowledge Base

Prop(s). ................................................................ Proposition(s)

Sec(s). .................................................................. Section(s)

Thm(s). . ................................................................ Theorem(s)

[x, Ch. y] . . .. . . . . .. . .. . .. . .. . .. . . . . . . . .. . .. .... .. . .. . .. .... . . .... .. ... . . Reference x of Chapter y

xiii