FUZZY RELATION EQUATIONS AND THEIR APPLICATIONS TO KNOWLEDGE ENGINEERING

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1 FUZZY RELATION EQUATIONS AND THEIR APPLICATIONS TO KNOWLEDGE ENGINEERING

2 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.

3 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.

4 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. ISBN ISBN (ebook) DOI / Expert systems (Computer sclence) 2. Fuzzy systems. 1. DI Nola, Antonlo. II. Serles. QA76.76.E95F '3' dc ISBN printed on acid free paper AII Rights Reserved 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.

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

6 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

7 Table of Contents Preface Acknow1edgements List of Abbreviations xi xii xiii CHAPTER 1: INTRODUCTORY REMARKS ON FUZZY SETS Remarks on Fuzzy Sets Some Usefu1 Bib1iographies CHAPTER 2: FUZZY RELATION EQUATIONS IN RESIDUATED LATTICES Basic Definitions Max-0 Composite Fuzzy Equations Another Characterization Theorem Some Theoretical Results Dual Composite Fuzzy Equations CHAPTER 3: LOWER SOLUTIONS OF MAX-MIN FUZZY EQUATIONS Lower Solutions of Eq.(2.5) in Linear Lattices Lower So1utions of Eq.(2.6) in Linear Lattices Further Lattice Results in Further Properties of the Fuzzy Relation L On Lower Solutions ofeq.(3.3) in CompIete Brouwerian Lattices On Lower Solutions of Eq.(3.3) in Complete CompIetely Distributive Lattices Concluding Comments CHAPTER 4: MEASURES OF FUZZINESS OF SOLUTIONS OF MAX-MIN FUZZY RELATION EQUATIONS ON LINEAR LATTICES Basic Preliminaries Two Optimization Problems Some Measures of Fuzziness Further Lattice Results Concluding Remarks

8 viii CHAPTER 5: BOOLEAN SOLUTIONS OF MAX-MIN FUZZY EQUATIONS A Fundamental Theorem Minimal Boo1ean Solutions Solutions with Maximum Boolean Degree CHAPTER 6: a-fuzzy RELATION EQUATIONS AND DECOMPOSABLE FUZZY RELATIONS a-fuzzy Relation Equations on Complete Brouwerian Lattices Re1ated Results on Linear Lattices Composite a-fuzzy Relation Equations Fuzzy Relations Decomposable into Two Fuzzy Sets Decomposable Fuzzy Relations and the Transitivity Property Convergence of Powers of Decomposable Fuzzy Relations CHAPTER 7: MAX-MIN DECOMPOSITION PROBLEM OF A FUZZY RELATION IN LINEAR LATTICES Preliminaries Solution of the Decomposition Problem Two N umerica1 Examples Max-min Decomposition of Transitive Fuzzy Relations CHAPTER 8: FUZZY RELATION EQUATIONS WITH LOWER AND UPPER SEMICONTINUOUS TRIANGULAR NORMS Triangular Norms as Logica1 Connectives in Fuzzy Set Theory Fuzzy Relation Equations with Lower Semicontinuous Triangular Norms and Upper Semicontinuous Conorms Fuzzy Relation Equations of Comp1ex Structure A Re1ated Adjoint Fuzzy Relation Equation Fuzzy Relation Equations with Upper Semicontinuous Triangular Norms CHAPTER 9: FUZZY RELATION EQUATIONS WITH EQUALITY AND DIFFERENCE COMPOSITION OPERATORS Equality and Difference Operators Basic Lemmas Resolution of the Fuzzy Equation (9.4) Resolution of the Fuzzy Equation (9.4') Resolution of Fuzzy Equations with Difference Operator Concluding Remarks

9 CHAPTER 10: APPROXIMATE SOLUTIONSOF FUZZY RELATION EQUATIONS Preliminaries On Expressing a Property of Solvability offuzzy Relation Equations On Solving Fuzzr Relation Equations via Modification offuzzy Sets: a Role ofy-leve Fuzzy Sets Hierarchy in the Data Set 1. Preliminaries. Determination of the Hierarchy in the Families of 153 Fuzzy Sets On Measuring the Representation Power of the Fuzzy Data On Solving a System ofmax-min Fuzzy Relation Equations by Gathering Statistics of Partial Solutions On Solving a Max-min Fuzzy Relation Equation Concluding Remarks ix CHAPTER 11: HANDLING FUZZINESS IN KNOWLEDGE-BASED SYSTEMS Handling of Factors of Uncertainty in Knowledge-Based Systems: Approaches and Requirements Selected Aspects of Reasoning with Fuzziness Principle of Consistency of Aggregation and Inference Mechanisms Information Granularity and Fuzzy Sets Conclusions CHAPTER 12: CONSTRUCTION OF KNOWLEDGE BASE, ITS VALIDATION AND OPTIMIZATION Preliminaries Validation of Production Rules Distributed Knowledge Bases Reduction Problem in Knowledge Bases Reconstruction Problem of Knowledge Bases Concluding Remarks CHAPTER 13: INFERENCE ALGORITHMS IN KNOWLEDGE-BASED SYSTEMS Preliminaries Representation ofmodus Ponens with Fuzzy Premises Expressing the Scheme of Modus Ponens in Terms of Fuzzy Relation Equations Backward Chaining Modus Ponens Performed in the Framework offuzzy Logic Detachment Principle and Its Implementation with the Aid offuzzy 216

10 x Equations CHAPTER 14: A FUZZY CONTROLLER AND ITS REALIZATION Theoretical Preliminaries Schemes ofinference with Fuzzy Information Realization of the Fuzzy Controller and an Applicational Example Concluding Remarks CHAPTER 15: BIBLIOGRAPHIES A List of Papers on Fuzzy Relation Equations and Related Topics A List of Papers on Fuzzy Relations and Related Topics Author Index Subject Index

11 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 ~ ~ 13 xi

12 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

13 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