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Fuzzy Relational Calculus Theory, Applications and Software ADVANCES IN FUZZY SYSTEMS — APPLICATIONS AND THEORY Honorary Editor: Lotfi A. Zadeh (Univ. of California, Berkeley) Series Editors: Kaoru Hirota (Tokyo Inst. of Tech.), George J. Klir (Binghamton Univ.-SUNY), Elie Sanchez (Neuhnfo), Pei-Zhuang Wang (West Texas A&M Univ.), Ronald R. Yager (lona College) Vol. 8: Foundations and Applications of Possibility Theory (Eds. G. de Cooman, D. Ruan and E. E. Kerre) Vol. 9: Fuzzy Topology (Y. M. Liu and M. K. Luo) Vol. 10: Fuzzy Algorithms: With Applications to Image Processing and Pattern Recognition (Z Chi, H. Yan and T. D. Pham) Vol. 11: Hybrid Intelligent Engineering Systems (Eds. L. C. Jain and R. K. Jain) Vol. 12: Fuzzy Logic for Business, Finance, and Management (G. Bojadziev and M. Bojadziev) Vol. 13: Fuzzy and Uncertain Object-Oriented Databases: Concepts and Models (Ed. R. de Caluwe) Vol. 14: Automatic Generation of Neural Network Architecture Using Evolutionary Computing (Eds. E. Vonk, L C. Jain and R. P. Johnson) Vol. 15: Fuzzy-Logic-Based Programming (Chin-Liang Chang) Vol. 16: Computational Intelligence in Software Engineering (W. Pedrycz and J. F. Peters) Vol. 17: Non-additive Set Functions and Nonlinear Integrals (Forthcoming) (Z. Y. Wang) Vol. 18: Factor Space, Fuzzy Statistics, and Uncertainty Inference (Forthcoming) (P. Z. Wang and X. H. Zhang) Vol. 19: Genetic Fuzzy Systems, Evolutionary Tuning and Learning of Fuzzy Knowledge Bases (O. Cordon, F. Herrera, F. Hoffmann and L Magdalena) Vol. 20: Uncertainty in Intelligent and Information Systems (Eds. B. Bouchon-Meunier, R. R. Yager and L. A. Zadeh) Vol. 21: Machine Intelligence: Quo Vadis? (Eds. P. Sincak, J. VaScak and K. Hirota) Vol. 22: Fuzzy Relational Calculus: Theory, Applications and Software (With CD-ROM) (K. Peeva and Y. Kyosev) Advances in Fuzzy Systems - Applications and Theory - Vol. 22 HSHKHV I THJi I IZL • CXX^K^ "Si^ • ^ | r ^ ^ W P T M I if.f JMW9?K& ^mB3EBHSl3ltj '111 t i l *** • ••^ Fuzzy Relational Calculus Theory, Applications and Software (With CD-Rom) Ketty Peeva & Yordan Kyosev Technical University of Sofia, Bulgaria \Hp World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • H O N G K O N G • TAIPEI • CHENNA! Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401^102, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Advances in Fuzzy Systems — Applications and Theory, Vol. 22 FUZZY RELATIONAL CALCULUS Theory, Applications and Software (With CD-ROM) Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-076-9 Printed by Multiprint Services Preface Fuzzy relations appear as a natural generalization of crisp relations. While a crisp relation determines the presence or absence of interconnectedness between the elements of two or more sets, fuzzy relations supply additional information for degrees of membership, strengths of associations, i