SYSTEMS: APPROACHES, THEORIES, APPLICATIONS EPISTEME A SERIES IN THE FOUNDA TIONAL, METHODOLOGICAL, PHILOSOPHICAL, PSYCHOLOGICAL, SOCIOLOGICAL AND POLITICAL ASPECTS OF THE SCIENCES, PURE AND APPLIED Editor: MARIO BUNGE Foundations and Philosophy of Science Unit, McGill University Advisory Editorial Board: RUTHERFORD ARIS, Chemistry, University of Minnesota DANIEL E. BERLYNE, Psychology, University of Toronto HUBERT M. BLALOCK, Sociology, University of Washington GEORGE BUG LIARELLO, Engineering, Poly technic Institute of New York No AM CHOMSKY, Linguistics,MIT KARL W. DE UTSCH, Political science, Harvard University B RUN 0 F R IT S C H, Economics, E. T.H. Zurich E R WI N HIE BER T, History of science, Harvard University ARISTID LINDENMA YER, Biology, University of Utrecht JOHN MYHILL, Mathematics, SUNY at Buffalo JOHN MAYNARD SMITH,Biology, University of Sussex RAIMO TUOMELA, Philosophy, University of Helsinki VOLUME 3 SYSTEMS: APPROACHES, THEORIES, APPLICATIONS INCLUDING THE PROCEEDINGS OF THE EIGHTH GEORGE HUDSON SYMPOSIUM HELD AT PLATTSBURGH, NEW YORK, APRIL 11-12,1975 Edited by WILLIAM E. HARTNETT Dept. of Mathematics, State University College, Plattsburgh, N. Y., U. S.A. D. REIDEL PUBLISHING COMPANY DORDRECHT-HOLLAND / BOSTON-U.S.A. Library of Congress Cataloging in Publication Data George H. Hudson Symposium, 8th, Plattsburgh, N. Y., 1975. Systems: approaches, theories, applications. (Episteme; v. 3) Includes bibliographies and index. 1. System theory - Congresses. I. Hartnett, William, E. II. Title. Q295.G461975 003 77-22238 ISBN-13: 978-94-010-1241-6 e-ISBN-13: 978-94-010-1239-3 DOl: 10.1 007/978-94-010-1239-3 Published by D. Reidel Publishing Company, P. O. Box 17, Dordrecht, Holland Sold and distributed in the U. S. A., Canada, and Mexico by D. Reidel Publishing Company, Inc. Lincoln Building, 160 Old Derby Street, Hingham, Mass. 02043, U.S.A. All Rights Reserved Copyright © 1977 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 1s t edition 1977 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 informational storage and retrieval system, without written permission from the copyright owner TABLE OF CONTENTS PREFACE IX ACKNOWLEDGEMENTS XI NOTES ON THE CONTRIBUTORS XIII 1. A CATEGORY-THEORETIC APPROACH TO SYSTEMS IN A FUZZY WORLD by Michael A. Arbib and Ernest G. Manes 1. Machines in a Category 2 2. Fuzzy Machines 16 2. PARALLELISM, SLIDES, SCHEMAS, AND FRAMES by Michael A. Arbib 27 1. Parallelism 27 2. Slides and Schemas 30 3. Frames and Schemas 33 4. Development 37 5. More on Parallelism 39 3. THE FUNDAMENTAL DUALITY OF SYSTEM THEORY by E. S. Bainbridge 45 1. Introduction 45 2. Networks 47 3. Duality 51 4. Conclusion 60 4. TOWARDS A SYSTEMS METHODOLOGY OF SOCIAL CONTROL PROCESSES by Walter Buckley 63 VI TABLE OF CONTENTS 5 . STATES AND EVENTS by Mario Bunge 71 1. Introduction 71 2. Properties and Predicates 73 3. Definition of a State Function 76 4. Law Statements 78 5. Lagrangian Law Schemata 80 6. State Spaces 82 7. Law Statements and Transformation Formulas 86 8. Events and Processes 89 9. Event Space 91 10. The Category of Events 92 11. Concluding Remarks 94 6. UNDERSTANDING SOCIAL AND ECONOMIC CHANGE IN THE UNITED STATES by Jay W. Forrester 97 1. System DynamiCS 98 2. Dynamics to Be Represented 104 3. Social and Economic Issues 109 4. Structure of the Model 114 5. Status, Schedule, Procedure 118 7. PATTERN DISCOVERY IN ACTIVITY ARRAYS by George J. Klir 121 1. Introduction 121 2. Sampling Procedure 129 3. Evaluation of Masks 141 4. Reduction of ST-Structures 153 5. Conclusions 154 8. A PURPOSIVE BEHAVIOR MODEL by Peter Milner 159 9. COMPLEXITY AND SYSTEM DESCRIPTIONS by Robert Rosen 169 TABLE OF CONTENTS VII 10. CONCERNS, COMMENTS, AND SUGGESTIONS by William E. Hartnett 177 1. Educational Concerns 177 2. Useful Mathematical Models 180 3. Problems of Applied Mathematics 180 4. Modeling 182 5. State Modeling of Objects 189 6. Questions 195 INDEX 199 PREFACE For many years I have believed in a particular style of education for myself. The idea is to focus on matters that you want to learn about, find a modest amount of money, and then organize a symposium of those matters, inviting knowledgeable individuals to participate - and, by extension - to come and help with my education. The Eighth George Hudson Symposium held at Plattsburgh, New York on April 11-12, 1975 was another attempt on my part to learn something. The ostensible reason for the Symposium was explained in the Announce ment of the Symposium as follows: Systems Theory is currently one of the exciting areas of intellectual activity, attracting persons from diverse disciplines. In fact, it has almost become the prototype of inter disciplinary effort. As such, it needs the interchange of ideas, viewpoints, and opinions as a necessary condition for growth. This Symposium was convened to bring together a number of persons- some of them experts and some beginners - for two days of con centrated interaction on Systems Theory. The breadth of the interests of the invited speakers can be noted from their "home" disciplines but space limitations forestall any attempt to document their actual current interests which range from brain function to political institutions to technoethics. The speakers were chosen for their expository and interactive ability as well as for their work in Systems Theory and ample time has been allowed for discussion with them. The Symposium provided some measure of local interchange of ideas, view points, and opinions; this publication is the global counterpart. For the most part, the papers are those which were presented at the Symposium, in some cases revised by the authors in the light of reactions and subsequent discus sions. The paper by Bunge was prepared for the Symposium but he was not able to attend. The papers by Forrester and by Bainbridge were added after the Symposium. E. Manes gave a talk entitled 'A categorical catalyst: new mathematical ideas in system theory' at the Symposium; by mutual agree ment, no manuscript was prepared. The Arbib-Manes paper was background for the Manes paper. No central theme was set for the Symposium and hence the editor is spared the task of showing (and the reader of accepting) how each paper fits in with the central theme. Each paper speaks for itself and despite the x PREFACE diversity of themes there are remarkable similarities of notions. One comment should perhaps be made about the Arbib-Manes paper - more precisely, about the title. In their paper - and elsewhere - fuzzy sets, fuzzy automata, fuzzy machines, and fuzzy categories abound. In each case, fuzzy is an adjec tive which adds something to the noun it modified. Reading again all of the papers of this volume, I am convinced that "fuzzy" as in "fuzzy world" is redundant. One might almost be tempted to say that the world is inherently fuzzy. If such is indeed the case, I hope that this volume will help us deal with the fuzz. Plattsburgh, New York, US.A. WILLIAM E. HAR TNETT March 1976 ACKNOWLEDGEMENTS To D. Reidel Publishing Company and to Jaakko Hintikka, Editor of Synthese, for permission to reprint 'A Category-Theoretic Approach to Systems in a Fuzzy World', Michael A. Arbib and Ernest G. Manes, Synthese 30 (1975) 381-406. To Jay W. Forrester for permission to print copyrighted material. To the Faculty of Science and Mathematics, State University College, Plattsburgh, New York, for funding the Eighth George Hudson Symposium on Systems: Approaches, Theories, Applications, held at Plattsburgh on April 11-12,1975. To Mario Bunge and to the Canada Council. To Bunge for haVing me as a Research Associate during a very educational year and to the Council for underwriting my weekly visits to McGill University. Their support provided the gestation period for my paper. To Carol Burnam, Secretary of the Department of Mathematics, for many things, particularly for the loving care and attention she has given to my work over the years. I am especially grateful for all the onerous tasks she has undertaken in connection with the Eighth George Hudson Symposium and the publication of this volume. W.E.H.