Topics in the General Theory of Structures PDF
Preview Topics in the General Theory of Structures
TOPICS IN THE GENERAL THEORY OF STRUCTURES THEORY AND DECISION LIBRARY General Editors: W. Leinfellner and G. Eberlein Series A: Philosophy and Methodology of the Social Sciences Editors: W. Leinfellner (Technical University of Vienna) G. Eberlein (Technical University of Munich) Series B: Mathematical and Statistical Methods Editor: H. Skala (University of Paderborn) Series C: Game Theory, Mathematical Programming and Mathematical Economics Editor: S. Tijs (University of Nijmegen) Series D: System Theory, Knowledge Engineering and Problem Solving Editor: W. Janko (University of Vienna) SERIES D: SYSTEM THEORY, KNOWLEDGE ENGINEERING AND PROBLEM SOLVING 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 including computational and algorithmic aspects and the investigation of the empirical behaviour of systems and organisations will be an essential part of this library. The study of problems related to the interface of systems andorga'nisations to their environment is supported. Interdisciplinary considerations are welcome. The publication of recent and original results will be favoured. TOPICS IN THEGENERALTHEORY OF STRUCTURES Edited by E. R. CAIANIELLO Dipartimento di Fisica Teorica, UniversitiL di Salerno, Italy and M. A. AIZERMAN Institute of Control Sciences, Moscow, U.S.S.R. D. REIDEL PUBLISHING COMPANY A MEMBER OF THE KLUWER ACADEMICPUBUSHERSGROUP DORDRECHT / BOSTON / LANCASTER / TOKYO Library of Congress Cataloging in Publication Data Topics in the general theory of structures. (Theory and decision library. Series D, System theory, knowledge enginee- ring, and problem solving) Includes index. I. System analysis. 2. Computational complexity. I. Caianiello, Eduardo R., 1921- . II. Alzerman, M. A. (Mark Aronovich), 1913- II I. Series. QA402.T65 1987 003 87-4329 ISBN -13: 978-94-010-8199-3 e-ISBN-13:978-94-009-3819-9 DOl: 10.1007/978-94-009-3819-9 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland. All Rights Reserved © 1987 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 1st edition 1987 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 permission from the copyright owner ACKNOWLEDGMENT The Editors express their warm thanks to the Istituto Italiano per gli Studi Fi1osofici and to the International Institute for Advanced Scientific Studies, Naples, for generously promoting and organizing a series of Meetings in Naples, Capri and Ama1fi, from which the collaboration between Italian and Soviet scientists, the object of this volume, has greatly benefited. v TABLE OF CONTENTS 1 • INTRODUCTION 2. STRUCTURE AND MODULARITY IN SELF-ORGANIZING COMPLEX SYSTEMS E. R. Caianiello, M. Marinaro, G. Scarpetta, and G. Simoncelli 5 3. HIERARCHY AND MODULARITY IN NATURAL LANGUAGES A. Negro, R. Tagliaferri, and S. Tagliaferri 59 4. DYNAMIC APPROACH TO ANALYSIS OF STRUCTURES DESCRIBED BY GRAPHS (FOUNDATIONS OF GRAPH-DYNAMICS) M. A. Aizerman, L. A. Gusev, S. V. Petrov, I. M. Smirnova, and L. A. Tenenbaum 69 5. STRUCTURAL PROPERTIES OF VOTING SYSTEMS M. A. Aizerman and F. T. Aleskerov 137 6. APPLICATION OF' PREDICATE CALCULUS TO THE STUDY OF STRUCTURES OF SYSTEMS S. V. Petrov 149 7. TOURNAMENT FUNCTIONS IN PROBLEMS OF COLLECTIVE CHOICES V. 1. Vol' skij 157 8. C-CALCULUS: AN OVERVIEH E. R. Caianiello 163 9. ON SOME ANALYTIC ASPECTS OF C-CALCULUS E. R. Caianiello and A; G. S. Ventre 175 10. A NEH METHOD BASED ON C-CALCULUS FOR SOME PROBLEMS OF CELL MOVEMENT ANALYSIS E. R. Caianiello and Lu Huimin 183 11. SYSTEMS AND UNCERTAINTY: A GEOMETRICAL APPROACH E. R. Caianiello 199 Subject Index 207 INTRODUCTION This volume is about "Structure". The search for "structure", always the pursuit of sciences within their specific areas and perspectives, is witnessing these days a dra- matic revolution. The coexistence and interaction of so many structures (atoms, hu- mans, cosmos and all that there is in between) would be unconceivable according to many experts, if there were not, behind it all, some gen- eral organizational principle.s that (at least in some asymptotic way) make possible so many equilibria among species and natural objects, fan- tastically tuned to an extremely high degree of precision. The evidence accumulates to an increasingly impressive degree; a concrete example comes from physics, whose constant aim always was and is that of searching for "ultimate laws", out of which everything should follow, from quarks to the cosmos. Our notions and philosophy have un- dergone major revolutions, whenever the "unthinkable" has been changed by its wonderful endeavours into "fact". Well, it is just from physics that evidence comes: even if the "ultimate" could be reached, it would not in any way be a terminal point. When "complexity" comes into the game, entirely new notions have to be invented; they all have to do with "structure", though this time in a much wider sense than would have been understood a decade or so ago. Clear indications show the difference: the laws that determine "structures" in physics (in this volume we shall quote some examples of them) are not of purely physical nature, such as Newton's or Maxwell's laws: they are rather logical, connected with in- formation, entropy, the structuring of complex systems into levels. Such laws, which are by some claimed to hold from hadron jets to galaxy clu- sterings, do not belong to mechanics, whether classic or quantum. This view is enforced by the fact that exactly the same laws emerge from a variety of fields which have nothing to do with physics: to name one, mathematical linguistics. The challenge in the study of structure is epitomized by the word "complexity". We have to learn "simple" ways to catch what is really essential in phenomena which occur only in systems with large numbers of components and interactions. E. R. Caianiello and M. A. Aizerman (eds.), Topics in the General TheoryoiStructures, 1-3 © 1987 by D. Reidel Publishing Company. 2 INTRODUCTION To give an example from a different field, suppose a situation in which a large number of voters will determine, with their free vote, the election of one among a number of competing candidates; such a situation cannot be reduced to the familiar methodology of physics, of descending to ultimate constituents and ascending back to macroscopic objects. What happens is something that. has to be studied by considering the system in its actual complexity; models that will render its handling amenable to mathematical treatment must not destroy that complexity, but only ex- purge from it all finer details of the real structure that are not re- levant in the global perspective. The works collected iri the present volume all point toward this goal. About a decade ago the emerging importance of this subject was re- cognized in an official agreement between the Italian C.N.R. and the Soviet Academy of Sciences •. Two teams were to cooperate in this widely interdisciplinary effort, led respectively by the two authors of this volume. Each year there were meetings and general discussions, from which it soon emerged that the important thing to attempt was not to try to build a general theory of things of which basic features are still unknown, but rather to assault the fortress by a riumber of well.defined inroads, aiming at some concrete, though modest, knowledge which might provide a basis for further speculation in a quest that no sane person can expect to see achieved within his lifetime. The task is far from finished, the cooperation continues; enough material has been gathered however, to make desirable its publication as a set of contributions, each addressing an aspect of the problem up to a degree of completion permitting tests and applications. The contributions cover the following aspects of general structure theory. 1. HIERARCHICAL SYSTEMS All complex systems are structured into levels, each containing eleme- nts characterized by a value which depends upon the level; "value", ig- nored in information theory, appears to be essential to understand the structure of systems. As the simplest model, we can consider a counting (decimal or binary) system or a monetary system, which are characterized by their modular structure (e.g. a power of ten, or two). A general the- ory of such modular hierarchical systems is made,which yields results 'in surprising agreement with many more facts than one could a priori expect. Urban population, army hierarchies, the Zipf law of linguistics etc. follow straightforwardly. Two ways of change are possible for such systems, "evolution" and "revolution" (change of module); equilibrium between two such systems is possible only if they have. the same struc- ture. 2. GRAPHODYNAMICS The structure of systems can in most cases of interest be described by INTRODUCTION means of oriented graphs (such as armies, corporation structures, com- puter memories etc.). These graphs contain a relevant amount of inform- ation about the system. It therefore becomes of interest to have simple mathematical techniques which may describe with a minimum of formal com- plication the .change of structure through the change of the correspond- ing graphs. 3. STRUCTURAL PROPERTIES OF VOTING SYSTEMS When the components of a system can express their individual op~n~on with a decision which will select one out ofa set of choices (projects, nomination.s,plans), many subtle questions of logic and paradoxical situ- ations are encountered; "optimization" may not be possible or even desirable. A survey and study of this situation, which is of the high- est interest to social scientists, is made in chapter 4 and in appendix A. 4. C-CALCULUS Complex systems, for instance a written text, a set of images, an assem- bly of biological cells, are the object of a huge variety of ad hoc mathematical studies and techniques. A simple calculus, simpler in fact than .arithmetics because it uses only the direct operations of sum and product, is proposed and tested in various such cases with satisfactory results, as a "natural" language of rather general applicability in many such situations. To this "calculus" are dedicated chapters 5,6 and 7. 5. UNCERTAINTY IN MODEL BUILDING It has always been assumed that uncertainty is typical of quantum phy- sics. It has been proved that this is not true; the famous Cramer-Rao inequality of statistics has the same content as Heisenberg's uncertain- ty relation, and a geometrical approach is open which can work in two ways: quantum mechanics. can be ·derived from it, as shown here; converse- ly, the well proven methods of quantum mechanics become conceivable in many other systems in which uncertainty plays a role. STRUCTURE AND MODULARITY IN SELF-ORGANIZING COMPLEX SYSTEMS E.R.Caianiello l M. Marinaro 1 G.Scarpetta1,2 G.Simoncelli3 (1) Dipartimento di Fisica Teorica e sue Metodologie per Ie Scienze Applicate - Universita di Salerno - Salerno - Italy (2) Istituto Nazionale di Fisica Nucleare - Sezlone di Napoli - Italy (3) Dipartimento di Scienze di Base - Accademia Aeronautica - Pozzuoli 1. SYSTEM AND PATTERN The mathematical notion of set is a primitive concept, i.e. it cannot be defined in terms of other, simpler concepts. If we say that a set is a collection, or an aggregate, all we are actually doing is just giving its synonyms. The concept of set, in other words, is as primitive as the concept of element, even though the two logical concepts are obviously quite different and indeed even, to some extent, antithetical; one and the some object can be a set and at the same time can be considered an element of a larger set. More or less the some applies to the term system. Etymologically, system is the Greek equivalent of the Latin composition; it therefore implies, first of all, the simultaneous presence of more than one comp- onent, part or organ. A system however is not merely a set of indepen- dent elments. The word implies an interaction of the parts such that the totality of the parts presents characteristics and properties which are not the result of a simple addition or juxtaposition. The following example helps to clarify the concept. Let us take a set of stones (hewn stones) normally used in the building of arches. With the aid of an arch centre as support, the hewn stones are placed one next to the .other. Once a keystone (1. e. the central hewn stone) is in- serted, the set of stones acquires, thanks to the elements' capacity of supporting one another, the static capacity of supporting not only itself, but even superimposed loads: a structure is born. The load sup- porting capacity is not a property of each stone, or even of all the stones put together, it comes into being the moment the stones interact in a given arrangement. Broadly speaking, a system may have dynamic properties, i.e. prop- erties which entail, in time, a change not only in the interactions of the parts, but also in the components themselves. In this sense, living organisms, as well as social, economic or political organizations, are E. R. Caianiello and M. A. Aizerman (eds.), Topics in the General Theory ofS tructures, 5-57. © 1987 by D. Reidel Publishing Company.
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