Kybernetes
ISSN 0368-492X
The International Journal of Systems & Cybernetics
Volume 31 Number 3/4 2002
Contemporary cybernetic and systemic researches Editor-in-Chief Brian H. Rudall Paper format Kybernetes includes ten issues in traditional paper format. The contents of this issue are detailed below.
Internet Online Publishing with Archive, Active Reference Linking, Key Readings, Non-article Content, Institution-wide Licence, E-mail Alerting Service and Usage Statistics. Access via the Emerald Web site: http://www.emeraldinsight.com/ft See p. 351 for full details of subscriber entitlements.
Access to Kybernetes online ______________________
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Editorial advisory board __________________________
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Abstracts and keywords __________________________
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Preface __________________________________________
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Part I: Research studies of intelligence and the brain Evolution of intelligence: direct modeling of temporal effects of environment on a global absolute scale vs statistics H.M. Hubey ___________________________________________________
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Temporal and simultaneous processing in the brain: a possible cellular basis of cognition Uri Fidelman__________________________________________________
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Part II: Research studies with interdisciplinary methodologies Direct and indirect causality in living systems Miguel Lloret-Climent ___________________________________________
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Physical system theory: fundamentals, recent developments and relationships with system dynamics Sushil________________________________________________________
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This issue is part of a comprehensive multiple access information service
CONTENTS
CONTENTS continued
Neural network process vision systems for flotation process Harry Coomar Shumsher Rughooputh and Soonil Dutt Dharam Vir Rughooputh ______________________________
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A new approach to solve a diffusion-convection problem M. Inc and Y. Cherruault ________________________________________
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An expert system for the selection of strategic planning technique Suresh Subramoniam and K.V. Krishnankutty_______________________
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Semantics of L(MT): a language for ecological modelling J.L. Uso´-Dome`nech and P. Sastre-Vazquez __________________________
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Approximation of the solution for a class of first order p.d.e. by Adomian method S. Khelifa and Y. Cherruault _____________________________________
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A cybernetic approach to the multiscale minimization of energy function: Grey level image segmentation Pilar Arques, Patricia Compan˜, Rafael Molina, Mar Pujol and Ramo´n Rizo___________________________________________________
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Part III News, conferences and technical reports ___________
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30th Anniversary Cyber-profiles___________________
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Internet commentary Alex M. Andrew _______________________________________________
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Book reviews C.J.H. Mann __________________________________________________
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Book reports C.J.H. Mann __________________________________________________
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Software review Andrew Adamatzky ____________________________________________
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Announcements __________________________________
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Special announcements ___________________________
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EDITORIAL ADVISORY BOARD A. Bensoussan President of INRIA, France E. Billeter University of Fribourg, Switzerland V. Chavchanidze Institute of Cybernetics, Tbilisi University, Georgia, (former) USSR A.B. Engel IMECC-Unicamp, Universidad Estadual de Campinas, Brazil R.L. Flood Hull University, UK F. Geyer The Netherlands Universities Institute for Co-ordination of Research in Social Sciences, Amsterdam, The Netherlands A. Ghosal Honorary Fellow, World Organisation of Systems and Cybernetics R.W. Grubbstro¨m Linko¨ping University, Sweden Chen Hanfu Institute of Systems Science, Academia Sinica, People’s Republic of China G.J. Klir State University of New York, USA K.E. McKee IIT Research Institute, Chicago, IL, USA
M. Ma˘nescu Academician Professor, Bucharest, Romania M. Mansour Swiss Federal Institute of Technology, Switzerland K.S. Narendra Yale University, New Haven, CT, USA C.V. Negoita City University of New York, USA W. Pearlman Technion Haifa, Israel A. Raouf Pro-Rector, Ghulam Ishaq Khan (GIK) Institute of Engineering Sciences & Technology, Topi, Pakistan Y. Sawaragi Kyoto University, Japan B. Scott University of the Islands Project Lews Castle College Scotland, UK D.J. Stewart Human Factors Research UK I.A. Ushakov Moscow, (former) USSR J. van der Zouwen Free University, Amsterdam, The Netherlands
Evolution of intelligence: direct modeling of temporal effects of environment on a global absolute scale vs statistics H.M. Hubey Keywords Cybernetics, Intelligence, Computers, Brain The social sciences are really the ‘‘hard sciences’’ and the physical sciences are the ‘‘easy’’ sciences. One of the great contributors to making the job of the social scientist very difficult is the lack of fundamental dimensions on the basis of which absolute (i.e. ratio) scales can be formulated and in which relationships could be realized as the [allegedly] coveted equations of physics. This deficiency leads directly to the uses of statistical methods of various types. However it is possible, as shown, to formulate equations and to use them to obtain ratio/absolute scales and relationships based on them. This paper uses differential/ integral equations, fundamental ideas from the processing view of the brain-mind, multiple scale approximation via Taylor series, and basic reasoning some of which may be formulated as infinite-valued logic, and which is related to probability theory (the theoretical basis of statistics) to resolve some of the basic issues relating to learning theory, the roles of nature and nurture in intelligence, the measurement of intelligence itself, and leads to the correct formulation of the potential-actual type behaviors (specifically intelligence) and dynamical-temporal model of intelligence development. Specifically, it is shown that the: (1) basic model for intelligence in terms of genetics and environment has to be multiplicative, which corresponds to a logical-AND, and is not additive; (2) related concept of ‘‘genetics’’ creating its own environment is simply another way of saying that the interaction of genetics and environment is multiplicative as in (1); (3) timing of environmental richness is critical and must be modeled dynamically, e.g. in the form of a differential equation; (4) path functions, not point functions, must be used to model such phenomena; (5) integral equation formulation shows that intelligence at any time t, is a a sum over time of the past interaction of intelligence with environmental and genetic factors; (6) intelligence is about
100% inherited on a global absolute (ratio) scale which is the natural (dimensionless) scale for measuring variables in social science; (7) nature of the approximation assumptions implicit in statistical methods leads to ‘‘heritability’’ calculations in the neighborhood of 0.5. and that short of having controlled randomized experiments such as in animal studies these are expected sheerely due to the methods used; (8) concepts from AI, psychology, epistemology and physics coincide in many respects except for the terminology used, and these concepts can be modeled nonlinearly.
Temporal and simultaneous processing in the brain: a possible cellular basis of cognition Uri Fidelman Keywords Cybernetics, Brain, Cognition, Quantum mechanics It is suggested that the left hemispheric neurons and the magnocellular visual system are specialized in tasks requiring a relatively small number of large neurons having a fast reaction time due to a high firing rate or many dendritic synapses of the same neuron which are activated simultaneously. On the other hand the right hemispheric neurons and the neurons of the parvocellular visual system are specialized in tasks requiring a relatively larger number of short term memory (STM) Hebbian engrams (neural networks). This larger number of engrams is achieved by a combination of two strategies. The first is evolving a larger number of neurons, which may be smaller and have a lower firing rate. The second is evolving longer and more branching axons and thus producing more engrams, including engrams comprising neurons located at cortical areas distant from each other. This model explains why verbal functions of the brain are related to the left hemisphere, and the division of semantic tasks between the left hemisphere and the right one. This explanation is extended to other cognitive functions like visual search, ontological cognition, the detection of temporal order, and the dual cognitive interpretation of the perceived physical phenomena.
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Direct and indirect causality in living systems Miguel Lloret-Climent Keywords Cybernetics, Living systems Not all cells are equal: each tissue and organ has its own type of cell. Although the nucleus of each cell in a living system has the same genetic information, each one dispenses of the lion’s share of that information and only those genes that are necessary for carrying out the function of the particular organ or tissue to which they belong remain active. Despite the fact that in specific scientific fields, such as ecosystem studies, it is possible to measure the relationships between different variables and to compare the various direct and indirect effects they may have on one another, there has been no such development in the wider context of a General Systems Theory. This paper sets out to address the question of cellular change by interpreting processes such as direct and indirect causality, cellular meiosis and mutation of cells.
Physical system theory: fundamentals, recent developments and relationships with system dynamics Sushil Keywords Cybernetics, Systems theory, Modelling Two important methodologies having some common grounds, but based on differing contexts and paradigms are Physical System Theory (PST) and System Dynamics (SD). The developments in both the fields have taken place almost independently, and attempts have been made to integrate the two to complement their strengths and limitations. This paper provides an overview of PST in terms of its foundations, philosophy, fundamental postulates, recent developments on its simplification and enlargement, and applications to socioeconomic and managerial systems. A comparison of PST is made with SD on different fronts so as to understand their similarities and differences for carving out their place in modelling of managerial and socio-economic systems and integrating the two more meaningfully and flexibly. The
paper is concluded emphasizing the need for a ‘Flexible System Theory’ which can relate many such systems based approaches and techniques on the whole continuum from hard to soft systems thinking to cater the whole spectrum of problem situations effectively. Neural network process vision systems for flotation process Harry Coomar Shumsher Rughooputh and Soonil Dutt Dharam Vir Rughooputh Keywords Cybernetics, Neural networks, Modelling Froth flotation is a process whereby valuable minerals are separated from waste by exploiting natural differences or by chemically inducing differences in hydrophobicity. Flotation processes are difficult to model because of the stochastic nature of the froth structures and the ill-defined chemorheology of these systems. In this paper a hierarchical configuration hybrid neural network has been used to interpret froth images in a copper flotation process. This hierarchical neural network uses two Pulse-Coupled Neural Networks (PCNNs) as preprocessors that ‘convert’ the froth images into corresponding binary barcodes. Our technique demonstrates the effectiveness of the hybrid neural network for process vision, and hence, its potential for use for real time automated interpretation of froth images and for flotation process control in the mining industry. The system is simple, inexpensive and is very reliable. A new approach to solve a diffusion-convection problem M. Inc and Y. Cherruault Keywords Cybernetics, Decomposition method, Noise We use the Adomian decomposition method to study a non-linear diffusion-convection problem (NDCP). The decomposition method has been applied recently to a wide class of non-linear stochastic and deterministic operator equations involving algebraic, differential, integro-differential and partial differential equations and systems. The method provides a solution without
linearization, perturbation, or unjustified assumptions. An analytic solution of NDCP in the form of a series with easily computable components using the decomposition method will be determined. The non-homogeneous equation is effectively solved by employing the phenomena of the self-cancelling ‘noise terms’ whose sum vanishes in the limit. Comparing the methodology with some known techniques shows that the present approach is highly accurate.
An expert system for the selection of strategic planning technique Suresh Subramoniam and K.V. Krishnankutty Keywords Cybernetics, Strategic planning, Expert systems The expert system for the selection of strategic planning technique was developed with a view of assisting an executive in selecting a strategic planning tool based on constraints such as: time, skill, finance, computer, expert advice, form and content of input, and form and content of output required to be generated. The expert system runs on IBM PC/AT or compatible, which is the minimum system requirement, and contains approximately 115 rules in the rule base. The expert system was developed in Turbo PrologTM. The premises which help in concluding a particular strategic planning tool are stored in separate file called ‘STRATEGY.DAT’ and can be consulted at the end of the session by the user. This file saves current session’s premises which concluded the particular strategic planning tool. The system, after recommending a tool, outputs a list of planning phases where the implementation of such a strategic planning tool will succeed. It runs on a backward chaining inference mechanism which is a feature of Turbo PrologTM.
Semantics of L(MT): a language for ecological modelling J.L. Uso´-Dome`nech and P. Sastre-Vazquez Keywords Cybernetics, Systems theory, Language The authors continue developing a Linguistic Theory of the Ecological Systems, but in
terms of the Semantics. Each symbol (transformed function) is syntactically a lexeme, carrying associate a sememe or atomic semantic unit. Each sememe can be decomposed in semes or quantic semantic unities. They are studied the semantic systems, associated with the syntactic, and that serve them as suprastructure, in terms of two levels: the quantic and the atomic. Also, it is demonstrated that for all model of the complex reality, will exist always a most complex model from the syntactic and semantic point of view.
Approximation of the solution for a class of first order p.d.e. by Adomian method S. Khelifa and Y. Cherruault Keywords Adomian polynomials, Cybernetics, Decomposition method Aim’s to show how to approach the solution for a class of first order p.d.e. using Adomian decomposition method. Discusses the generalities of the method and a-dense curves. Outlines the new approach and provides application’s of its use.
A cybernetic approach to the multiscale minimization of energy function: Grey level image segmentation Pilar Arques, Patricia Compan˜, Rafael Molina, Mar Pujol and Ramo´n Rizo Keywords Cybernetics, Computers, Image processing Segmentation is an important topic in computer vision and image processing. In this paper, we sketch a scheme for a multiscale segmentation algorithm and prove its validity on some real images. We propose an approach to the model based on MRF (Markov Random Field) as a systematic way for integrating constraints for robust image segmentation. To do that, robust features and their integration in the energy function, which directs the
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process, have been defined. In this approach, the image is first transformed to different scales to determine which one fits better to our purposes. Then, it is segmented into a set of disjoint regions, the adjacent graph (AG) is determined and a MRF model is defined on the corresponding AG. Robust features are incorporated to the energy function by means of clique functions and optimal segmentation is then achieved by finding a labeling configuration that minimizes the energy function using Simulated Annealing.
Internet commentary Alex M. Andrew Keywords Cyberculture, London-AI, Cyrillic alphabet, Homophonic keyboard Two metasites that give access to previouslyneglected cyberculture topics are reviewed. Changes in the arrangements for the valuable free London-AI events calendar are reported. The use of a homophonic keyboard layout to produce Cyrillic text in e-mails and other documents is discussed with recommendation of a particular package.
Preface Special double issue Contemporary cybernetic and systemic researches
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This Special Double Issue is designed to publicise the work of researchers in many of the specific areas of the study of systems and cybernetics. All, however, have a common bond in that they illustrate an interdisciplinary approach and communicate important research studies. In response to the feedback of our readers after the publication of some of our more recent special issues the studies presented cover a wide-ranging spectrum of current research endeavours in the fields of systems and cybernetics. In addition we have responded to readers comments and some of our authors were allowed to develop their research themes to a much greater depth than the space allocated for previous special issues permitted. The requests that our normal sections of the journal should also be included in these special compilations have also be addressed, and, indeed, welcomed by the Editorial Team. To meet the new criteria this issue is divided into three parts. The first is concerned with research studies in intelligence and the brain. Both of the topics presented have been considered in depth by two well-known and respected researchers, Mark Hubey and Uri Fidelman. In the second part we include a variety of research studies that use interdisciplinary methodologies and many of these contributions have been selected from the large number of papers submitted. We are sorry that space does not allow the inclusion of all of these excellent studies but all accepted, peer-reviewed contributions will be published in later issues of this journal. The third part contains the familiar journal sections that enable our readership to keep in touch with the activities of the flourishing cybernetics and systems communities. Also included in this section is the first of our 30th Anniversary Cyber-Profiles which will initially introduce the members of the journal’s editorial team who compile and present the ten issues (some 1500 pp) of this journal each year. Brian H. Rudall Editor-in-Chief
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File name{KYB}Issues/45373/Prelims/KYB 31_3-4 Robotica.3d 14:01 – 18/6/02 – Total page(s) 1
International Journal of Information, Education and Research in Robotics and Artificial Intelligence Editor J. Rose, Hon. Director of the World Organisation of Systems and Cybernetics VISITING PROFESSOR, UNIVERSITY OF CENTRAL LANCASHIRE Robotica aims to endow robotics with an authoritative, competent and dedicated journal to serve industry, research and education. It provides an international forum for the multidisciplinary subject of robotics and helps to encourage development in this important field of automation. It covers the many aspects of robotics, including sensory perception, software (in the widest sense), particularly in regard to programming languages and links with master computers and CAD/CAM systems, control devices, the study of kinematics and dynamics involved in robot design, design of effectors and ancillary manipulators, problem solving, world model representation, development of relevant educational courses, training methods, analysis of managerial and social policy, economic and cost problems, and items of theoretical and practical interest. As well as original papers, the journal publishes research notes, book reviews, conference reports and letters to the editor. Robotica is of interest to academics, research workers and industry. In manufacturing industry the robot plays a fundamental part in increasing productivity, quality of products, and safety in hostile environments. In this era of advanced automation this publication is of primary importance to both theoretical researchers and practitioners. ‘Robotics will offer exciting solutions to some of the most challenging problems of modern society. Robotica has dedicated itself to this, and l wish the journal every success in the exciting years ahead.’ Lord Henry Chilver FEng FRS, (UK) ‘So often students will ask ‘‘What courses of study do you recommend as preparation for a career in robotics?’’ . . . read Robotica.’ J.F. Engelberger, Late President, Unimation Inc., Connecticut (USA) Volume 20, 2002, Parts 1-6 with Special Issues An official journal of the International Federation of Robotics If you would like further information on this prestigious journal please contact the publishers at the address below or on the Internet: http://www.cup.cam.ac.uk or http://www.cup.org
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PART I
Research studies of intelligence and the brain
The current issue and full text archive of this journal is available at http://www.emeraldinsight.com/0368-492X.htm
Evolution of intelligence Direct modeling of temporal effects of environment on a global absolute scale vs statistics H.M. Hubey Department of Computer Science, Montclair State University, Upper Montclair, USA
Evolution of intelligence
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Keywords Cybernetics, Intelligence, Computers, Brain Abstract The social sciences are really the “hard sciences” and the physical sciences are the “easy” sciences. One of the great contributors to making the job of the social scientist very difficult is the lack of fundamental dimensions on the basis of which absolute (i.e. ratio) scales can be formulated and in which relationships could be realized as the [allegedly] coveted equations of physics. This deficiency leads directly to the uses of statistical methods of various types. However it is possible, as shown, to formulate equations and to use them to obtain ratio/absolute scales and relationships based on them. This paper uses differential/integral equations, fundamental ideas from the processing view of the brain-mind, multiple scale approximation via Taylor series, and basic reasoning some of which may be formulated as infinite-valued logic, and which is related to probability theory (the theoretical basis of statistics) to resolve some of the basic issues relating to learning theory, the roles of nature and nurture in intelligence, the measurement of intelligence itself, and leads to the correct formulation of the potential-actual type behaviors (specifically intelligence) and dynamical-temporal model of intelligence development. Specifically, it is shown that the: (1) basic model for intelligence in terms of genetics and environment has to be multiplicative, which corresponds to a logical-AND, and is not additive; (2) related concept of “genetics” creating its own environment is simply another way of saying that the interaction of genetics and environment is multiplicative as in (1); (3) timing of environmental richness is critical and must be modeled dynamically, e.g. in the form of a differential equation; (4) path functions, not point functions, must be used to model such phenomena; (5) integral equation formulation shows that intelligence at any time t, is a a sum over time of the past interaction of intelligence with environmental and genetic factors; (6) intelligence is about 100 per cent inherited on a global absolute (ratio) scale which is the natural (dimensionless) scale for measuring variables in social science; (7) nature of the approximation assumptions implicit in statistical methods leads to “heritability” calculations in the neighborhood of 0.5. and that short of having controlled randomized experiments such as in animal studies these are expected sheerely due to the methods used; (8) concepts from AI, psychology, epistemology and physics coincide in many respects except for the terminology used, and these concepts can be modeled nonlinearly.
1. Introduction There is an old issue going back to Aristotle (who thought that slaves were slavish by birth), and which has become a heated debate in recent years by Burt, Spearman, Thurstone, Jensen, and Gould, having to do with the role of genetics and environment in intelligence, revived only a few years ago by Herrnstein and Murray (H&M). During this century the discussion has become
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more “scientific” via the use of mathematical models and technique, the evidence consisting of tests, grading, and statistical analysis of such tests. Because of the importance of the history of the subject, the various incommensurable views adhered to by various parties, and sweeping breadth of discussion, the paper treads over known territory, some of it in standard/classical fashion and some with original twists, at the risk of boring some readers, in order to be accessible to the broad readership some of whom may not have any familiarity with some of the mathematical techniques. The attitude of the workers in this field, according to Herrnstein and Murray (1994) without oversimplifying, and without taking too long can be put into three groups: Classic: Intelligence is a structure. Whether there is a single number, two or several is not as important as the fact that there’s a structure to it, and this structure can be captured in a single number, which Spearman called g, general intelligence. Thurstone claimed about a half-dozen PMAs (Primary Mental Abilities). According to Vernon, they are hierarchical. According to Guilford there are 120 or so components in this structure. Computational-AI model (revisionist): Intelligence is a process. This seems to be an evidently more modern attitude encompassing the information processing view. According to Sternberg there are three aspects of human information processing; the transducers or our sensory organs that change real world inputs into special forms for our brain, classifying the real world problems into groups, and actually making use of the apparatus in living (and hopefully being successful) in the real world via the use of the schemes of adapting (to the environment), shaping (the environment), and selecting (a new environment). Scalar vs. Tensor (radical): There are different kinds of things called intelligence. For example, according to Gardner there are linguistic, musical, logical-mathematical, spatial, bodily,musical, interpersonal and intrapersonal forms of intelligence. Of course, the phrase cognitive ability (CA) has now replaced intelligence and according to H&M, it is substantially heritable, apparently no less that 40 per cent and no more than 80 per cent. The importance of personal skills and emotional issues already clouds the definition of intelligence. Is it possible that all of these views have part of the truth and like the men who fought over what to do with their money without knowing that they all wanted to purchase grapes, they are fundamentally more in agreement as far as the facts are concerned just as men and women are more alike than unlike? Is there a unified view of which all of these are components? 1.1 Properties of intelligence: classical One simple way of invalidating the results seems to be to deny the existence of race. The arguments are from biology; There’s no scientific definition of race!
It’s too silly to be of much use. For one thing, it won’t stop the racists; another word will take its place. For another, biology is hardly in a position to be arguing about what is science and what is not since it is still rather low on the totem pole. And thirdly, the definition of race as given doesn’t say anything more than what it’s supposed to be: arguing that there’s no such thing as beauty because it’s only skin deep is silly. Who said it’s supposed to be any deeper? We could, of course, see all kinds of beauty and in everything including in intelligence; indeed it exists everywhere. We might make up a simple table of words and phrases used in the literature for describing the intelligence or the CA debate as below (Table I):
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Man came first to the realm of the minerals, and from them he fell in among plants. For years he lived among the plants and remembered nothing of the vegetative state. In the same way he passed from realm to realm, until now he is intelligent, knowledgeable, and strong. He remembers not his first intellects, and he will leave his present intellect behind. He will be delivered from this intellect full of avarice and cupidity and see hundreds of thousands of marvelous intellects. Rumi (Chittick, 1983)
We can also use the standard terminology of nature-nurture, innate vs. cultural, but they all seem to boil down to a discussion of whether there is biological determinism. We might expand upon the standard arguments against the thesis that human intelligence is unfairly distributed to the different races by summarizing the arguments as making some version of the statement that, the IQ tests measure a .
.
.
structure (vector or tensor) AND a scalar should not or can not be derived from the tensor. scalar (single number) BUT cannot used to linearly order or rank humans. component that’s race-based AND is immutable.
The last part has been changed slightly from the ones expressed by Gould (1981). Some people might say ‘genetically-based’ (i.e. genetically inheritable) and slightly mutable. According to both detractors and proponents the IQ tests and their conclusions are about what is called biological determinism (BD); the idea that intelligence or cognitive ability/capacity is/are innate, intrinsic,
Degree of freedom Type-kind Mechanism Sources Elastic/plastic
Pole 1
Pole 2
Potential-virtual Darwinist Genetic-biological Absolute-immutable
Real-actual Lamarckian Learned-environmental Relative-mutable
Table I.
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inherited (biologically or genetically). A small problem revolves around the definition of heritability. Heritable means capable of being inherited. Inheriting has to do with coming to possess certain characteristics and not necessarily genetically although it is often meant that way. So the fact that statistical techniques such as correlation-regression, and analysis of variance have been used to define inheritance or heritability means that somehow we are to assume that we know exactly what it is and is clearly defined, but it is simply not the case. It is as if we went to a doctor’s office complaining of our heart beating too fast and he told us that we had tachycardia. He hasn’t diagnosed the problem but only given it a name, and we should not be impressed. Several definitions of heritability are possible (Table II). Of these it is probably WAH and WAE are the most commonly held views in opposition. So intelligence could be inherited, but only socially. Superficially the basis of statistical inference and correlation-regression analysis seems secure. Who would fight it? Most people would head for the intellectual hills whenever faced with squiggly symbols of mathematics so the battle lines for the Bell Curve would seem at a glance as if they resemble the Scopes monkey trial with science once again about to triumph over its emotional opponents, who naturally once again seem to be the ‘bleeding heart liberals’. It would be strange to hear someone say that it is all for nothing but that is essentially what it is about. race: A local geographic or global human population distinguished as a more or less distinct group by genetically transmitted physical characteristics. species: A fundamental taxonomic classificiation category, ranking after genus, and consisting of organisms capable of interbreeding. subspecies: a subdivision of a taxanomic species, usually based on geographic distribution.
SAE (Strong Anti-Environmentalism) all intelligence is due to heredity and has nothing at all to do with environment SAH (Strong Anti-Hereditarianism) all intelligence can be accounted for by environmental factors and especially learning
Table II.
WAE (Weak Anti-Environmentalism) some (very little) has to do with the environment and learning but not much. Almost all intelligence can be accounted for simply by genetics WAH (Weak Anti-Hereditarianism) some of the differences in IQ tests and thus intelligence is due to hereditary factors but this factor must be distributed uniformly across all races so that the differences between the races is due only to environmental factors
The theories of statistical testing (and CA testing debates) are replete with oblique axes, multicollinearity, orthogonal regression, covariant vs. contravariant tensors and to this we could add others such as rates of cultural vs. biological change. Some of this is done in detail in later sections. For a brief and intuitive tour of the relevant ideas we should turn to a short history of the vector vs. scalar theory of intelligence and thus to the pioneers of this century. The real vectors of mind, Thurstone reasoned, must represent independent primary abilities (PMAs). If they are truly independent they should be orthogonal (that is, perpendicular) to each other. But whatever these PMAs are, they are correlated; that is they tend cluster. The problem is called multicollinearity in statistics. Not all sets of vectors have a definable simple structure. A random array without clusters cannot be fit by a set of factors. The discovery of a simple structure implies that vectors are grouped into clusters and that clusters are relatively independent of each other; that is they represent, however inaccurately some aspect of some primary mental abilities, or PMAs. Thurstone identified seven of them: Verbal comprehension, Word fluency, Number(computational), Spatial visualization, M (associative memory), Perceptual speed, and Reasoning. Thurstone admitted strong potential influence for environment but emphasized inborn biology and also refused to reduce these to a single number, hence was an advocate of the structuralist school, it might be said. He claimed that Spearman’s scalar g (general intelligence factor of some sort) was imply an artifact of the tests Spearman gave and nothing more. Spearman’s retort was that Thurstone’s PMAs were also artifacts of chosen tests, not invariant vectors of mind, which is also as true as Thurstone’s claim. 1.2 Vector/tensor vs. scalar controversy: distance metrics & normalizations Suppose we want to represent physical agility or physical capability of athletes from various different tests. Suppose we only use three tests; (i)endurance/stamina; (ii) reflex, reaction-time, and (iii)strength. How should we represent these three qualities (as quantities)? As the simplest such measure we can simply make three separate bits (i.e. zero or one) which will represent the possession or lack of the relevant property (such as a pass/fail grade) which we can write as 000,001,010,011,100,101,110, and 111 (see Figure 1). Or we can decide to give them grades in the normalized interval [0,1] for each of the three separate tests, and thus implicitly switch to using some kind of reasoning related to fuzzy logic or probability theory. Of course, we can easily increase the number of such tests to five or ten, and we can also increase the dimensionality of the problem but plotting more than 3 dimensions is very difficult. Hence, it is easy to deal with such high dimensional problems using only symbols and logic. To continue the example of 3 dimensions, we can make bar charts, pie charts or we can plot them on a 3-dimensional graph. Then we can represent each person as a point in three dimensions {x,y,z}. We call such
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ordered n-tuples or vectors. A vector is obviously a simpler case of a matrix. It is a 1 by n matrix. Matrices are also called tensors of rank 2, and vectors are tensors of rank 1. Therefore the ordinary single numbers are called tensors of rank 0 or simply scalars. Consider the case of colors. Colors are produced from three so-called primary colors, Red, Green and Blue (RGB) or their complements, Cyan, Yellow, and Magenta (CYM) depending on whether an additive or subtractive process is used. No one would really argue that a color is not a single indivisible quantity if we think of it as something our perceptual/visual system is able to transduce. So then the natural question is whether a color is a single number, multiple numbers, a vector, a structure or a dynamic thing that causes our perceptual system to process the input data. It depends on our perceptual abilities and our knowledge. For sure it is all of them depending on what we want to do with it, and there’s no contradiction. As we know all the colors (for all practical purposes) can be obtained (additively) from the three basic primaries, Red, Green and Blue, (RGB) and Figure 2. The gray scale runs from black to white along the diagonal. The great advantage of using multiple dimensional space is the accuracy of such representations of much phenomena. We all know what colors are but they would be virtually impossible to explain to someone who was congenitally blind. If we did attempt to “explain” colors by explaining that “black is the absence of color and white is a mixture of all the colors” it is likely that the blind person would think of colors as what we call “gray scale”. We can write the primary colors as vectors 2 3 2 3 2 3 1 0 0 6 7 6 7 6 7 7 6 7 6 7 r¼6 ð1Þ 4 0 5 g ¼ 4 1 5 and b ¼ 4 0 5 0 0 1 Since a vector consists of ordered elements, the first entry refers to redness, second to greenness and the third to blueness. Thus the red vector r has only a
Figure 1. Pass/Fail Physical Agility Space: If we use only {0,1} we can have a discrete distance metric which we can use for binary pass/fail or {0,1} scorling
1 in the redness-place and zeroes elsewhere. Similarly for the other primary colors, g, and b. We suspect, then, that the other colors will be some combination of these primary colors. What this boils down to is that we want to add different proportions of the primaries to create other colors so that we will multiply the primary colors by some number less than one (so that it is a small proportion) and then add them all to get some other color cany, so that 2 3 2 3 2 3 1 0 0 6 7 6 7 6 7 7 6 7 6 7 ð2Þ c any ¼ pr 6 4 0 5 þ pg 4 1 5 þ pb 4 0 5 0 0 1
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where pr ¼ proportion of red, pg ¼ proportion of green and pb ¼ proportion of blue. If we had pr ¼ pg ¼ pb ¼ 0.5 we will obtain a gray since the diagonal of the color space that runs from black to white is called gray-scale. We can represent this particular gray as 3 3 2 3 2 3 2 2 3 2 2 3 2 3 0:5 0 0 0:5 0 1 0 7 7 6 7 6 7 6 6 7 6 6 7 6 7 7 7 6 7 6 7 6 6 7 6 6 7 7 c gray ¼ 0:56 4 0 5 þ 0:54 1 5 þ 0:54 0 5 ¼ 4 0 5 þ 4 0:5 5 þ 4 0 5 ¼ 4 0:5 5 0:5 0:5 0 0 0 0 1 ð3Þ In the example above we saw the rules for scalar-vector multiplication and vector addition, but not vector multiplication. The final result for this particular gray is that it has 0.5 proportion of red, green and blue since those are the vector components. However, if we do make an analogy to the 3D space in which we live with the exception that the dimensions of color are not homogenous like our space dimensions, it is more likely to be understood
Figure 2. Color Space: Almost all colors can be produced additively via the three colors Red, Green, and Blue. (RGB). See Banks (1990).
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better. For a more detailed look at color, see Hubey (1997). There is a simple way to obtain magnitude from the PA space (instead of using the Hamming metric) by treating as the color space except that the meaning may not be intuitive. Simply define 1
PA ¼ ð1E 2e þ rR 2r þ sS 2s Þ2n
ð4Þ
For the special case of 1 ¼ r ¼ s ¼ e ¼ r ¼ s ¼ n ¼ 1 this is simply the Euclidean distance metric that we use for our ordinary three-dimensional space [please see appendix A.1).Although it’s just as obvious that the color vector is being produced from the primary component colors our mind’s eye sees a single color. Indeed this is done all the time; the colors on the computer monitors are produced directly by energizing the red, green and blue strips of phosphors to varying degrees of intensity. The eye in the case of the high resolution monitors (0.28 mm dot pitch) is unable to resolve the different components and produces instead what we see as a single recognizable color on the color wheel. The number does mean something. We can all see it. But naturally we will not be able to assign a linear ranking since it’s pointless. We can see that the Euclidean norm of the color vector will be a section of a sphere in the positive orthant but it could be one of an infinite number of colors on the surface of this sphere. Making the analogy to colors what we can immediately see is that our unaided intuition, if we only considered this color space to be a homogeneous space like that in physics, would not be able to tell us that what we perceive subjectively as color often does not seem to have any obvious connection to the constituent components of the color vector, since we now know that what looks to be a distinct “thing” is merely a shorter/longer wavelength in the visible bandwidth of the electromagnetic spectrum. However, there is no doubt that any given color can be comprised of the basis colors RGB. Therefore we have to reason to insist that a vector created from the components of intelligence will not possess intuitive properties totally different than the basis vectors. At the same time, the scalar quantity obtained from the vector certainly is missing much information. The real question is how different from each other are the components of the intelligence vector. However along the diagonal from black to white, we can indeed assign a single scale the so called grey scale. And everyone will be able to visually compare them. It will take some training to be able to estimate the color vector components for various colors however in these days of computers it should not be too difficult to be able to find a program with which to play around. And indeed the results will be what we imagined above for the physical case, there are differences and they are quite noticeable. So the whole idea of whether to combine the components to produce a single number or to leave them alone may not be much more than a matter of taste. In fact, if anything they should both be done.
I died from the mineral kingdom and became a plant; I died to vegetative nature and attained to animality. I died to animality and became a man. Why should I fear? When did I become less through dying? Next time I will die to human nature, so that I may spread my wings and lift up my head among the angels. Once again, I will be sacrificed from angelic nature and become that which enters not the imagination. . . Rumi (Chittick, 1983).
And different weightings should be used just to see what kinds of differences it would make. In the specific case of CA or PQ since the various alleged factors or components of the tests are or would be highly correlated and not independent as in the case of the three primary colors they would all be increasing more or less together and it would correspond almost exactly to the case of the grey scale, so there is something after all to what the classicists claim. Since they are correlated (i.e. tendency to increase or decrease together) then this resembles something like the grey scale and we can make use of this idea to comprehend what these tests purport to measure. So there’s no serious difficulty with making sense of a scalar measure (i.e. a single number, say, Spearman’s g ). We can use analogical reasoning now to try to comprehend what this single number could mean, if we had a mind to produce such a single number. Indeed, it is an excellent example of the fact that although we can ‘see’ the grey number as something clearly related to black, we would not have been able to imagine that it is really being produced from red, green and blue. It is one of the miracles of the natural world; strange but true, just like finding order in randomness in chaos. But there is another simple way in which we can produce scalars from which we can get an idea of the colors. The problem of structure vs. process vs. multiple intelligences is a pseudo problem since the arguments are really about definition of intelligence. From the way it is explained it seems that by structure is meant really a state. In some ways the cognition view seems to be an attempt to solve the problem of intelligence by avoiding it, and the multiple intelligences view seems to be taking the vectors of mind view as is and refusing to go further. The only one that causes immediate grief is the difficulty of connecting the state view with process view since this is in general very difficult even in relatively well-tread fields such as thermodynamics. 2. Anomaly or contradiction? (data is evaluated according to a theory) A more serious problem is the apparent paradox of the fact that we have the largest b/B for all animals (where b ¼ brain mass, B ¼ body mass) and yet brain size among humans doesn’t seem to matter much from the evidence. Evidently either (a) there is something analogous to flab for the brain so that massive brains don’t necessarily imply high intelligence
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Figure 3. (a) Global pattern but lack of correlation at local scale: A situation where using an absolute (ratio) scale would yield correlation over a large scale as expected but fail over small scales (magnitudes). The figure is not drawn to scale but is only meant to be suggestive. The simulated data points would be much more closely clustered (horizontally) in real life. (b) The evolution of information content in genes and brains (after Britten & Davidsen (1969); see also Sagan (1977). Compare this to (a)
(b) or whatever intelligence is, the test doesn’t measure it but rather a narrow set of skills taught to students who are expected to have this core knowledge just to survive and be a reasonably productive member of this society during this century (c) or it’s the connectivity that is important so that the more efficient connections may be present in the brains of some individuals who have small brains and are anecdotally said to have been ‘smart’.
It seems as if the correlation between the Encephalization Index (EI) (Eccles, 1989) and intelligence holds at large scales (i.e. global scale) and does not hold at small scales (local scale). We do definitely find that the larger the EI the more intelligent the species. Why then doesn’t the relationship hold at local scales? Superficially there could be two reasons; the tests (instruments) do not possess the resolving power required or that the relationship is not linear thus linearcorrelation -regression (LCR) analysis does not divulge any information. However there are other reasons why the EI does not seem to correlate with intelligence at local scales (i.e. only for humans). We do know that more complex organisms also have larger brain/body mass ratio (Britten and Davidson, 1969; Sagan, 1977) 2.1 Artificial (machine) intelligence perspective: form, mode, and type As for the taxonomic structure of the skills that comprise what we call intelligence the first thing we note is that like a database there are different possible classifications, and that if they all seem to be just as attractive then they must be different conceptual views of the same thing which can possibly be all accounted for some day when we have better mathematical models. The standard models were reviewed in the beginning and we have yet more possibly taxonomies and also other pieces of evidence that points in the direction of a logarithmic scale. Some skills of problem solving are serial which would include what we call formal logic [and definitely its informal version that shows up constantly in verbal comprehension type questions], and some of the simple arithmetic (i.e word) problems. Others the most obvious of which is spatial visualization require a parallel mode of processing. The visual [nonverbal and parallel] mode of thinking was probably best expressed by Einstein. Since the number of brain states (in analogy with computer science sense of the word i.e. say, a state of a set of flip-flops of a real machine or the internal states of an abstract machine such as a Turing machine) increases exponentially with the number of neurons, and we expect that ability, in some sense, also increases exponentially so that we should use a logarithmic scale. As for the complexity (in the sense of number of components of the number of operations a machine executes in algorithmic complexity) of the brain and expressive power of a language there are good reasons to think that they should be multiplicative and that there are trade-offs in time vs. space complexity, for languages (see Hubey (1994)). Going back to standard computer paradigms, if we concede that animals can think (although at some lower level) we must also concede that thinking doesn’t require language [if the few tens of words that animals can recognize is not counted as language]. There may be natural spaces in which to represent intelligence, which means that we may yet provide some kind of a structure to it. For example it would be possible to represent many of the ideas in terms of a simplified three-dimensional space whose axes are
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Some kinds of questions require explicit knowledge such as mathematics, geography, verbal comprehension [grammar], and word fluency. Others are implicitly learned such as personal and interpersonal skills, physical coordination, and much of language. We might also call explicit knowledge much of what is taught in schools, and the implicit, what is learned without a formal education [which would include the so-called street smarts and also certain personality skills which would make for good manager or salesperson.] The last axis of the 3D space has to do with what might be called the difference in computation between batch vs. real-time or between I/O bound vs. computebound processes; it’s really a combination of both. Into this last category (axis) would fall such things as bodily-kinetic intelligence of Gardner, musical talents (i.e. ability to play an instrument), being athletically minded, and perhaps some aspects of personality. Those involved in real-time programming know that it is a task of difficult constraints. Similarly, coordinating physical activity and mental tasks (i.e as in team sports) is a rather difficult task i.e. high complexity. It is for this reason that music and dancing have calming effects; it stops the internal dialogue of Casteneda. It is for this reason that music might break some people’s concentration but improve others’. We can try to include what should really be a fourth dimension, that is essentially a memory fetch vs. computation in this third dimension, but only to make its comprehension easier since representing more than three dimensions is very difficult except purely mathematically. The possible fourth dimension [only for the purposes of simplification and exposition], that of the difference between a compute-bound process vs. one of memory fetch in computer science would be the difference between a complex algorithm vs. that of table look-up. In the real world of humans, the table- lookup has the analog of word fluency, and perceptual speed [of Thurstone’s PMAs]. Clearly it has to do with the organizational skills of the person, which naturally is about the organization of the knowledge of his brain, and hence his past which includes both formal and informal education. It is this which Thurstone probably calls M (associative memory). In Gardner’s world view, this would get split into spatial (since perceptual speed might have to do with spatial resolution and manipulation of objects in space), and logicalmathematical would also be in this category. Since all memory in the brain seems to be associative, and analogically based, this particular component is probably what we might call efficiency in another setting instead of the intelligence debate and is probably what we are measuring along with some basic knowledge that we presume every human should know. Continuing with this idea we can see that things are often measured as a product of two variables in which one is intensive and the other extensive. For example, in
thermodynamics/ physics work done is dW ¼ p·dV; entropy is dQ ¼ T·dS: The idea of intensive vs. extensive variables do have uses in many different areas. Training or education is probably something like dt ¼ x·dT where x is the intensity or quality of the training program and dT the extensive variable which is the amount of time spent in it. Problem solving ability is dp ¼ 1·dK where 1 has something to do with the inferencing mechanism or engine used, and K the knowledge base, for despite all claims to the contrary and protestations, we cannot separate the two completely, at least in the human brain, and at least for the time being. Knowledge of the world comes from our senses, and our inferencing about the world at large comes from our observations. In fact, we can see the same ideas being used in scoring in gymnastics and diving. The score is calculated by multiplying the raw score (how well performed) by an inherent degree of difficulty of the routine of the dive. Hence the measurement is really about a product of an intensive parameter (organizational effectiveness of the brain or its efficiency) multiplied by an extensive parameter which is knowledge. Please see appendix A.3 on Path Integrals and their connection to these ideas.
2.2 Potential and its realization In articles on intelligence (indeed almost any other characteristically human trait such as language) we often run into words which talk about human potential which has not been realized. It is often thought to be a single dimension in which the actual realization is simply a proportion of the potential (capacity). What seem like two poles of a continuum often turn out to be separate dimensions. The case of language turns out to be one of these. There are really two variables; capacity & existence of instruction. There is a window of opportunity for picking up language. We see that we are dealing with a product of variables since it is only a product which can create this. And in this case the simplest approximation is just a logical-AND. In other words, there must be both language-capacity (i.e. innate, inherited, potential) AND also there must be proper environment (i.e. instruction), so that language can be learned. The next level of approximation is simply using fuzzy logic concepts. As long as the potential is there (for example in mentally retarded children) and there is instruction, there will be some form of language. Indeed, IQ tests do measure language competence to various degrees and use it as a part of the test of intelligence. Combining the two, (i.e Knowledge Form, Computation Mode, Bound Type ) and the concept of potential from physics we might try a potentials of form C ¼ ae wF
f
þmM m þtT t
Q ¼ aF f M m T t
ð5Þ ð6Þ
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from which we can compute the vectors of the mind, and also derive single or multiple scalars using any of the ideas shown in earlier sections. The potential in Equation (6) is already multiplicative and Equation (5) becomes additive as lnðCÞ ¼ wF f þ mM m þ tT t after taking logarithms so that if we are interested only in adding up scores on various sections of the test without any compelling reason not to do so, the logarithm will relate these numbers to the potential. In the later case, the logarithm produces the standard form for linear regression. Without some data on what these mean it be pointless to speculate on the choice of functions however we should note that Q is multiplicative so that if any one of the components is zero Q will be zero, so that it already has built-in correlatedness for the components. It would tend to produce high scores for more well-rounded informal low level education [i.e. cognitive intelligence] whereas if there is a limit to what is possible because our brain is finite after all, then the high achievers would certainly be deficient in some areas and stronger in others which would be exaggerated by the exponential form of F given the right coefficients, so both forms are flexible enough for creative use. Even a simple multiplicative model is much closer to the truth than the standard linear regression models. Many things having to do with psychophysics is best modeled by a power law, and the sigmoidal functions, as pioneered by Rasch (1980) seem to have much success. Other sigmoidal models can be seen in Hubey (1987) and stochastic models in Hubey (1991a) and also below. We still have the problem of obtaining the actual/real from the potential (which is the concern of learning theory).
2.3 Functional view of the psychology/biology of learning, and intelligence It’s reasonably clear from all the evidence that whatever it is that intelligence tests measure whether it should be called the Intelligence Quotient or Cognitive Ability or Problem Solving Capability, or Problem Solving and Creativity Scale or whatever can be changed/affected via training, emotion, poverty in other words environmental influences. Even after scaling things correctly, we are still left with variation among humans. It might be argued despite the evidence that it still means something and needs an “explanation”, in other words, some simple theoretical model. It is not difficult to produce a very simple model that hopefully will not do terrible injustice to the idea of intelligence. We know that our memory is associative. Memory events seem to be linked to other events and we can recall almost everything with some prompting. We might make an oblique reference here to artificial intelligence programs in which an inference engine is working on data so that we may liken problem solving to having an inference engine (naturally not necessarily localized in some part of the brain but possibly scattered about) fetching data and doing some kind of a search (breadth first, depth first or some combination thereof or something completely unknown to us yet). Of course it will take time to do all this.
Let us call the time it takes to do this Tc, for complete_search_time, without implying that the search does not include conventional computation i.e. problem solving. Suppose now that over a period of time we’ve built up (via formal or informal education) a large bag of cheap tricks which is also kept in storage someplace. We can think of reasoning as analogical reasoning in which we solve problems via analogy to problems resembling the given problem in one or more dimensions and that we kind of keep a mental template of such solved problems in memory (which we might imagine is functionally being kept someplace separate from the rest of the memory). Thus if we first are able to find the template for the given problem in this memory of pre-solved problems [premem] we can ‘solve’ the problem much faster than we could have if we never had encountered problems of this type. The truth of the matter is that there are really no completely original or novel problems that can be presented at any of these tests. And solving some of them really revolves around guessing what the tester wants to get as an answer. Therefore the time to solve a problem if we can find an analogical match in our pre-solved memory is much shorter than if we treated the problem as completely original and tried to be creative in its solution in which case might never be able to solve it at all. The time to solve a problem, then having this highly-simplified two-tier memory system then drops to T s ¼ HT p þ ð1 2 H ÞðT p þ T c Þ
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ð7Þ
where H is a probability of finding this solution in premem [pre-solved memory]. This is essentially what is being referred to as “chunking” in learning theory and in artificial intelligence.Thus if we do find it there, the answer is very quickly found, which takes time Tp. The time to find the solution if it’s not found in this pre-memory fetch is that time spent doing this plus the time spent actually solving it via supposedly original methods. Naturally, this simplification is so gross that we should not expect anything beyond the simplest kind of description of matching reality. First, there really is no such
Figure 4. Memory levels: A simplified functional view of memory needed for solving problems and the role of learning
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thing as these two memories locatable anywhere in memory but there’s no need for it; the connections must behave something like it. Secondly, we’d have a tough time solving very original problems; if anything the problem we have is in finding a good match for the problem at hand and trying to force fit couple of problems together or cobbling solutions from several such virtual templates; it is this efficient time that we’ve called Tp. In any case, the tests don’t give us time to find the solutions but rather give us a fixed amount of time in which to solve such problems, so the assumption behind this idea is that we’ll be able to solve less problems in this fixed amount of time if we cannot find many of them in our premem. In truth all of us who are alive have some small virtual memory in which already solved problems from life are stored (naturally not necessarily in some localized region of the brain) so that the time it takes for us to solve the problem compared to some hypothetical baseline of the finely-tuned problemsolving brain would be of the form
l ¼ T ¼ T s =T p ¼ H þ ð1 2 H Þð1 þ T c =T p Þ
ð8Þ
We should note that this solution time, T, is equal to H if H ¼ 1 and is equal to 1 þ T c =T p if H ¼ 0: The case of H ¼ 0 corresponds hypothetically to a situation in which we are faced with a problem for which we have no handles. In this case the factor Tc/Tp is something that corresponds to the inherent originality of the problem, at least to the subject. We should suspect that l should be a large number since very few people [almost no one] are actually creative but rather partially creative; we may cobble together solutions to new problems by combining several old ones. We solve large problems by cobbling together solutions to a bunch of smaller component problems. The process is iterative, and hierarchical since the same types of solutions can be used at different scales, hierarchies or levels. Most people probably cannot even do that unless they’ve been trained to do so. In any case, highly educated people, especially those who’ve studied mathematical sciences will have high H values since they probably have already solved symbolic problems of the type found on tests many times over. Similarly, questions such as “what is the opposite of. . .” will be easier for those children raised by parents who are highly literate than those living in “tarzan neighborhoods”. We should really consider not T but another quantity t ¼ 1=T if we want to consider the values as normalized. We then have
t¼
1
1 ¼ Tc fH þ ð1 2 HÞð1 þ lÞg H þ ð1 2 H Þ 1 þ Tp
ð9Þ
The plot shows that for H ¼ 1; t ¼ 1 since at that time we have reached maximum efficiency since every problem presented to us in the test is already present in our pre-solved memory and we need only to fetch the answer. We
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should also note that the most rapid increases occur for large l, which is exactly as it should be since it implies that the ratio of the learned solutions to the searching/groping for solutions using all ingenuity and creativity is high, meaning that if the problems are of the type that would be very difficult to solve without being exposed to problems of this type, then the steepest increases come near H ¼ 1 when we can find the solutions in the pre-solved memory. It’s possible that the large brain individuals may be capable of more original lines of thought, capable of more creative lines of thought, have more memories built in. It’s also possible that the so- called intelligent beings such as mathematicians or novelists, or philosophers were merely one-dimensional experts in small domains and managed to score high on these tests particularly because they were trained for these tests. In particular the tests might overemphasize classification which is a large component of education, especially in the ‘soft sciences’. It is said that an expert knows everything about nothing and the generalist knows nothing about everything. This is simply an example of trade-off as can be observed in many fields (Hubey, 1996), and also Appendix A.2. Most people would naturally have to fall somewhere in between. In yet another sense we can consider the effect of H plotted against the amount Sp/Sc where SP and Sc are proportions of the memory devoted to the two different types of problem solving modes and their associated memories, where we’ve assumed that there must be some kind of a parameter V which has to do with the organization of the brain. If knowledge is organized so that there’s a method to the solution searching mechanism instead of being a cut & try method that an unsophisticated person might attempt, the probability of finding the answer (or something close to it) in the faster premem H will increase. Hence we might think of V as a kind of efficiency of the brain as far as
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its organization goes. It’s also possible that this could point to overorganization in the sense that it will be good only for solving the types of problems given on such tests. As can be seen if there was absolutely no efficiency raising mechanisms or learning by experience, hence no localization of memory (i.e. associativity) then the increase in H should be about linear with SP/Sc. There should be a higher rate of increase of H with SP/Sc if the learning mechanism was efficient instead of simply being rote training. In all likelihood memory (that is, neural net) organizes itself in some manner which is captured in an extremely simple way by these equations. The early methods of solving problems are much closer to the parts of the triune brain (MacLean, 1973; Jerison, 1973) so that they become automatic means or fall-back methods, and thus the increase in the likelihood of finding the solution to problems such as those given in various IQ/CA tests greatly increases performance. This “organizational efficiency” of the brain has been captured in the single parameter V. Other thoughts on functional descriptions of memories of living entities include the procedural vs. declarative memory (Squire, 1983), working and reference memory (Olton, 1983], and associative and recognition memory (Gaffan and Weiskrantz, 1980), which like the present work is borrowed directly from computer science. For trade-off type relationships in many fields of science, and epistemology, see Hubey (1996). 3. Mathematical analysis of proposals The previous section was a purely functional view of the role of learning in problem-solving, but IQ/CA is not supposed to be learned but innate/hereditary/genetic. If intelligence cannot be learned, what exactly, then, is IQ? To answer this we must first ask what intelligence is. IQ is a normalized version of intelligence. The question has obviously been asked and answered in different ways in the past. In binary form the answer is the Turing Test. To know what intelligence is in nonbinary form we should try to
Figure 6. Effect of localization of memory and specialization: With early learning there is more efficient organization of the brain for certain types of tasks and thus leading to higher V than for late learning
delineate its properties. Some of this was already done in the beginning in the literature review. In this section we can try to produce answers from other points of view, ignoring the previous section and re-starting new thread by examining the standard arguments but evaluating them from different perspectives. Historically, the brain/mind was always described by using the highest technology available as metaphors for understanding and what we are attempting to understand or describe is a function of the brain/mind. The mind was likened to clockworks, then the telephone switch, then the digital computer and finally artificial neural networks. The memory part was likened to holograms, and the associative memories of computer science are still used as analogies. The computational paradigm is still rampant, and the concepts of state and process come from this view. However since the brain/mind is a very complex thing, there is yet one more analogy we can make, and that is to databases which have different conceptual (often called logical) views. The multiple view perspective is taking hold these days even in operating systems. Since analogies are always single dimensional, it is not surprising that something as complex as the human brain/mind (the three-pound universe ) can be seen to be like so many things. Since we don’t yet understand it whole but only its parts, we can liken ourselves to the story of the four blind men and the elephant. There are other questions we can ask regarding its properties. Is it an extensive property or an intensive one? Is it like temperature or pressure (i.e. an intensive function) or is it like volume/capacity/mass/internal energy? The answer to both is that it is probably a product of both! Not only is the problem solving ability a function of some kind of an efficiency of neurons or organization of the brain but also of the pure mass or amount of neurons. If it were not so, animals such as reptiles would be as intelligent as humans. On the other hand if we claim that since we are only considering humans, and since the brain masses all fall into the same range, we should consider this constant, then we still have to deal with whether IQ is intensive or extensive purely from the consideration of whether it depends on knowledge (extensive) and also on some kind of an efficiency of processing or creativity in solution finding (intensive). Therefore we still cannot escape the bind of choosing one or the other. It is most likely a function of both and hence it must still be multiplicative function, aside from the problem of being a path function and not a point function. On the basis of the foregoing we can find at least four serious problems with the attempts by which psychologists so far have tried to capture the idea of intelligence, aside from the ones that have already been discussed in the literature and earlier in this text. .
What kind of a quantity is intelligence? Is it binary or measurable on some scale? What kind of a scale is appropriate? Is it an ordinal, interval, or an absolute (ratio) scale?
.
Is it an additive function of its constituents, the most important ones for purposes of simplification being hereditary(nature) and environment
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(nurture)? Or is it a multiplicative function? Is it logarithmic function, an exponential function or a polynomial function of its variables? .
Is it a vector/tensor function or a scalar?
.
Is it a point function, or a path function? In other words is it a state or a process? Is it a quality or a quantity? Is it an extensive variable or an intensive variable?
We all recognize that genetic influence can be spread diffusely among many genes, and that genes set limits to ranges; they do not provide blueprints for exact replicas. In one sense, the debate between sociobiologists and their critics is an argument about breadth of ranges. For sociobiologists, the ranges are narrow enough to program a specific behavior as the predicatable result of possessing certain genes. Critics argue that ranges permitted by these genetic factors are wide enough to include all behaviors that sociobiologists atomize into distinct traits coded by separate genes.Gould (1981), 329.
It’s clear that all of these questions are not independent of each other but related to one another. If this thing called intelligence is to make any sense it should be comprehended and comprehensible in a broader context. It is paradoxically true that sometimes one can find solutions to problems by generalizing them and looking for more general solutions since that enables us not only to locate the phenomena in its proper space relative to related ideas or objects but also allows us to use more data as evidence to better grasp the constraints to be imposed on the phenomena. This intelligence scale should encompass and allow us to measure intelligence of fleas, as well as that of chimps, humans and also machines. Common sense says that the scale should be logarithmic in order to accommodate the vast differences in intelligence but also because many laws in psychophysics are power laws. Logarithmic transduction of inputs allows for a greater range of sense perception without a proportional increase in the size of the organs. Furthermore, if this scale is to be something like the temperature scale, then absolute zero should belong to something like viruses or simple computer programs. Furthermore ideally this scale should be an absolute/ratio scale instead of simply an interval or an ordinal scale. A highly mathematical treatment of the subject of scaling going back to Campbell (1920) can be found in Suppes & Zinnes (Luce et al., 1963). 3.1 Heritability: why is the “intelligence function” not additive? First problem with the Linear Correlation-Regression Models (LCRM) is that it is highly unlikely that intelligence is an additive function of environment and heredity since additive means logical OR and not AND. So therefore the verbal expression that intelligence is a function of both environment and heredity is being twisted out of shape as soon as we try a linear additive model. As is well known, AND is represented as multiplication, and not necessarily only in
bivalent logic or even fuzzy logic but even in modeling via differential equations, for example in the nonlinear Lotka-Volterra models the interaction is multiplicative (see Appendix A.5). Various types of infinite- valued AND functions can be found in Hubey (1998). The sigmoidal function is produced quite naturally in the nonlinear differential equation modeling of forced binary discrimination of phonemes in Hubey (1994). Biological determinsim] is fundamentally a theory about limits. . . Why should human behavioral ranges be so broad, when anatomical ranges are generally narrower?. . .[I] conclude that wide behvioral ranges should arise as consequences of the evolution and structural organization of the brain. Human uniqueness lies in the flexibility of what our brain can do. What is intelligence, if not the ability to face problems in an unprogrammed (or as we often say, creative) manner? Gould (1981):331.
Additivity implies that the environmental and hereditary components are grossly substitutable for one another which is simply untrue. No amount of teaching will make a chimp into a human. There is no question that the model should be multiplicative. The model cannot be additive since additivity logical translates to OR and nobody would really dispute that environment and heredity are not grossly substitutable for one another. If it were so, we could be teaching calculus to dogs by enriching their environment to make up for their genetic deficiency. The coefficients no longer mean what they meant in linear regression. If we are looking for the magnitude of variation of intelligence with the factors the two cases give fundamentally different results because if we have I ¼ f ðE; GÞ then
dI ¼
›f ›f ·dE þ ·dG ›E ›G
ð10Þ
For the linear case I ¼ F þ 1E þ kG
ð11Þ
dI ¼ 1·dE þ k·dG
ð12Þ
I ¼ aE e G h
ð13Þ
dI ¼ ðaeE e21 G h Þ·dE þ ðahE e G h21 Þ·dG
ð14Þ
the differential (i.e. variation)
For the nonlinear case
the variation/differential is
As can plainly be seen from the form of the multiplicative (i.e. AND) dependence the powers of G and E essentially determine the sensitivity of the
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intelligence to variations in environment and heredity. For the linear case the respective coefficients do determine the sensitivity of intelligence to the factors, but for the nonlinear case (which is the correct case) the respective coefficients no longer mean what they meant for the linear case. The model must be multiplicative. (See Appendix A.7 for some paradoxes.) The simplest such model accounting for environment and heredity would be of the multiplicative type which is interpreted as a logical-AND (i.e. conjunction) Therefore the linear regression could be done via using the logarithms /would have the form lnðI Þ ¼ lnðaÞ þ e·lnðEÞ þ h·lnðGÞ
ð15Þ
Immediately, we would see that all the numbers that were measured would get smaller and hence the variances. However, that is not the only problem (see Appendix A6 for correct computation of variation and Appendix A4 for the conditions on the functional form). The argument that the present testing methods and models are only for “human level intelligence” where the linearity is valid does not hold water for there are standard mathematical methods to deal with such approximations. We simply expand the function in a Taylor series and attempt to regress about some point which we may claim is some average human level genetic and environmental condition and that the function is approximately linear about that point. For example, if we suspected some general form such as I ¼ f ðE; GÞ; then we can expand ›f ›f I ¼ f ðE h ; Gh Þ þ ðE 2 E h Þ þ. . . ð16Þ E¼E h þ ðG 2 Gh Þ ›E ›G G¼Gh which for Equation 13 above is h e h21 I ¼ aE eh Ghh þ ðE 2 E h ÞðaeE e21 h Gh Þ þ ðG 2 Gh ÞðahE h Gh Þ þ . . .
ð17Þ
Rearranging terms and simplifying we obtain I ¼ F þ DE þ LG
ð18Þ
h e h21 where F ¼ aE eh Ghh ð1 þ e þ hÞ; D ¼ aeE e21 h Gh ; L ¼ ahE h Gh : In order to make Equation (16) linear we dropped the higher order terms in the Taylor series to obtain Equation (18). However the linear correlation-regression analysis computes the values of the constants F,D and L which parameters are no longer indicative of the effect of the self-variables since they are now functions of the other variable. In order to offset this dependence we would have to use the normalization E h ¼ Gh ¼ 1 thereby computing the coefficients að1 þ e þ hÞ; ae and ah in the linear regression. We can then solve for a, e and h from the three equations. If we do solve for these coefficients in terms of the regression values F, D and L we obtain the results:
a¼F2D2L D e¼ F2D2L D h¼ F2D2L
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If we had, say F ¼ 2 and if D þ L ¼ 1; then the above works out only to rescale the parameters since we would then have e ¼ D and h ¼ L so nothing would really change. If F , 1 we’d obtain negative correlation and we cannot allow F ¼ 1 since the numerator would then be zero. However, if we had used another scale, say the one in use right now (i.e. E h ¼ Gh ¼ 100) everything would not work as above. Something which depends on a particular choice of interval scaling for its truth cannot be correct. We do not know if the present IQ scaling is meant to be an interval scale or an absolute scale. It is through problems like this that Kelvin’s research led to the postulation of an absolute temperature scale. (Please see appendix A1 and Appendix A6). For more on fuzzy logic and differential /and the meaning of multiplication and nonlinearity, please see appendix A. 4.) 3.2 Problem of dynamics in measurement and attribution of causality We should note there is another complication since the real complexity of the problem is in the dependence of the variables on one another since they can be functions of one another. For example, if we are traveling in an airplane from Maine to Florida starting up at around 9:00 AM and taking measurements of the ambient temperature, the rate of change of temperature we’d measure is that not only of the spatial variations in temperature (north-south) and also the temporal variations since the air would start to warm up after the sun comes up and will be reaching a peak say around noon. Since we have the temperature q ¼ qðxðtÞ; tÞ where x is the distance traveled starting form Maine then the rate of change of the temperature measured (recorded by the instrument across time) is dq ›q ›q ¼ þ v· ›t ›x dt
ð20Þ
where v is the velocity of the airplane. The first term is the purely temporal rate of change of the temperature (which is due to the warming of the earth because of the sun), and the second in which the term v· ››qx appears [which is the spatial change], multiplied by the velocity of the airplane gives the change due to both the actual spatial variation and the rate of sampling of this spatial thermocline. For the case of measuring intelligence (whatever it may be) we don’t know that the variables we have selected are really independent. For example suppose we have y ¼ yðM ðtÞ; V ðtÞ; t) [where M¼ mathematical, V ¼ Verbal, and t=
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training i.e.formal or informal education]. We know that verbal ability is important because without it we can’t even give these tests. But are we sure that mathematical/symbolic/quantitative reasoning is not important for verbal comprehension? What exactly is the relationship between the two? In terms of the neural networks underlying these they are both handled by neurons, although there is lots of evidence of localization of speech, spatial reasoning etc. (for example Gazzaniga, 1985; Sperry, 1988; LeDoux, 1977). However our main concern now is in mathematical formulation of the problem. Since speech and visual ability are developing in infants simultaneously, in all likelihood threedimensional spatial comprehension and its verbal articulation probably go hand in hand although people seem to start early into developing some modes more than others, for example, spatial orientation, verbal fluency, physical development. In the study of any scientific discipline it is necessary, in the beginning stages, to use words whose precise meanings may not be defined, but are accepted as defined in an intuitive sense as starting points. On the basis of the ideas and concepts derived from these basic terms, a theory begins to develop and then it is possible to retrogress and give precise quantitative definitions to the words and terms defined only verbally. Perhaps, the best example of this process is in the field of thermodynamics. Concepts such as heat, temperature and pressure were properties only physically felt and intuitively understood. After thermodynamics was put on a theoretical footing, the concepts of temperture, heat and pressure were defined operationally (mathematically on the basis of the developed micro (kinetic-statistical) theory of thermodynamics. Hubey (1979).
4. Putting it all together: effect of learning and timing of learning on potential Many things which are accepted to be a part of the “natural” (how this word is abused probably will take a book to explain) growth/maturation of humans are all due to learning. For example, in very early ages, we are told that it is quite “natural” for children to have pretense play and to invent objects and people. In all likelihood, this is due simply to the fact that the infant still has not made the strong differentiation between sleep/dreams and wakefulness. The child falls asleep in one place and wakes up in another (for example in the car, at the beach, or in someone’s arms). This is probably no more mysterious at this time than being in one wonderland (in sleep i.e. dreaming) and then waking up to another reality in another place. At the same time, if it is talking to dolls or toys or dogs, it is still learning that somethings are alive and move on their own accord, some are toys and are run by batteries and that somethings that move (i.e. toys or animals) do not speak. Another stage in growth/development is when it still does not understand for example the concept of picture so that if we tell it to “do a truck” it might mimic driving it instead of drawing the picture (Gardner, 1991). But of course, does the child at that age understand that small
iconic representations of objects which it sees on TV, or a book got there by various means such as a camera or being “drawn” by other human beings? It is simply ignorance, nothing more. If someone draws a picture in front of his eyes (not a bad picture since it might not be able to make the connection well at that stage) it might think that the pictures on TV are also drawn or it might think that there is a little guy inside a Polaroid camera like the German peasants during the last century who thought there were horses inside the locomotive. For the case of measuring intelligence (whatever it may be) we don’t know that the variables we have selected are really independent. For example suppose we have c ¼ cðM ðtÞ; V ðtÞ; t) [where V ¼ Verbal, M ¼ mathematical, and t = training i.e.formal or informal education], then the variation in the potential is
dC ¼
›C ›C ›C dM þ dV þ dt ¼ CM dM þ CV dV þ Ct dt ›M ›V ›t
ð21Þ
where we denote the partial derivatives by subscripts, so that if we wanted to know the change in this potential with respect to training (which would naturally affect the measured intelligence) we’d need to compute the total derivative dC ›C dM ›C dV ›C ¼ þ þ ›M dt ›V dt ›t dt
ð22Þ
It’s possible that dC=dt ¼ 0 since we cannot think now of how C could change if neither M nor V changes (assuming that these are the only factors/variables we’ve identified). In the general case, naturally, there’d be more variables. But in truth things are more complicated; it may be more like c ¼ cðM ðV ðtÞÞ; V ðtÞ; tÞ: It’s obvious that we can’t even get across the problem let alone the solution without language so that V will definitely affect M. In this case we have dC ›C dM dV ›C dV ›C dM dV dV ¼ CM ¼ þ þ þ CV þ ct ›M dV dt ›V dt ›t dt dV dt dt
ð23Þ
where t is a proxy for the environmental richness. It seems at this point that we can get stuck in infinite regress since if M ¼ M ðV Þ we may then need to write V ¼ V ðM Þ ¼ V ðM ðV ÞÞ ¼ V ðM ðV ðM . . .ÞÞÞÞ if we cannot separate the influence of V directly and via M and if V is also a function of M. In some problems we can measure this, say, in industry analysis in economics. In any case, we can see the effect that this will have on computing the gradient of the potential to derive the vector function. Or we might have a more complex case such as that of having a potential of the form C(M(V,t ),V(t ),t ) or even C(M(V,t ),V(M,t ),t ). There is even a more serious objection to linear correlationregression analysis. We can see immediately that new memories are built on
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top of old ones, and that learning to solve problems is just as good as being creative, and in many cases, the learning eventually outstrips creativity and that is naturally the reason why the testing for IQ stops at adulthood since everyone has already pretty much learned what there is to learn, and that if the same trends continued, we should be asking questions on algebra, trigonometry and calculus on the IQ tests given to adults, say college students or graduates. The fact that it is not done is a testimony to the simple fact that the tests also test for knowledge. Furthermore the earliest memories should count more heavily, even in using the standard IQ tests since new memories are built on old ones, and thrusting someone into a new socioeconomic (SE) class is not the same as having someone in that SE class since childhood. In fact, there is probably a lag of several generations at least, and probably centuries as can be seen in the long cycles of histories of countries and empires. Therefore we already know that IQ is a path function and not a point function. Again it is not a state function but a process function, where state and process are used in a more general sense than in computer science or in psychology, or in mental testing (please see Appendix A.3). The earliest such models come from thermodynamics. And it is also from thermodynamics that we have the ideas of extensive vs. intensive properties of systems. The standard example of a path function is the length of a curve on a plane. The standard example in physics comes from thermodynamics in which the heat rejected or the work done of a system depends on the path that the process took and not a function of the end states of the process. We can thus surmise that intelligence will be a function of the time, and the path that the environment variable, say, E(t ) took during this time. Advantages of mathematical or analytical models-unambiguity, possibility of strict deduction, verifiability by observed data-are well known and make them highly desirable in systems engineering. This does not mean that models formulated in ordinary language (e.g. verbal descriptions) are to be despised. A verbal model is better than no model at all, or a model which because it can be formulated mathematically falsifies reality. Indeed theroies of enormous influence such as Darwin’s Theory of Selection, were originally verbal. Much of psychological, sociology and even economics today is still descriptive. Models in ordinary languaeg, therefore, have their place in system theory. The system idea retains its value even where it cannot be formulated mathematically, or remains a “guide” rather than being a mathematical construct. Hubey (1979)
Thus in addition to the problem that what we purport to measure, say, verbal skill V which may be a function of mathematical or spatial skills M or vice versa, we now have a bigger problem of the form of the function itself. The significance of this cross-dependence of variables is obvious if we think of the fact that some proposals have been put forward that the richer environment itself is a function of genetics, i.e. I ¼ f ðEðGÞ; G; tÞ (see for example Plomin and Bergeman (1991) for a review). Obviously, this is true on a global scale and the
derivatives of this function will have to be calculated in the same way as that of Evolution of Equation (23). More on this can be found in Appendix A6 and in the conclusion intelligence section. We can approach this problem a little differently: if we have say some learning ability, L, we can see that it will in all likelihood be a function of time, since the earliest years are the most important and by old age very rarely can people retain their mind’s elasticity of their early years. However, here 387 intelligence is a function of this learning ability for which we use time, t as a proxy. Furthermore an enriched environment is essential, and the earlier this rich environment is provided the better it is, so that we can attempt to surmise the form of the functional dependence of intelligence on environment. We have I ¼ f ðG; EðtÞ; LðtÞÞ ¼ f ðG; EðtÞ; tÞ where we have accepted that t is a proxy for the learning ability, and that E(t ) and G are representative of or proxies for environmental and genetic variables, respectively. We expect that we should have f E . 0 since with a more enriched environment we expect increases in intelligence. Similarly we expect that f t . 0 if people are measured on the same scale since problem solving ability should increase with age (please see Appendix A.4). Note that we are not discussing IQ which can be obtained from intelligence via normalization. The intelligences of people have been increasing over the past half a century or so. One of the reasons, of course, is that the environment itself has been changing. Not only has the educational level gone up, but the environment itself (i.e. the standard of living) has gone up, and thus children are being exposed to more things and are getting better care both healthwise and nutritionwise. Consequently, not only do IQ tests measure knowledge [albeit claimed not to measure knowledge at all but some kind of “innate/genetic” capacity] but whether it means an intensive variable or an extensive variable or a product is not clear in the literature since it hasn’t been discussed with respect to any model except in terms of regression or correlation coefficients. The tests also measure a quantity which is a function of the path (the history of the individual in a particular changing environment). In other words, the words process and Figure 7. state are not necessarily to be understood only in the sense made popular by Realization of potential: the emergence of the digital computer as the metaphor of choice among The potential (heredity) philosophers and scientists working in the intelligence/knowledge field but for speech is there for all humans but if the window opportunity for learning language passes language cannot be learned. Meanwhile, although instruction (environment) is given to animals they cannot learn to speak. The dividing lines are arbitrary and merely suggestive
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rather in a more general sense in which thermodynamics made popular. We should construct a path function for the dependence of intelligence on environment. The simplest path function is the length of a curve, and it is an integral. Taking a cue from this we may try a very simple function of this form for intelligence to be of form; Z I ðE; tÞ ¼ hðEðtÞ; tÞdt ð24Þ However it would be preferable to derive such a function from more basic considerations instead of producing it out of thin air, like the standard linear correlation-regression analysis. Otherwise we could be accused of behaving like the man who was looking for his lost keys in his yard because there was more light there. Even worse, we could be accused of behaving like the little boy, who, given a toy hammer, discovered that everything looks like a nail. Since the process of acquiring intelligence (as measured in some fashion by standard intelligence tests) is a dynamic process we should turn to differential equations. In the case of a simple differential equation, a first order ordinary differential equation, given by y0 ðtÞ þ bðtÞyðtÞ ¼ f ðtÞ
ð25Þ
has the solution yðtÞ ¼ e
2
R· 0
bðsÞds
Z
t
f ðtÞe
2
R· 0
bðsÞds
dt þ yð0Þe
2
R· t
bðsÞds
ð26Þ
0
We note that for a constant a, it reduces to Z yðt 2 tÞ ¼ f ðtÞe 2bðt2tÞ dt þ yð0Þe 2bðt2tÞ
ð27Þ
so that the Green’s function (the exponential function which is the kernel of the integral) of the convolution integral is, in a sense, a weighting function, since it assigns an exponentially decreasing weight to the earlier forces that affect the system. In contrast, the weighting function that should be used for the effects of environment on intelligence should give greater weight to the earlier times since it is now common knowledge that brain damage can occur even in the womb due to effects of simple things like smoking cigarettes. Since the fastest growth for the brain, as well as the body occurs during the early years, it is a foregone conclusion that it should be so. In this case, in addition to the fact that the basic model should be multiplicative, the environmental factors should be a path function, something that accounts for the effects of the environment at different phases of development.
Clearly the differential equation model which had “feedback” can be modeled as a “black box” and we can find out from the inputs and outputs how the internal mechanism of the “black box” works. The black box models in the social sciences are strawman arguments, not against behaviorism but against science. This area of “identification” of the system (i.e. black box) and prediction based on its is a well-developed science. The “black box” model (i.e. Equation 27) is “time-invariant” in that the parameters of the differential equation are not time-dependent. In the real world, the behavior of intelligent entities changes with time; it is a part of learning. These changes require nonlinear differential equations and this topic is discussed briefly in appendix A.6 (Meaning of Nonlinearity). We know that as children grow, not only do their bodies develop but so do their brains and minds. A simple example of growth is given by the differential equation dy ¼ kð1 2 yðtÞÞ dt
ð28Þ
We assume that the growth rate of the child is proportional to the size yet to be achieved; that is, it grows faster when it’s smaller because the size yet to be achieved is much larger than in adolescence when it has reached close to its adult height. The brain also grows at similar rates and we can take this equation to be a simple model for the development of intelligence for a start. As is well known the solution consists of an exponential approach to a final constant value. The coefficient g can be seen in this plot to control the rate at which the child approaches its final adult height, and would grow faster for larger values of k. Of course, this is a simplified model and does not take into account the fact that there are spurts in growth rates around puberty. If anything a large g would indicate a precocious child, especially if its intelligence were to increase at the same rate. At this point we need to consider other global effects in what to expect. On a large scale, as can be evidenced every day, we see that except for humans all other animals seem to have a limit of intelligence and capability/capacity beyond which they cannot advance. We already know that a multiplicative formulation is needed, therefore we need to combine this idea with the dynamics of intelligence. On large scale over time we expect to see what is shown in Figure 9(a). It would seem on a global (large) scale that intelligence is definitely genetic. No dog will ever talk and no chimp will ever do calculus. So then why do the statistical tests give results that intelligence is 40 per cent to 80 per cent genetic instead of more like 99.99999 per cent? As a simple example of the kind of mathematical equation for the above we can try I ¼ Ax ð1 2 e 2kt Þ
ð29Þ
The real question then becomes, exactly what kind of function of heredity or environment are the parameters Ax and k? As can be seen from the plot and the
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Figure 8. Various “Black Box” Models: It is thought by some that the last two models with a feedback loop from the environment is not a part of the “black box” method of science and cannot be handled by standard mathematical tools. Of course, that is not how it is practiced in real life. See Appendix 17
equation, the parameter k determines how fast the intelligence of the subject increases in time toward its potential which is apparently mostly genetically determined. But if we examine these plots at small scales (or higher resolutions) say only for humans then we see something like this (again simplified). Of course, in reality both Ax and k fluctuate and although the plots do not show them a typical sample function could crisscross across others. In any case, now that we look at it at small scale (and high resolution) we have other means of interpreting the differential equation that gave rise to this
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Figure 9. (9a) Increase in intelligence of various species after birth on an absolute intelligence scale. The slow increases up to some limit are typical of exponential curves. (see appendices .3 and .4). The initial condition is not really zero but is drawn this way for simplicity. (9b) Variations in the parameters of the intelligence model for a given species. It is evident that both parameters Ax, and k are functions of both heredity and environment. The arrows show the “track jumping” behavior of Head-Start type programs in which change in the environment puts the child into a different sample function (i.e. path)
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dx ¼ 2kx þ kAx ¼ kðAx 2 xÞ dt
ð30Þ
The rate of increase of intelligence is proportional to the intelligence yet to be achieved with g being the constant of proportionality. The intelligence limit on the whole is determined genetically but it acts as a kind of an attractor of expectation from the child. In other words the difference ðAx 2 xÞ can easily be thought of as motivation. Does k then denote the genetically determined factor (i.e. a rate of increase)? Since we already have much evidence that this constant (is not constant but varies) can be changed with more attention and greater quality and quality of teaching and practice it cannot be a purely genetic factor either. It could be genetically determined on the whole but it is also a factor of both heredity and environment. So then we are led toward the complete model which could be of form; I ¼ aE 1e G hh ð1 2 le 2lE
1
G ht
Þ
ð31Þ
This is clearly the solution of the differential equation for intelligence d I ðtÞ þ lE 1 G h I ðtÞ ¼ laE 1eþ1 G hhþh dt
ð32Þ
which although simple and linear still has basically all the right ingredients to be a model of dynamical learning in the nature-nurture environment. From the solutions of first order ODEs of this type (e.g. Equation 25 through Equation 31) the coefficient of I(t) (i.e. l E1Gh) determines the rate of increase therefore it is the part that represents the interaction of the environment with genetics. The limit intelligence (l Ee1Ghh) is achieved eventually but if this were completely independent of the coefficients e and h it would mean that all this interaction has nothing to do with intelligence and two other coefficients representing something else (i.e. e and h ) determine final components of intelligence. The important part is that the multiplicative interaction of G and E is modeled. Variations on this theme can be seen below and in Appendix A.4. 4.1 Much ado about nothing? In mathematical modeling, it is really the equations that talk. However the meanings of these equations has been discussed throughout the exposition. By changing the scales to the natural (absolute) scales, and by making the global intelligence/behavioral parameter multiplicative with genetics factors and by examining the behavior of this function in the neighborhood of human level behavior we can unify much of the work done on such topics as has been done. We can improve the model above by making appropriate changes, pointing out its relationship to the constraints that must be satisfied by such models, and by connecting it to the standard analysis of such problems in the literature. We see
clearly that Equation (31) is simple linear (but dynamic) realization of a more general form I ¼ bG h ·Fn ðEððtÞ; tÞÞ
ð33Þ
where f(E(t ),t ) is a path function examples of which are given in Appendix A.4. The reasoning to obtain the differential Equation (32) was already given, but there are criteria/constraints that it must satisfy to be a good representation of the genetics-environment interaction. Since E is not constant but heredity is fixed at conception (at least at present) a more general (and slightly different) version of Equation (32) is d I ðtÞ þ lG h E 1 ðtÞI ðtÞ ¼ laG hþh E eþ1 ðtÞ dt which by virtue of Equation (26) has the solution Z t Rt RD lG h E 1 ðtÞdt lG h E 1 ðtÞdt 0 s I ðtÞ ¼ laG hþh e E eþ1 ðsÞe ds
ð34Þ
ð35Þ
0
There is yet more to the power that this simple linear differential equation hides. Integrating it once and rearranging terms we obtain the integral equation Z t h I ðtÞ ¼ KðtÞ 2 lG E 1 ðsÞI ðsÞds where 0
KðtÞ ¼ alG hþh
Z
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ð36Þ
t
E eþ1 ðsÞds: 0
The interpretation of this equation is exactly what is claimed by most researchers in the field, namely, that intelligence at time t, that is I(t ) is a function of the past interaction of intelligence with environment summed up over time from time zero to the present time t. The K(t ) term is also a multiplicative function of environment and genetic interaction and its position is reminiscent of the differential equation formulation in that it seems to be some ultimate potential for a given environment and genetic makeup (for all humans) toward which all humans grow. Obviously, Equation (32) can easily be cast in integral form as above and has the same interpretation. This Equation (36) is a simple version of a more general integral equation, for example, in Appendix A4. A more convenient form of this equation (especially for purposes of testing, which is discussed in Appendix A6) is in Appendix A4. The solution of this integral Equation (36) [and the equivalent differential Equation (32)) is given in Equation (31)] and in it we can see that after a
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sufficient amount of time elapsed, the effects of environment and genetics wears off so that the limit I ¼ aE e G h ð37Þ is reached [where we have made the change of variables e1 ! e and hh ! h from Equation (31)] which is the original multiplicative form that was posited based on fundamental reasons valid for all intelligent organisms. Indeed, the complexity of the real world is much beyond what can be captured by these linear and deterministic equations. More on this train of thought can be seen in Appendices I.3 and I.4. Even if we did stick to these linear models (i.e. Equations (34) or (36)) we still have to consider that an ensemble of such /would be needed, one for each person with its own parameters. That would mean that we’d have to consider the parameters E and G as random variables thus turning Equation (34) into a stochastic process in which we would compute the probability density p(I,t ) of the process. For simple cases we can obtain solutions for example in Appendix A.8. That such methods are the wave of the future for highly complex problems of the social sciences has been argued in detail in Hubey (1996) and examples of simple solutions can be seen in Hubey (1993) and complex ones in Helbing (1995). Since the intelligence has been normalized to unity, then Equation (37) is really another expression of the relationship of environment to genetics, which can be written as E e G h ¼ cons: In other words, the nonlinear formulation as in (37) is not necessarily the only one, but rather an example. There are other formulations possible, for example E e ¼ Ce 2mG or Ee vG ¼ cons ð38aÞ eE eE
a þG b
aG b
¼ cons a
b
¼ e E e G ¼ cons
ð38bÞ ð38cÞ
Whatever the case, a linear approximation via the Taylor series in the neighborhood of human level genetic endowment and cultural/environmental achievement will all lead to the same linear approximation results. Since the linearity is obtained from an approximation the numbers are only good in the neighborhood where E þ G < 2 (see appendix A6). But then we have E < 2 2 G or
dE < dG
ð39Þ
which leads inexorably to the conclusion
dE 2 < dG 2
ð40Þ
Since these have to add up to the total variation of unity, it is not a surprise that the environmental variance or genetic variance hovers around 0.5 in studies (Rowe, 1994; Rushton, 1997; Plomin and Daniels, 1987; Plomin and Bergeman,
1991). Similar to the situation in economics (see Appendix A4) this fact is really an indication that the socioeconomic and technological systems and the educational systems that support such systems are created via complex interactions so that we (humanity) work near our optimal limits. It is interesting that similar results hold in the case of the production functions of economics theory, for example, the Cobb-Douglas type, which is multiplicative as here, that the “share of capital” and “share of labor” in the production process and function are about 0.5. In the case of the production function, the numbers are merely a reflection of how the socioeconomic system is setup. For example, in LDCs where they must buy machines from other more developed countries, the “share of labor” is much less than the “capital’s share”. The reason that in advanced societies, the shares of labor and capital are about equivalent merely reflects the fact that the machines which are in the production process are built by other workers in industry and machine costs reflect the salaries and wages of those involved in the production of these machines. Furthermore it is a sign of the power of workers since if capitalists were really that powerful, they could conceivably pay pittances to all the workers, and claim large shares of the profits for themselves. In a similar vein, the reason the heritability is about 50 per cent is really a reflection of the way tests are created and is an indication of the importances that society attaches to skills in various ways. It would be quite easy to create tests in which word recall, reading would be rated low, and mathematical-symbolic-logical capability rated greatly and skew the results of tests so that the heritability results are more skewed than they are now. If these intelligence tests given today were given to our ancestors 5,000 years ago they probably would have scored about the same as those in less developed countries for most of what passes as intelligence is really knowledge of the shared environment which in advanced societies is shared through the popular mass media organs such as television, and propagated through our educational institutions. If anything, the scores less than 0.5 are a testimony to the unequal environment in our societies. Similar views have been voiced by others, for example very strongly by Lewontin (1975). To show explicitly the effect of the nonlinearity on one of the infamous variables of cognitive ability studies, we can show that in the linear (additive) case such as I ¼ a þ eE þ gG ð41aÞ the heritability coefficient calculations are rather easy since
dI ¼ e·dE þ g·dG
ð41bÞ
and therefore H¼
g 2 dG 2 g 2 dG 2 g2 ¼ ¼ dI 2 e 2 dE 2 þ g 2 dG 2 e 2 þ g 2
ð42Þ
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The last step was obtained assuming that the virtual variations (displacements) are equal i.e. dE 2 ¼ dG 2 and by assuming the crossproduct term to be zero as is usually done when the interaction term variance is ignored. We can do this because we want to know how much of the variation in I is due to unit variations in G and E. Behind this approximation are really stochastic differentials. In other words, alternatively, we may treat the variations to be random quantities and then average over the ensemble in which case assuming that the variations dE and dG are independent (which is the assumption used when ignoring the VGE in the standard calculations in ANOVA) the crossvariation is zero. The result is clearly what is expected, and that the heritability is really the ration of the variation due to genes to total variation. What is hidden here but assumed is that both G and E are measured on the same ratio scale since if it were not so the equality of the small variations would not be assumed. For the nonlinear dependence of intelligence on both the environment and heredity as given by I ¼ aE 1 G g
ð43Þ
dI ¼ ðaeE e21 G g Þ·dE þ ðagE e G g21 Þ·dG
ð44Þ
the variation/differential is
To compute the heritability as usually done we would have to divide the variation due to the variation due to genetics by the total variation. In this case we compute it (ignoring the cross-product as is usually done to be H2 ¼
H2 ¼
ðagE e G g21 Þ2 ·dG 2 ðaeE e21 G g Þ2 ·dE 2 þ ðagE e G g21 Þ2 ·dG 2
ð45Þ
dG 2 ·ða 2 E 2e G 2g Þg 2 G 22 g 2 G 22 ¼ dG 2 ·ða 2 E 2e G 2g Þðe 2 E 22 þ g 2 G 22 Þ e 2 E 22 þ g 2 G 22
The result can be put in final form which can be analyzed for semantics. H2 ¼
g2 2 G e2 þg 2 E
ð46Þ
Clearly, it is now even more obvious that both G and E must be measured on a ratio scale (i.e. absolute scale) since now the ratio shows up explicitly in the calculations. Simple analysis of variance calculations based on a linear (additive) model of the influence of genetics and environment in which they can be used as substitutes for one another, is clearly false. The fact that only equations for variance are almost always used in the discussions such as
V P ¼ V E þ V G þ V GE
ð47aÞ
VG ¼ VA þ VD þ VI
ð47bÞ
hides the fact that these ad hoc derivations rely and are based on linear/additive models as shown above. If E is measured on a scale, even if it is a ratio scale, in which it is, say, 10 times larger than the scale on which G is measured, the product will make the contribution from the environment seem small so that the heritability coefficient will get larger. To see the devastating effects of nonlinearity on computations of heritability as is usually done, we can examine a simple case of nonlinearity in which I ¼ aEG: Substituting e ¼ g ¼ 1 into Equations (45) or (46) we obtain H2 ¼
G 22 E2 ¼ E 22 þ G 22 G 2 þ E 2
ð48Þ
It is impossible under these conditions to claim that H 2 really measures (the genetic component) heritability! Obviously, to obtain true heritability we must compute 1 2 H 2 : Furthermore it can be shown that if the dependence of e g intelligence on the environment and genetics is I ¼ e E G the results for H 2 (or h 2 as appropriate) are still the same as above. In the case of the dynamical equation or the solution [Equations (31) and (32)] the equation for h 2 is still the same form as above, therefore the heritability calculations based on the linear model are incorrect. Furthermore, such calculations need to be made on measurements based on a ratio scale so that arbitrary scales for socioeconomic factors cannot be used. We can obtain similar results via Taylor series. Expanding Equation (43) about E0 and H0, we obtain ›I ›I þ ðH 2 H 0 Þ þ... ð49Þ I ¼ E 0 H 0 þ ðE 2 E 0 Þ ›E E 0 ;H 0 ›H E 0; H 0 ðI 2 I 0 Þ2 ðI 2 I 0 Þ2 H 0 E0 ¼ ¼ ðE 2 E 0 Þ2 þ ðH 2 H 0 Þ2 I0 E 0H 0 E0 H0
ð50Þ
where I 0 ¼ E 0 H 0 which is the average/normal human intelligence. Obviously then we can identify dH 2 ¼ ðH 2 H 0 Þ2 and dE 2 ¼ ðE 2 E 0 Þ2 : It is clearer from these derivations that the previous analysis was basically the equivalent of analysis of variance, and thus the results have been demonstrated for a variety of ways. If these intelligence tests were given to our ancestors 5,000 years ago they probably would have scored about the same as the semiliterate peoples in the less developed countries for much of what passes as intelligence is really knowledge of the shared environment which in advanced societies is propagated through the popular mass media organs such as television, and through our educational institutions. If anything, the wide variations on these
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tests are a testimony to the unequal environment in our societies. Similar views have been voiced by others, for example, Lewontin (1975). In terms of the processes which give rise to such scores, there is basic agreement among many workers in the field, except for some lingering confusion. For a more detailed exposition of the ideas, see Hubey (1996). The fundamental concepts are shown in the Table III below. Most of the time by “qualitative” people mean “not well understood” because many intensive variables are quite easily quantifiable. It is an unfortunate accident of history and sloganeering that a word like “quality” has come to be the basis of a word like “qualitative” which is used in opposition to quantitative to disparage the physical sciences. The most correct version of all of these is the intensive-extensive dichotomy (Hubey, 1996) which is what the psychological division of associative vs cognitive/conceptual signifies, as can be easily seen via extrapolation from the AI concepts of knowledge-base and the inference-engine which operates on it. In humans both of these are stored in the brain using neurons. The earlier some of the inferencing mechanisms (i.e. intensive variable) are learned the more they become a natural part of the human reasoning process similar to talking and walking and the more easily they are able to masquerade as intrinsic/genetic factors. The increase in the probability of finding similar problems already solved in memory (section 2.3) greatly increases performance (Figure 5). The earlier these are learned, the more efficient the brain organization for problemsolving (Figure 6). Therefore more accurate measurement these effects requires models in which time is explicit. This also explains why brain size does not correlate more strongly with intelligence tests. Problem-solving techniques, whether learned informally during early childhood or formally in school present themselves in studies as “intelligence”. It is for this reason that more difficult questions are not asked on such tests and especially to adults. Many people in the physical sciences and mathematics would score very high on such tests, but then the learned component would be very obvious to every researcher and layperson alike. However, when fundamental concepts learned early in childhood and which adds to the efficiency of brain organization are asked on such tests we are instead left with “controversy”. It is for this reason that some researchers have put forward ideas such as musical talent, body intelligence and the like. This argument misses the point if one can retort “are Philosophy
Table III.
quantitative, e.g. almost any physical science qualitative, e.g. most “soft sciences”, philosophy
Physics
Artificial intelligence
Psychology
extensive variable, e.g. volume, entropy Intensive variable, e.g. pressure, temperature
knowledge-base, e.g. any field-specific database of knowledge inference-engine, e.g. breadth-first vs depthfirst searching strategies
associative, e.g. memorization of data and information cognitive/conceptual, e.g. fluid intelligence, chunking by experts, logical, mathematical
there music neurons” or if music is noncomputational. Clearly, music is also computational (Johnson, 1997). In the past some researchers took refuge in such arguments are creativity, and originality (in anti-AI arguments) and musicalkinetic intelligence (in anti-math orientation of test questions). However, the validity of the argument stands if it is about the lack of natural dimensions and the weighting of the distance metric in n-dimensional space. (Appendix A.1). At this point in time unless tests which are stringently controlled can be given, and tests which explicitly take into account nonlinear interaction of genetics and environment, there is not sufficient reason to attribute differences in performance on standardized tests to genetic differences which is not to say that what the questions test for are not important to society. If however, motivation is a key factor, one might ask why people with PhDs cannot learn 2 semesters worth of calculus or physics over a period of 50–60 years. In general the differences between humans and other animals (say chimps) in all measurable behavioral characteristics are likely differences of degree and not differences of kind unless there are definite physiological constraints. This means that the interval [0,1] is the natural absolute scale of measurement, and the maximum will be achieved by our species which provides for normalization at the upper end of the scale. Furthermore the natural kind of relationship is multiplicative, which can still be tested using standard methods using logarithms in which there is a trade-off between order of magnitude and the nonlinearity of the logarithmic transformation. If Taylor series approximations are used to obtain linear relationships to be tested, the necessity of using the natural scales is obvious, otherwise the interaction of the different factors cannot be separated from each other.
References Banks, S. (1990), Signal Processing, Image Processing and Pattern Recognition, Prentice Hall, Englewood Cliffs. Britten, R.J. and Davidson, E.H. (1969), “Gene Regulation for Higher Cells: A Theory”, Science, 165, pp. 349-57. Chittick, W. (1983), The Sufi Path of Love: The Spiritual Teachings of Rumi, State University of New York Press, Albany. Eccles, J. (1989), Evolution of the Brain: Creation of the Self, Routledge, New York. Gardner, H. (1993a), The Unschooled Mind, Basic Books, New York. Gould, S.J. (1981), The Mismeasure of Man, W.W. Norton, New York. Gaffan, D. and Weisenkrantz, L. (1980), “Recency effecs and lesion effects in delayed nonmatching to randomly baited samples by monkeys”, Brain Res, 196, pp. 373-86. Gazzaniga, M. (1985), The Social Brain, Basic Books, New York. Helbing, D. (1995), Quantitative sociodynamics: stochastic methods and models of social interaction processes, Kluwer Academic Publishers, Dordrecht. Hubey, H.M. (1987) Software Lifecycle Models, Proceedings of the IASTED conference, Applied Identification Modelling and Simulation: AIMS87, New Orleans, Louisiana 10-13 1987.
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Hubey, H.M. (1991a) Stochastic Putnam-Parr-Hubey Type Software Engineering Models, Proceedings of the Twenty-Second Annual Pittsburgh Conference on Modeling and Simulation, 2-3 May 1991. Hubey, H.M. (1993), “Psycho-socio-economic Evolution of Human Systems, Math. Modelling and Sci. Computing”, Principia Scientia, 2, pp. 320-5. Hubey, H.M. (1994), Mathematical and Computational Linguistics, Mir Domu Tvoemu, Moscow. Hubey, H.M. (1996), “Topology of Thought, CC-AI: The Journal for the Integrated Study of Artificial Intelligence”, Cognitive Science, and Applied Epistemology, 13 No. 2-3, pp. 225-92. Hubey, H.M. (1997), “Logic, physics, physiology, and topology of color”, Behavioral and Brain Sciences, 20 No. 2, pp. 191-4. Hubey, H.M. (1998), The Diagonal Infinity: Problems of Multiple Scales, World Scientific, Singapore. Hubey, H.M. (1999b), Mathematical Foundations of Linguistics, LINCOM EUROPA, Muenchen, Germany. Herrnstein, R. and Murray, C. (1994), The Bell Curve, The Free Press, New York. Jerison, H.J. (1973), Evolution of the Brain and Intelligence, Academic Press, New York. Johnson, George (1997) Music by the numbers, New York Times, 29 November 1997. Kojima, Ken-ichi Mathematical Topics in Population Genetics, Springer-Verlag, New York. Klir, G. and Yuan, B. (1995), Fuzzy Sets and Logic, Prentice-Hall, Englewood Cliffs. Lewontin, R.C. (1975), “Genetic Aspects of Intelligence”, Annual Review of Genetics, 9, pp. 387-405. Luce, R., Bush, R. and Galanter, E. (Eds) (1963), Handbook of Mathematical Psychology, John Wiley & Sons, New York vol Volume I. MacLean, P.D. (1973), A Triune Concept of the Brain and Behavior, University of Toronto Press, Toronto. Maranell, G. (Ed.) (1974), Scaling: A Sourcebook for Behavioral Scientists, Aldine Publishing Company, Chicago. Olton, D. (1983), “Memory functions and the hippocampus“, in, Molecular, Cellular and Behavioral Neurobiology of the Hippocampus, Seifert (Eds), Academic Press, New York pp. 335-73. Papoulis, A. (1984), Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York. Plomin, R. and Daniels, D. (1987), “Why are children in the same family so different from one another”, Behavioral and Brain Sciences, 10, pp. 1-60. Plomin, R. and Bergeman, C.S. (1991), “The nature of nurture: Genetic influence on environmental measures”, Brain and Behavioral Sciences, 14, pp. 373-427. Rasch, G. (1980), Probabilistic Models for Some Intelligence and Attainment Tests, University of Chicago Press, Chicago. Roughgarden, J. (1979), Theory of Population Genetics And Evolutionary Ecology: An Introduction, MacMillan, New York. Rowe, D. (1994), The Limits of Family Influence, Guildford Press, New York. Rushton, J.P. (1997), Race, Evolution, and Behavior, Transaction Publishers, New Brunswick. Sagan, Carl (1977), Dragons of Eden: Speculations on the Evolution of Human Intelligence, Ballantine Books, New York. Shrager, J., Hogg, T. and Huberman, B.A. (1988), “A graph-dynamical model of the power law of practice and the problem-solving fan-effect”, Science, 242, pp. 414-6. Tricomi, F.G. (1985) Integral Equations, Dover, New York. Zadeh, L. (1963) Fuzzy Sets, Inf. Control, Vol 8, 1965 pp. 338-353.
Zadeh, L. (1978) Fuzzy Sets as a Basis for a Theory of Possibility, Fuzzy Sets and Systems, Vol. 1, pp. 3-28. Zadeh, L (1987) A Computational Theory of Dispositions, Int. Journal of Intelligent Systems, Vol. 2, pp. 39-63. Further reading Arbib, M. (1964), Brains, Machines, and Mathematics, McGraw-Hill, New York. Arnold, L. and Lefever, R. (1985), Stochastic Nonlinear Systems, Springer-Verlag, New York. Bell, D.J. (1990), Mathematics of Linear and Nonlinear Systems, Oxford, New York. Bennett, J.G. (1973), Gurdjieff: Making of a New World, Harper & Row, New York. Brockman, J. (Ed.) (1993), Creativity, Simon & Schuster, New York. Block, N.J. and Dworkin, G. (Eds) (1976), The IQ Controversy, Random House, New York. Brooks, D. and Wiley, E. (1986), Evolution as Entropy: Toward a Unified Theory of Biology, University of Chicago Press. Carroll, R. (1988), Vertebrate Paleontology and Evolution, Freeman, New York. Carrier, G. and Pearson, C. (1976), Partial Differential Equations, Academic Press, New York. Casdagli, M. and Eubank, S. (1992), Nonlinear Modeling and Forecasting, Addison-Wesley, Reading, MA. Davis, A. (1962), Introduction to Nonlinear Differential and Integral Equations, Dover, New York. Dennett, D. (1991), Consciousness Explained, Little, Brown and Co., New York. Drazin, P.G. (1992), Nonlinear Systems, Cambridge University Press. D’Souza, D. (1991), The End of Racism, Free Press, New York. Duff, G. (1956), Partial Differential Equations, University of Toronto Press, Toronto. Davis, M. and Weyuker, E. (1983), Computability, Complexity, and Languages, Academic Press, New York. Dunn, O.J. and Clark, V.A. (1987), Applied Statistics: Analysis of Variance and Regression, Wiley, New York. Durden-Smith, J. and DiSimone, D. (1983), Sex and the Brain, Warner Books, New York. Devlin, B., Fienberg, S., Resnick, D. and Roeder, K. (Eds) (1997), Intelligence, Genes, and Success, Copernicus, New York. Eaves, L.J. (1973), “Assortative Mating and Intelligence: An Analysis of Pedigree Data”, Heredity, 30 No. 2, pp. 199-210. Eldredge, N. and Cracraft, J. (1980), Phylogenetic Patterns and the Evolutionary Process, Columbia University Press, New York. Elkana, Y. et al. (Eds) (1978), Toward a Metric of Science, Wiley, New York. Ferguson, M. (1973), The Brain Revolution: Frontiers of Mind Research, Bantam Books, New York. Forrest, S. (1991), Emergent Computation, MIT/North Holland, Cambridge. Fraser, S. (Ed.) (1995), The Bell Curve Wars, Harper Collins, New York. Fallside, F. and Woods, Wm. (Eds) (1983), Computer Speech Processing, Prentice-Hall, Englewood-Cliffs.
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Fischer, M. and Firschein, O. (1987), Intelligence: The Eye, the Brain and the Computer, AddisonWesley, Reading, Massachusetts. Gamut, L.T.F. (1991), Logic, Language, and Meaning, Chicago University Press, Chicago. Gardiner, C.W. (1983), Handbook of Stochastic Methods, Springer-Verlag, New York. Gardner, H. (1993b), Frames of Mind, Basic Books, New York.
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Giri, N. (1977), Multivariate Statistical Inference, Academic Press, New York. Gregory, R. (1973), Eye and Brain, McGraw-Hill, New York. Gregory, R. (1987), Oxford Companion to The Mind, Oxford University Press, Oxford. Guastello, S. (1995), Chaos, Catastrophe, and Human Affairs, Lawrence Erlbaum Associates, Mahwah, NJ. Guilford, J. (1967), The Nature of Human Intelligence, McGraw-Hill, New York. Gorsuch, Richard (1974), Factor Analysis, W.B. Saunders Company, Philadelphia. Hacking, I. (1965), Logic of Statistical Inference, Cambridge University Press, London. Holton, G. (1993), Science and Anti-Science, Harvard University Press, Cambridge. Hubey, H.M. (1991b) Catastrophe Theory and Human Sexual Response, presented at the Third International Symposium on Systems Research, Informatics, & Cybernetics, Baden-Baden, Germany, August 12-18. Submitted to the Journal of Sex Research. Hubey, H.M. (1999a), Mathematical and Computational Linguistics, LINCOM EUROPA, Muenchen, Germany. Hardcastle, Wm. and Marchal, A. (1990), Speech Production and Speech Modeling, Kluwer Academic Publishers, Boston. Hooper, J. and Teresi, D. (1986), The 3-Pound Universe, Tarcher, Los Angeles. Jackson, E.A. (1991), Perspectives on Nonlinear Dynamics, Cambridge University Press, New York. Jazwinski (1970), Stochastic Processes and Filtering Theory, Academic Press, New York. Jacoby, R. and Glauberman, N. (Eds) (1995), The Bell Curve Debate, Random House, New York. Jones, S. et al. (Eds) (1992), Cambridge Encyclopedia of Human Evolution, Cambridge UP, Cambridge. Kosko, B. (1993), Fuzzy Thinking, Hyperion, New York. Krantz, D., Luce, R.D., Suppes, P. and Tversky, A. (Eds) (1971), Foundations of Measurement, Academic Press, New York, Volume I. Lepore, E. and Van Gulick, R. (Eds) (1991), John Searle and His Critics, Blackwell, Oxford. Li, M. and Vitanyi, P. (1993), An Introduction to Kolmogorov Complexity and Its Applications, Springer-Verlag, New York. Lord, F. and Novick, M. (1968), Statistical Theories of Mental Tests, Addison-Wesley, Reading, MA. Lowley, D.N. and Maxwell, A.E. (1971), Factor Analysis as a Statistical Method, American Elsevier, New York. Murray, Charles (1984), Losing Ground, Basic Books, New York. Mattingly, I. and Studdert-Kennedy, M. (1991), Modularity and the Motor Theory of Speech Perception, Lawrence Erlbaum Associates, Hillside, New Jersey. Ornstein, R. (1975), Psychology of Consciousness, Penguin Books, New York.
Owen, D. (1984), A First Course in the Mathematical Foundations of Thermodynamics, SpringerVerlag, New York.
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Ornstein, R. and Sobel, D. (1987), The Healing Brain, Simon & Schuster, New York. Ornstein, R., Thompson, R. and Macaulay, D. (1984), The Amazing Brain, Houghton-Mifflin, Boston. Papadimitriou (1994), Computational Complexity, Addison-Wesley, Reading, MA. Penrose, R. (1989), The Emperor’s New Mind, Oxford University Press, Oxford. Penrose, R. (1994), Shadows of the Mind, Oxford University Press, Oxford. Pinker, S. (1995), The Language Instinct: How the Mind Creates Language, Harper Perennial, New York. Quine, W. (1960), Word & Object, MIT Press, Cambridge, MA. Quine, W. (1970), Philosophy of Logic, Harvard University Press, Cambridge, MA. Rescher, N. (1969), Many-Valued Logic, McGraw-Hill, New York. Restak, R. (1979), The Brain: The Last Frontier, Warner Books, New York. Restak, R. (1988), The Mind, Bantam Books, New York. Rose, S. (1976), The Conscious Brain, Vintage Books, New York. Rosenthal, D. (1970), Genetic Theory and Abnormal Behavior, McGraw-Hill, New York. Schmidt-Nielsen, K. (1984), Scaling: Why is Animal Size So Important?, University of Cambridge Press, Cambridge. Schroeder, M. (1991), Fractals, Chaos, Power Laws, Freeman and Company, New York. Searle, J. (1984), Minds, Brains, and Science, Harvard University Press, Cambridge, MA. Skinner, B.F. (1974), About Behaviorism, Vintage Books, New York. Soong, T.T. (1973), Random Differential Equations in Science and Engineering, Academic Press, New York. Speeth, K.R. (1976), The Gurdijieff Work, Pocket Books, New York. Sternberg, R. (1977), Intelligence, Information Processing, and Analogical Reasoning, Wiley and Sons, New York. Stevens, S.S. (Eds), (1966), Handbook of Experimental Psychology, John Wiley & Sons, New York. Strogatz, S. (1994), Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA. Scientific American (1993), Mind and Brain, Freeman, New York. Temple, C. (1993), The Brain, Penguin Books, London. Thurstone, L. (1935), The Vectors of Mind, University of Chicago Press, Chicago. Thurstone, L. (1947), Multiple Factor Analysis, University of Chicago Press, Chicago. Torgerson, W. (1958), Theory and Methods of Scaling, Wiley and Sons, New York. Van de Geer, J.P. (1971), Introduction to Multivariate Analysis for the SocialSciences, W.H. Freeman and Company, San Francisco. Wang, Wm. (Ed.) (1986), The Emergence of Language: Development and Evolution, Freeman, New York. Young, E. (1972), Partial Differential Equations, Allyn and Bacon, Boston.
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Zeeman, E.C. (1977), Catastrophe Theory: Selected Papers, Addison-Wesley, Reading, MA. Zurek, W. (Ed.) (1995), Complexity, Entropy, and the Physics of Information, Addison-Wesley, New York. Appendix A.1: Scales or levels of measurment Before we try to measure or normalize quantities we should know what kinds of measurements we have. They determine if we can multiply those numbers, add them, rank them etc. Accordingly measurements are classified as: (i) Ratio scale, (ii) Interval scale, (iii) Ordinal scale, or (iv) Nominal scale. According to Thurstone, the decision to use factor analysis as a primary method implies a deep ignorance of principles and causes Gould (1981: 316) Absolute (ratio) scale: The highest level of measurement scale is that of ratio scale. A ratio scale requires an absolute or nonarbitrary zero, and on such a scale we can multiply (and divide) numbers knowing that the result is meaningful. The standard length measurement using a ruler is an absolute or ratio scale. Distance Probably the most common measurement that people are familiar with is that of distance. It is such a general and common-sensical idea that mathematicians have abstracted from it whatever properties it has that makes it so useful and have extended it to mathematical spaces so that this idea, is in fact, used and useful in the previous ideas of measurements. The requirement that the concept of distance satisfies is this: dðx; zÞ # dðx; yÞ þ dðy; zÞ
ðA:1Þ
The concept of “distance” or “distance metric” or “metric spaces” is motivated by the simple concept illustrated Figure A1. If we substitute from the figure above we can see that the distance from LA to NYC can never be greater than the distance from LA to some intermediate city plus the distance from that intermediate city to NYC. The most commonly known distance measure is the Euclidean metric which most of us have seen in some form or have heard about as the Pythagorean Theorem. In the plane the Euclidean metric follows directly from the Pythagorean theorem as shown. Hamming distance Hamming distance is the number of bits by which two bitstrings differ. For example the distance between the bitstring 1111 and 0000 is 4 since the corresponding bits of the two bitstrings differ in 4 places. The distance between 1010 and 1111 is two, and the distance between 1010 and 0000 is also two. Phonological distance: distinctive features In phonology, the basic primitive objects are phonemes. They are descriptions of the basic building blocks of speech and are usually described in binary as the presence or absences of specific characteristics such as voicing, rounding, frication, plosivity etc. Since we can represent these as bitstrings the Hamming distance can be used to measure the distance between phonemes. One simple way to describe a subset of phonemes (vocalic phonemes) especially of the Turkic, Mongolian and Uralic languages is via the Hubey Ordinal Vowel system (Hubey, 1994) below Figure A3.
Figure A1. The concept of distance
Such hypercubes can be constructed also for sets of consonants. We need higher dimensional or fractional spaces to represent them all on the same graph or hypercube or same space. In order to represent them all there are several ways. What’s a bird? The concept of distinctive features can also be used in conjunction with fuzzy logic in artificial intelligence to describe (or define) objects, such as a bird, fruit or a chair. For example, a set of simple properties such as “has feathers”, “is bipedal” and “flies” is generally sufficient to define a bird for intelligent entities (such as humans). We can use a simple Hamming distance or try a Euclidean distance. Interval scale: However, not everything that can be measured or represented with integers (or real numbers) is not a ratio/absolute scale. Ordinal scale: The next level on the measurement scale is the ordinal scale, a scale in which things can simply be ranked according to some number but the differences are not valid. In the ordinal scale we can make judgements such as A . B. Therefore if A . B and B . C, then we can conclude that A . C. In the ordinal scale there is no information about the magnitude of the differences between elements. We cannot use operations such as +, 2, * or / on the ordinal scale.
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Figure A2. Disance Metric: One would not expect, if the above were the real space of our existence, that the x and y dimensions would be measured on different scales, i.e. x in feet and y in meters before using the formula above for distance. It would be an unthinkable error, but we do not know how we should combine scores on many tests in psychology because the dimensions such as Space/Length, Time, Temperature and Charge of physics do not exist
Figure A3. The ordinal vowel cube (Hubey, 1994)
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Likert scale: It is possible to obtain an ordinal scale from questionnaires. One of the most common, if not the most common is the multiple-choice test which has the choices: extremely likely/agreeable, likely/ agreeable, neutral, unlikely/disagreeable, and extremely/very unlikely/disagreeable. Nominal scale: The lowest level of measurement and the simplest in science is that of classification. In classifying we attempt to sort elements into categories with respect to a particular attribute. This is the nominal scale. On this scale we can only say if some element possesses a particular attribute but cannot even rank them according to some scale on a hierarchy based on the intensity of possession of that attribute. We can only think of creating sets based on the possession of some property and apply the operations for sets. In this sense the set operations are the most primitive of operations of mathematics. It ranks so low on the scale or hierarchy that we all instinctively do it. Even an animal that can tell food from nonfood can be said to have learned or can be said to know about set operations instinctively. Whatever kind of logic that flows from this must obviously be related to set theory in some way. If it were not so, how would animals be able to do it? Consider now a simple question: Is 100F twice as hot as 50F? In a quick and unscientific poll via email in my GER class the students split approximately 50 –50 on this question. There are similar questions that can produce similar splits: What is the volume of stereo player? What is the average pitch of voice of a female? Is it twice as high as that of a male? What is different about these than say measuring distance or weight? The simplest reason is that we are not familiar with the scales used in measuring them. The second reason is that normally we are used to interval scales and not absolute scales say in measuring distance. Let’s explore the implications of the differences and the difficulties using the temperature scale. Heat transfer rate is usually approximated as linearly proportional to the temperature difference between the two bodies q0 ¼ cðT 1 2 T 2 Þ
ðA:1aÞ
and here it doesn’t make any difference whether we use the absolute (i.e. Kelvin or Rankine) or the relative (Fahrenheit or Celsius) temperature scale. However, the efficiency of a heat engine is proportional to the ratio of the two temperatures i.e. h ¼ 1 2 ðT 1 =T 2 Þ ¼ ðT 2 2 T 1 Þ=T 2
ðA:1bÞ
so that the absolute scale must be used. In order to understand clearly what this means, it is instructive to look into the history of development of the temperature scale. It was noticed that the level of some fluid put in a glass tube was correlated with the ambient temperature. Fahrenheit decided to put some numbers to this effect. In order to normalize it, he used the body temperature as a constant and marked that as 100 F. He then arbitrarily put some tick marks on the thermometer and on this scale we got water to freeze at 32F and to boil at 212 F. In order to put it into more logical/ scientific form Celsius decided to use the freezing and boiling point of water and marked these off as 0 C and 100 C. However, this wasn’t enough. Because of theoretical considerations from thermodynamics Lord Kelvin realized that an absolute scale was needed and hence we now have the absolute scale in which water freezes at 273 K and boils at 373 K, so that absolute zero is at 2273 C. In the absolute Fahrenheit scale also called the Rankine scale, absolute zero occurs at about 2 460 F. As a testimony to the insight of Kelvin we should note that no one to date has been able to produce any temperature colder than absolute zero, that is less than 2 460 F. Hence we can see the answer to our question since the ratio is ð460 þ 100Þ=ð460 þ 50Þ ¼ 560=510 ¼ 1:098 so that 100 F is not twice as hot as 50 F (Figure A4). We should think of intelligence as being more than simply what these tests purport to measure. It’s a process and the tests measure the end result of some computation or process. If we think about intelligence in a more general sense, as in animals, we see that survival techniques
are a big part of it. Animals adapt to the environment; so do humans. Humans adapt to slavery as well as to being masters; they adapt to the slums as well as to the suburbia. Deceit is part of intelligence according to some since an artificially intelligent entity (AIE) wouldn’t pass the Turing Test (TT) if it only told the truth. This we certainly don’t seem to measure directly in IQ tests. One of the highly stylized facts is that intelligence should be close to 100 per cent inherited since if it were not true we should be teaching statistics and calculus to chimps and dogs. And yet intelligence as a concept has to take into account both machine intelligence (MI) and intelligence of animals. The real question is if correlation analysis, regression analysis or factor analysis does what it’s supposed to do? In comparing intelligences of humans without an absolute intelligence scale common sense says that ratios should be used and not differences. And indeed they are used. The tests are structured so that the results are Gaussian and that the average is 100. However this is an interval [i.e. relative] scale and not an absolute scale. On this test a difference of 15 points really boils down to 15/100 or 15 per cent although this is not explicitly stated either because it’s too obvious or because of the preoccupation with making things scale free. But we lose information every time we reduce sets of numbers to a single number and we lose some information when we rescale the original variables. If intelligence is something that all living entities [and possibly even machines] possess because our basic concepts of intelligence dictate that some kind of intelligence is possessed by animals, then it has to be measured on an absolute intelligence scale. Now suppose we add some constant to the scores on the tests to make sure we are measuring them on an absolute intelligence scale, say, about 7,000,000. Then the average score for whites is 7,000,100 and for blacks 7,000,085. Now, what person in his right mind would claim that the difference ð7; 000; 1007; 000; 085Þ=7; 000; 100 ¼ 2:14291026 ¼ 0:0000021429 which is 0.00021429 per cent has any kind of significance? In fact, it doesn’t. And it is also completely consistent with the kinds of results we should expect. Almost all of our intelligence is inherited, and we should all score very high when compared against say dogs, salamanders, or even chimps.
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Appendix A.2: Potential and its realization Other types of potential models, with yet another way of creating realization from potential can be created. Two examples will be given to show the futility of the standard arguments against intelligence tests. For one thing, it’s easy to define some function say C ¼ 12 ð1E 2 þ aA 2 þ sS 2 Þ
ðA2:1Þ
Figure A4. The temperature scales: In physics the absolute temperature scale (Kelvin scale) must be used in order to be meaningful in equations involving ratios
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Meanwhile it is of interest to note that the problem of equalizing intervals gets a bigger play from the mental testers and other psychometricians than it does from the sensory psychologists. In psychometrics the need for equal units is distressingly acute, because the paraphraenillia of metric statistics (means, standard deviations, coefficients of correlation, etc.) seem to be essential tools of the trade. The assessor of human abilities is usually knee deep in statistical problems to which most statistics do not apply unless his units can be equalized. Out of this quandary he hoists himself by an act of faith in the “normal distribution” of nature’s errors. If this faith is firmly founded-in truth it is legitimate to sue the distribution of scores as a criterion for the sizes of units– the the equalization of units is possible. It is certainly not unreasonable to believe that his faith is often justified. What huants us is the difficulty of knowing when it isn’t. Stevens [1951:39] where for sake of argument E ¼ endurance, A ¼ agility and S ¼ strength or power. Let’s name Y the Fizikal Fitness Potential (FFP). Can we now derive a scalar (a single number) called Physicality Quotient (PQ) or a Fizikalness Vector (FV) from this potential? Can we not derive a single number for something having to do with physical ability, physicality, or something along these lines? What about the scoring system of the decathlon? Does it not produce a single number for the various events in the contest? Is it all part of what we think is about physical fitness? There are two simple answers to these questions: PQ, (or FFP or FV) is what the tests measure! In other words if the word physical was not defined in FFP, PQ or FV, well, then it is now via the scores on these tests. So what we have done is to prescribe an operational procedure which can be duplicated (as closely as possible) and this operational procedure is what defines whatever it is that we were trying to define. It’s an idea due to Bridgeman and comes from physics when the physicists had similar problems with what an electron was or if it really existed. If, on the other hand, we think that physical fitness or physical ability was something that we all know and that they were already defined then we’d fight over whether PQ or FFP or FV measure it. The real problem is then, is this what it’s supposed to be? The whole idea of using words like potential and actual are fraught with dangers so we should be careful. After these standard disclaimers about problems which will probably plague this field for a long time, if not forever, we can proceed by assuming that we can measure these variables {E,A,S} via some suitable tests and that these tests will have met some of the basic criteria required. We can then easily produce a vector from this potential 3 2 3 2 3 3 2 2 0 1E 0 1E 7 6 7 6 7 7 6 6 7 6 7 6 7 7 6 p ¼ GradðCÞ ¼ 7C ¼ u^e eE þ u^ a aA þ u^ s sS ¼ 6 4 0 5 þ 4 aA 5 þ 4 0 5 ¼ 4 aA 5 0 sS sS 0 ðA2:2Þ where the uˆ’s are the unit vectors in the proper directions. For what follows bold lower case Latin letters will always represent vectors or tensors of rank 1; bold upper case Latin letters matrices or tensors of rank 2, and normal Latin or Greek letters will be scalars. The letters i, j, k, n will be used for integers. For what reason we’d do this is not important for now. If we want now to produce a single scalar number from this vector we can use some general norm such as in Equation (4) r ¼ jpj ¼ ð1E 2i þ aA 2j þ sS 2k Þð1=nÞ ðA2:3aÞ For 1 ¼ a ¼ s ¼ i ¼ j ¼ k ¼ 1 and n ¼ 2 this becomes the standard Euclidean norm. Note that we want even powers just in case the numbers we measure may be negative assuming that the components of the vector are “positive” quantities so that we don’t want some components of the vectors diminishing the norm of the vector. Note that we could have used other components such as height, weight, etc. and some of these could be very important in predicting success in sports
such as basketball or football. And we can produce many kinds of measures from something of more complicated form such as
r ¼ jpj ¼ ðE s A I S d Þ þ ðE e A a S s Þð1E 2i þ aA 2j þ sS 2k Þ1=n
ðA2:3bÞ
from which we can produce measures such as EAS+ EAS(E+A+S) or as EAS +E+A+S or EAS(E+A+S). Since product terms are useful in equalizing the various components in the single scalar measure, the latter could be used to further weight the weighted sums. The real question is if we have some fundamental reasoning behind selecting any of these or even selecting some other versions using logarithms. The scalar measure (i.e. the single number) can be changed if any negative factors are being used as components of the vector. For example, one does not normally find tall persons in gymnastics but lots of them in basketball. So we can easily work out different scalars for different sports. The real question is what weights should be given for the FFQ or PQ and what the number really means. Assuming that we don’t have much data we can always use these numbers in statistical studies judging the success of the subjects in various sports. Other possibilities exist and come to mind. For example, we could have used cubic powers for the potential Y, then the vector would have had squares of the variables. We could have then calculated the divergence of the vector and obtained another scalar. F ¼ 7·7C ¼ 72 C ¼ 1 þ a þ s ¼ constant
ðA2:4Þ
But what could it mean? In this case we are at a loss, at least momentarily, to know what these single scalar numbers could mean. However it is not in general true that a single number has no interpretation or that it cannot be done, although in general and for results such as for CA, PQ or FFP, we can and probably will always have arguments as to the appropriateness of the given scalar. If it is to have any enduring value, sound debunking must do more than replace one social prejudice with another. It must use more adequate biology to drive out fallacious ideas. S. Jay Gould (1981: 322) In any case, the most important thing is whether they correlate with things in the real world. And in extending the analogy to the test of intelligence we have at least two related problems. One is that they do correlate with success in our society, and not too many people would or should argue with this. It measures something which society seems to value. It is the other problem that’s caused the hullabaloo. Is it possible that these scores mean something about the genetic characteristics of the subjects. And here we must get more technical. Suppose for simplification we decompose cognitive capability or ability into usual Verbal (V)[serial], and Spatial(S)[parallel] (ignoring others such as Mathematical, Logical or Symbolic ). We could, of course, use other fundamental criteria such as Serial or Parallel processing capability. We realize, of course, that there is no such neat division into any of theses categories. Story telling and following takes place in at least 4D (3 space dimensions which are conceived in parallel and one time dimension, which is Serial). Similarly problem solving involves 4D. Mathematical talent/knowledge consists of solving 4D problems in detail and precision. However suppose that we want to derive a single number from these scores. Assuming that there’s some kind of a maximum capacity for the human brain (and there is since it’s finite) we must have something like V+S¼k=constant. We can define the potential as ðA2:5Þ F ¼ Vu Ss : Then the actuality vector using the same ideas would be
2 u3 6V7 7 a ¼ 7F ¼ uV u21 S s e^v þ sV u S s21 e^ s ¼ F½ðu=V Þe^ v þ ðs=SÞe^ s ¼ VS 6 4 s 5: S
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ðA2:6Þ
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7·7F ¼ uðu 2 1ÞV u22 S s þ sðs 2 1ÞV u S s22
ðA2:7Þ
Evidently for this ‘model’ we’d need s; u . 2 in order for some meaning to be attached to these /in the standard sense. There could be good reasons for choosing a multiplicative version of computing a number. If we want to give equal weight to being rounded in all areas, we’d do well to use a multiplicative number since it’s maximized if the components are about equal, and since a serious deficiency in one component would seem to have a large negative impact on observed/ hypothesized intelligence we might do such a thing. For example, to maximize the surface area of a rectangle for a given perimeter we can see that we have to maximize A ¼ hw where 2h þ 2w ¼ P ¼ constant: Therefore since we have h ¼ ðP 2 2wÞ=2 we can substitute into A and differentiate A wrt to h to obtain the result that the maximum occurs at h ¼ w: So we can add a multiplicative component to the measurements of various components to derive a single number. If anything it is the lack of such measurements that are a greater hindrance to the resolution of problems in this field and not the existence of tests. Another example of “potential” and its realization is how people choose to “expend” (even if they do this slowly over time) their intellectual energy is in the dichotomy between the generalist and the expert. It is said that the expert knows everything about nothing. In the same way the generalist knows nothing about everything. These can be represented as in Figure A5. Appendix A.3: Exact differentials and path functions/integrals The distinction between the related concepts state and process is an important one. There are mathematical definitions and consequences of these ideas. A state (or property) is a point function. The state of any system is the values of its state vector (a bundle of properties which characterizes a system). If we use these variables as coordinates then any state of the system is a point in this n-dimensional space of properties/characteristics. Conversely each state of the system can be represented by a single point on the diagram (of this space). For example for an ideal gas the state variables are temperature, pressure, volume, etc. Each color can be represented as a point in the 3-D space spanned by the R, G and B vectors. Intelligence is commonly accepted to be a state variable, i.e. a point. The scalar, Spearman’s g, (single number, not a vector) can be obtained from this vector by using a distance metric. The argument that the values of the components cannot be obtained from the scalar, g, may be valid depending on the distance metric however, the distance metric may be devised in a way in which the components can be obtained from the scalar. Distance on a metric space is a function only of the end points i.e. between two states. However, the determination of some quantities requires more than the knowledge simply of the end states but requires a specification of a particular path between these points. These are called path functions. The commonest example of a path function is the length
Figure A5. The potential and the expert: In (a) the expert has more general knowledge than the expert in (b). The isoclines are types of knowledge
of a curve. Another example is the work done by an expanding gas is a path function. So is the heat (transferred). In that sense work and heat are interactions between systems, not characteristics of systems. Intuitively, when we talk about small changes or small quantities we we want to sum up an infinite number of these, and use use the differentials dx or dx. When R R integration we use the notation dF for an infinitesimal amount of or d q, for an infinitesimal R quantity of heat or ds for an infinitesimal length of a curve. However the crucial difference is that although there may exist a function f ðxÞ ¼ dF=dx so that Z
b
dF ¼
Z
a
a
f ðxÞdx ¼ b
there is no function Q, such that Z
a b
FðxÞjab
¼ FðaÞ 2 FðbÞ
dq ¼ QðxÞjab ¼ QðaÞ 2 QðbÞ
ðA3:1Þ
ðA3:2Þ
Rb Instead we write a dq ¼ Qab meaning that Qab is the quantity of heat transferred during the process from pointpaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi to point b. Similarly because the infinitesimal length of a curve in the plane is given by ds ¼ dy 2 þ dx 2 we cannot integrate ds and obtain SðbÞ 2 SðaÞ ¼
Z
b
ds ¼ a
Z
b
ds
ðA3:3Þ
a
but instead first the curve y ¼ f ðxÞ must be specified so that we can use compute dy/dx and then use) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dy ds ¼ dx 1 þ dx
ðA3:4Þ
Equivalently, if z is a function of two independent variables x and y, and this relationship is given by z ¼ f ðx; yÞ then z is a point function. The differential dz of a point function is an exact differential and given by ›z ›z dz ¼ dx þ dy ðA3:5Þ ›x ›y Consequently if a differential of form dz ¼ Mdx þ N dy
ðA3:6Þ
is given, it is an exact differential only if
›M ›N ¼ ›y ›z
ðA3:7Þ
Therefore in the mathematical function used for the simple two-factor (nature-nurture) Intelligence Function (Appendix A.4), the environmental path taken does make a difference in the final result which is assumed to be a state function (although computed from mental processes). As a very simple example of the kinds of dynamics we can expect we can try a simple path of increasing richness of environment by first writing the equation for the increase in intelligence as
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dI ¼ g ðaE 2 I ðtÞÞ dt
ðA3:8Þ
in which we now have used E to show that the intelligence level may be modeled as approaching some constant value determined by the environment since when y ¼ aE the derivative is zero and hence the intelligence has hit its maximum level (steady state) and will increase no longer. The solution of this is quite easily shown to be IðtÞ ¼ aE 2 Ce 2g t
ðA3:9Þ
where C is an arbitrary parameter. As time increases the second term goes to zero so that the maximum intelligence (a simple function proportional to the environmental level) is reached. It is already known that IQ scores have been increasing for about half a century and that the levels of education, nutrition and the general standard of living are usually thought to be the cause of it. Taking a cue from this fact and the equation above we can change the model to a more general form in which environmental richness is not constant dI ¼ g ðaEðtÞ 2 I ðtÞÞ dt
ðA3:10Þ
The general solution of this is I ðtÞ ¼ g e 2g t
Z
EðtÞe g t dt þ Ce 2g t
ðA3:11Þ
We note that this is of form Equation (24) and it came from very reasonable modeling of the brain/body growth. For short periods of time we can approximate the change in the environment as a linear function of the form EðtÞ ¼ A þ Bt where the A is the environment to which the child was born and the B coefficient determines the linear enrichment rate of the environment. The solution is 1 yðtÞ ¼ ·ðA þ Bt 2 Ce 2g t 2 BÞ g
ðA3:12Þ
The plots of the increase in intelligence of three children for various values of the parameters is shown. One is born to a low SE class ðA ¼ 150Þ but his environment is changing more rapidly ðB ¼ 0:43Þ: From the earlier equations we can try to give a meaning to the constant g; it simply controls the rate at which the child adapts so if anything it could be some measure of innate ability, however we have not explicitly chosen any genetic component in the equation. The form of the solution implies that this coefficient g has to do with what many would consider to be a genetic factor which controls the speed at which people learn new things and adapt to the environment. The environment then acts as a motivation factor in which each child realizes his/her potential. In the growth equation this coefficient controls the rate at which the child would reach his/her adult size. Therefore we assume that it could contain some “innate” factor. Thus for this child the constant g ¼ 0:15: The other has started off in a relatively advantaged position ðA ¼ 170Þ and is in a state where his environment is not being enriched as rapidly as the first ðB ¼ 0:34Þ: Furthermore g ¼ 0:11: For both we have chosen the arbitrary constant of integration C ¼ 100: As can be seen from the plots, although the first child, if anything, is of greater innate ability (given the meaning attached to the coefficient g ), he cannot reach the intelligence level of the other; the difference drops from a high of about 20 down to approximately 8 at around age 8 at which time the differences actually start to increase again (Figure A6).
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Figure A6. Difference in intelligence due to environment changes
In the second set the parameters have been changed again, but the differences in the innate learning ability i.e. g are almost equal since they are 0.03 and 0.038 respectively, however the intelligence differences actually starts to increase until approximately age 12 – 13 and then still persist at about 15 points until age 30. One quick lesson we can see from this model is that units of measurement or scales do matter. The coefficients were selected arbitrarily in order to produce numbers in the ranges in which many tests work i.e. IQ tests around 100, and various achievement tests which are scored between 200 and 800. These are exactly what kinds of results that we should expect because of the evidence that early experiences are important, and that what we are measuring (purportedly a function of genetic ability and not learning) is a function of the environmental path. As simple as they are these models are still much better than mindlessly regressing of everything against everything else, and then attempting to draw deep scientific conclusions from them. We don’t know exactly how the increase in the environment goes so that instead of linear it could be exponential or logistic/sigmoidal. One can easily change
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Figure A7. Changes in “Intelligence” over Time
the environmental effects so that they are of sigmoidal shape. Many such differential equations with their solutions can be found in Hubey (1987,1991a) related to software engineering. As an example of a function which weights early effects of the environment more heavily than the later ones, let us choose a simple exponential weighting wðtÞ ¼ e 2v t
ðA3:13Þ
In addition let us choose an increasing environmental richness path (normalized single variable). A computation of an accumulation (of any behavioral observable, for example intelligence) over paths j with the weighting as shown above, would be Z t Z t EðtÞe 2v t dt ¼ E j ð1 2 e 2d j t Þe 2v t dt ðA3:1eÞ XðtÞ ¼ 0
0
1
h
in which E(t ) is of form aE e G h ð1 2 e 2lE G t Þ with dj ¼ lE 1 G h and E j ¼ aE e G h : These can be integrated with different values of dj and Ej for a simulation of/and comparison of the effects of the early environment. The early problem solving techniques learned add appreciably to later scores which was shown from the perspective of memory utilization and access time in section 2.4 More examples of this can be found in Appendix A.4. Appendix A.4: The intelligence function constraints The Intelligence Function of form I ¼ aE e G h ¼ FðE; GÞ
415 ðA4:1Þ
was selected only because it was the simplest such form that satisfies the conditions we expect such functions to satisfy. For E . 0 we expect positive and diminishing marginal returns FE ¼
›F . 0; ›E
and;
F EE ¼
›2 F ,0 ›E 2
ðA4:2aÞ
That is, we expect increases in intelligence scores with increasingly rich environments. The IQ scores of children over the century has been increasing in all the developed countries. But we expect the marginal increases to slow down as we enrich the environment because there is a top limit on the capacity of the human brain. Similarly for G . 0 we expect FG ¼
›F .0 ›G
ðA4:2bÞ
F GG ¼
›2 F ,0 ›G 2
ðA4:2cÞ
but it is not yet clear if
because we do not yet understand enough about genetics. However, it seems a reasonable enough assumption for a simple beginning. At the same time we can see that if we let h ¼ 1 2 e; then all the conditions of Equations (4.2) are satisfied if h , 1: In addition we can also expect limðF E Þ ¼ limðF G Þ ¼ 1
ðA4:3Þ
limðF E Þ ¼ limðF G Þ ¼ 0
ðA4:4Þ
E!0
E!1
G!0
G!1
if Equation (A4.2c) is true. These are the time-independent restrictions and conditions on the Intelligence Function. As is clear the models (both the Intelligence Function of this appendix and the dynamic model in the body) still do not account explicitly for the variation in utility of early learning vs late learning. One would still have to weight early learning (i.e. environmental variable) more heavily than late learning. One such exponential weighting scheme is given as an example further below. Since heredity changes at glacial speeds for a species we do not need to bother making G as a function of time, however in the equation above, although time is explicit to account for increasing intelligence over time, there is no time in the environmental variable itself and there is no time-dependent weighting of the environmental variable. Since we want to be able to account for the cumulative effect of the change in the environmental variable we might have to change it to R 1 0 0 I ¼ aE e G h 1 2 e 2lG
or
h
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E ðt Þdt
ðA4:5Þ
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I ¼ aG h
Z
t
E e ðt 0 Þdt 0 ð1 2 e 2lE
e
G h t0
Þ
ðA4:6Þ
0
or even I ¼ aG h
Z
416
t
R 1 0 0 h E e ðtÞ 1 2 e 2lG E ðt Þdt dt
ðA4:7Þ
0
Since it has already been argued at length (everywhere and here) that what is accumulated happens across time so that the rich-get-richer scenario (the Matthew effect of education) (of the simple kind) is of form I ðtÞ 2 l
Z
t
Kðt; EðtÞ; GðtÞIðtÞdt ¼ f ðEðtÞ; GðtÞ; tÞ
t [ ½a; b
ðA4:8Þ
a
where the I(t )¼ intelligence, and it is the past [multiplicative] interaction of intelligence with both hereditary and environmental factors summed up over time (the past) and driven by again a [multiplicative] interaction of environmental/social and hereditary/genetic factors. Among these interactions are the elegant integrated theory of both the Power Law of Practice (i.e., the more you practice the faster you get, or the Matthew effect ), and the Fan Effect (i.e., the more you know the slower you go) (Schrager et al., 1988). Integral equations of the type Z t IðtÞ 2 l Kðt; tÞI ðtÞdt ¼ f ðtÞ
t [ ½a; b
ðA4:9Þ
a
are called Fredholm equations. Volterra equation of the second kind is a special case of the Fredholm equation (Cushing, 1975 or Tricomi, 1985). We can try normalizing the solution given in Equation (31) for convenience as I ðtÞ ¼
CH ðt 2 tÞð1 2 e 2mðt2tÞ Þ þ F CþF
ðA4:10Þ
where C¼ ceiling, F ¼ floor, H(t ) ¼ Heaviside Unit Step Function and 0 # I ðtÞ # 1: This is to be realistic since nobody is born in any society to “zero environment”, but there is a floor (which changes from society to society), and there is a practical ceiling which no matter how rich the person is cannot be exceeded. Some typical ones are plotted below. We can see that I early ð0Þ ¼
F CþF
and
I late ð0Þ ¼
I early ð1Þ ¼ 1 and
Cð1 2 e mt Þ þ F CþF
I late ð1Þ ¼ 1
ðA4:11Þ
ðA4:12Þ
It would not be difficult to use nonlinear equations, for example, the logistics equation (Verhulst equation) instead of these linear equations. Simple versions of the Verhulst equation is used in Rasch testing (1980). Mathematical modeling of forced binary discrimination is naturally explained via the Verhulst/logistic equation (Hubey, 1994).
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Appendix A.5: Statistical testing methods If we have a 1st order linear ODE like this d yðtÞ þ byðtÞ ¼ f ðtÞ dt
ðA45:1Þ
we can try several easy ways to test it. Method 1: Take the Laplace transform and turn it into the algebraic domain obtaining syðsÞ þ syð0Þ þ byðsÞ ¼ f ðsÞ
417
ðA5:2Þ
Method 2: Transform the solution. The convolution integral becomes a product in the transform domain, so for the steady-state part we have yðsÞ ¼ gðsÞ f ðsÞ Method 3: Start with the solution, the steady-state of which is of form: Z t gðt 2 tÞ f ðtÞdt yðtÞ ¼
ðA5:3Þ
ðA5:4Þ
0
Discretization of this yields a time series in which the rhs is a summation of the type n X
bj2k f j
ðA5:5Þ
i¼1
This is a simple version of the ARMA/ARIMA model. IT can be done for every linear differential equation since the solution of every linear DE is in terms of a convolution integral with the kernel as the Green’s function. However, there are even more apt analogs in which the concept of linearity can be used to create more complex outputs and thus simulate richer class of behaviors. For example, let L(t ) be a linear differential operator, then LðtÞxðtÞ ¼ f ðtÞ
ðA5:6Þ
is what we’d call an inhomogeneous linear differential equation. As earlier x(t ) is a vector of output/behavior/response and f(t ) is external forcing i.e. inputs/stimulus. In this sense L(t ) is an
Figure A8. The effect of timing delay on environmental variables: the rich environment early in life leads to greater intelligence scores
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operator which is a description of the system being simulated. Since if we had a simple firstdifference equation in xn with forcing fn as xnþ1 ¼ xn þ af n
ðA5:7Þ
X xnþ1 ¼ x0 þ a a n f n
ðA5:8Þ
the solution is
418
n¼0
therefore the input into the system (i.e. stimulus) completely determines the response of the system. What happens if we try this idea on coupled equations? As given even if L(t ) consists of constant coefficients the behavior is intimately related to the linear vector systems such as y ¼ Ax since it can be approximated discretely as shown below. dx ¼ ay dt
ðA5:9aÞ
dy ¼ 2bx dt
ðA5:9bÞ
xnþ1 ¼ xn þ ayn
ðA5:10aÞ
ynþ1 ¼ yn þ bxn
ðA5:10bÞ
can be written as
which can be written as the vector equation znþ1 ¼ Azn
ðA5:11Þ
We can, of course, add external forcing into the system by writing it as znþ1 ¼ Azn þ f n
ðA5:12Þ
For the differential equation formulation we can differentiate the first (A7.11) and substitute the second (A7.11) to obtain d 2x þ abx ¼ f ðtÞ dt 2
ðA5:13Þ
where we have added the forcing f(t ) to make it similar to the discrete version. This can also be written in the formalism of linear theory since the integral part is the inverse of the linear differential operator. Therefore we can see the same principles in operation in both the differential equation and linear algebra case. In order to show the correspondence more clearly we can eliminate y(n ) from the discrete equations to obtain xnþ1 ¼ 2xn 2 ð1 þ bÞxn21
ðA5:14Þ
which is a second order difference equation corresponding to the second order differential Equation (A5.13). Without developing more comprehensive theory we can show that both xn and
yn depend on their past because of the coupling. For example, simply iterating and substituting into (A5.11) we can see that X ðA5:15aÞ xnþ1 ¼ x0 þ a yn
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Similarly X ynþ1 ¼ y0 þ a xn
419 ðA5:15bÞ
n¼0
It is clear that such simple linear equations cannot model HEPSIs (Historico Econo Politico-Social Intelligent) since there is no learning in which the system characteristics changes in response to the inputs (stimulus). In light of the fact that we can write any order difference equation as a system of difference equations of the type then we can use the same iteration technique to obtain the solution to a multidimensional (vector) difference equation with the difference that we must iterate a vector equation X A j f n2j znþ1 ¼ A jþ1 zn2j þ ðA5:16aÞ i
substituting j ¼ n we obtain znþ1 ¼ A nþ1 z0 þ
X A j f n2j
ðA5:16bÞ
i
We see that the solution at any time is determined by the weighted average of input over time (modified by the characteristics of the system itself as exemplified by the matrix A ). The initial state z(0) of the system is propagated forward by the system characteristics matrix A which is modified by the sum of the forcing (input or stimulus). However, as can be easily seen, the basic characteristics of the system as manifested in A cannot and does not change. In other words this formalism cannot model/simulate a self-learning system. It models whose behavior or response to inputs is time invariant such as that of a spring or a string on a guitar. A string on a guitar does not learn anything from the last time it was struck, and does not change its behavior in response to it ideally speaking. Practically speaking, it might stretch, the stringing might loosen etc. but in physical analysis we ignore such small changes. Indeed, if the string is stretched over its elastic limit, its behavior is not linear at all but again we have taken the simplest idealized behavior. We should be careful here to mention the caveat that much of the rich behavior of such recurrence relations is obtained from the nonlinearity as is well known. And since many of these nonlinear systems are deterministic but chaotic we can already see the germ of the solution of the philosophers’ conundrum of determinism vs randomness. We do not want life to be deterministic since it implies fatalism but we cannot assert that everything is random since it would contradict the everyday real observation that much of what we do is purposive. We do not behave at random and we’d like to think that our actions are results rational(!) beings would make repeatedly under exactly the same conditions. Since we can never have exactly the same conditions (because of learning) we cannot use the basic methodology of the hard sciences in which we repeat experiments in a laboratory. At the same time we cannot be predictable for that would make us robots or automatons. What we want is to have both determinism and unpredictability. The recent advances in mathematics is exciting because we can have both rational/logical action and at the same time uncertainty of behavior. In one case, we have fuzzy logic in which we can use uncertainty not due to chance or probability which is a better
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philosophical idea. In the other case we have chaos in which we have deterministic algorithm but unpredicable output. In a sense, stochastic or random is infinite dimensional chaos implying figuratively that there are degrees of unpredictability which is the embedding dimension. However the differential equation formulation allows explanations of the type in which many of the criticisms made against the previous models are not valid. The main reasons why such models are not used is probably because of the philosophical attitude of those who work in the HEPSI systems field. The equations above were ordinary, linear, deterministic, forced (and homogeneous) differential equations with constant coefficients (scalar and vector types). The main thrust of this appendix is to show how and why what is needed or is thought to be needed to model/simulate HEPSI systems is or can be shown to be in differential equations. Naturally, those equations that are not solvable can always be solved numerically (i.e. simulated on a digital computer) to (almost) any degree of accuracy. It is possible that some of the complex behavior shown in numerical solutions of nonlinear differential equations is due to the inherent problems of error propagation in discretization (which is a form of quantization) noise. Appendix A.6: Absolute measurement scales and fuzzy-logic, heritability-multiplicativity paradoxes If the additive model of intelligence is used then we can use the method of virtual variation to analyze how the variations in environment and genetic makeup will influence intelligence [scores] by computing the total differential (or variation) as
dI ¼ 1·dE þ h·dG
ðA6:1Þ
The variation above, say in E, is amplified through the effect of the coefficient 1, and thus the share of the total variation in I, is due to 1d E. However since the variation could be both positive or negative, to obtain the magnitude of the variation, or a number with essentially the same dimensional significance as the variance we should compute
dI 2 ¼ 1 2 ·dE 2 þ h 2 ·dG 2 þ 2h1·dE·dG
ðA6:2Þ
For the nonlinear case explicitly considered in above the variation is
dI ¼ ðaeE e21 G h Þ·dE þ ðahE e G h21 Þ·dG
ðA6:3Þ
But now the total variation due to the variation of E, is amplified via the effect of coefficients which obviously have genetic components. Can we now compute d I2 and still claim that the coefficient of d E2 is still only the environment effect? Clearly no. The type of model does make a significant difference in the interpretation of results. Continuing with the computations, since the terms dE 2 and dG 2 are the virtual variations which are the equivalents of variances in statistics and since we explain the total variance statistically via V ¼ V E þ V G þ V GE the likely conclusion is that the statistical methods work on variables measured on the absolute (ratio) scale and furthermore on some “natural scale”. For the linear case we could not make a claim (unless the absolute scale were used i.e. E ¼ G ¼ 1) that the coefficient of dE was the variation due only to the change in the environment. For the nonlinear case, the change in genetic endowment is amplified by environment so that we could not really claim that the second term in Equation (A6.2) is the variation due to genes whereas in the linear case, the variation in G is merely amplified by a constant h, which we assume has to do with genetics. Therefore, even analysis of variance is not scale free and it is not model free. In other words, we are faced with two conflicting views. One of them is that we should introduce new “dimensionless” variables E~ and ~ and then thus compute the variation as dI ¼ dE~ þ dG~ whereby Equation (A6.2) becomes G;
dI 2 ¼ dE~ 2 þ dG~ 2 þ 2dE·~ dG~ ¼ V ¼ V E þ V G þ V GE
ðA6:4Þ
where the new variables are the natural (absolute/ratio) scale variables. The second view is that we must have something like 1 þ h # 0:5 since we already know that we should have 0 # I # 1 in order to have the additive model valid as the Taylor expansion of the multiplicative model. Since we also want 0 # E, G # 1; unless condition 1 þ h # 0:5 is valid the intelligence measured will not be in the normalized range. We have already produced an approximately linear model in the neighborhood of human level intelligence (i.e. Equation (18)) from the basic nonlinear-multiplicative model, in which we set E h ¼ Gh ¼ 1: We also noted that setting e ¼ 1 2 h (Appendix A.4) fulfills some of the conditions for the Intelligence Function. Now if we let a ¼ 1=2 (with the new variables we can see that Equation (18) becomes I ¼ E~ þ G~ 2 1
ðA6:5Þ
Now the variations and the variances are
dI ¼ dE~ þ dG~ and dI 2 ¼ dE~ 2 þ dG~ 2 þ 2dE·dG~ ¼ V ¼ V E þ V G þ V GE
ðA6:6Þ
exactly as they should be. Because we have obtained the linear approximation from the nonlinear one, it is valid only in the neighborhood of E~ < G~ < 1: Certainly we must have E~ þ G~ . 1 otherwise we will obtain negative intelligence. Even this condition is not stringent enough since it would imply that human genetic endowment ranks along that of viruses or that our environment is about equivalent to that of speechless beasts. In order to be valid because of the approximations inherent, we’d need the constraint E~ þ G~ < 2: The other kinds of nonlinear formulations repeated below for convenience in slightly different form I ¼ cE 1 e jG eE eE
a þG b
aG b
¼ cons a
b
¼ e E e G ¼ cons
ðA6:7aÞ ðA6:7bÞ ðA6:7cÞ
have the virtual variations
dI ¼ cð1E 121 e jG dE þ jE 1 e jG dGÞ dI ¼ e E
a
Gb
dI ¼ e E
ðG b E a21 dE þ E a G b21 dGÞ
a þG b
ðE a21 dE þ G b21 dGÞ
ðA6:8aÞ ðA6:8bÞ ðA6:8cÞ
All of these are restricted to the interval [0,1] in order to correctly model the environment-genetic interaction, and they are on an absolute (ratio) scale. As can be seen, the amount of “interaction” of genes and environment in Equation (A6.7b) is much greater than the others in that it is difficult to separate the variation due to one of the factors from the other. Any of these could have been used in the differential equations model of learning and development of intelligence. Only the simplest was used for convenience. The best fitting model should probably be used in studies. There are, as can be seen, great benefits from having measurements which are on an absolute “natural” scale and in which the nonlinear interaction is explicit. One of the biggest problems, if not the biggest, that of finding natural dimensions such as Time, Length, Mass, and Temperature in the physical sciences has been overcome with the combination of statistical analysis, differential calculus, and a fuzzy-logical AND (i.e. intersection). What is even more to the point is that we now have many choices of using various different types of infinite-valued
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logic (i.e. fuzzy-logic see Hubey (1998) or Klir & Yuan (1995) for examples) for modeling social phenomena of this type. As a simple example, a chimp raised in a human environment will essentially still have chimp-level intelligence, and a human raised in a chimp environment (feral children) will have chimp-level intelligence. The simplest expression of this idea is via the fuzzy-AND introduced by Zadeh (1963; 1978; 1987), which is a simple minimum function. Thus I chimp ¼ MinðE human ; Gchimp Þ ¼ MinðGchimp ; 1Þ ¼ Gchimp ! 1
ðA6:9aÞ
I human ¼ MinðE chimp ; Ghuman Þ ¼ MinðE chimp ; 1Þ ¼ E chimp ! 1
ðA6:9bÞ
Over a global scale, there is no doubt that the environment/culture created is mostly a function of genetics, and that intelligence is mostly determined by genetics, but on local scales, the environment acts essentially as a stochastic variable in the determination/realization of intelligence. There are many more such fuzzy-AND functions that could possibly be more appropriate. What is of further interest is that the multiplicative model (which is logically the correct model) produces anomalies of its own. The multiplicative model consists of a family of hyerperbolic curves as shown in Figure A9. Although the first person (E1,G1) scores higher, I3, his genetic endowment is lower than the second person (E2,G2) who scores lower, I2, on the intelligence test.
Appendix A.7: The Meaning of nonlinearity in dynamics & learning Starting from the perspective of dynamics there are 4 factors: (1) initial conditions (2) laws of evolution (i.e. the particular form of the DEs) (3) sources of randomness (a) parameters (b) forcing (c) initial conditions (4) the need to have a model which can incorporate learning All of these are satisfied with general nonlinear differential equations. Gellman calls (3c) “frozen accidents”. The step (3b) is needed to model outside influences (i.e convergence). Step (3a) (and nonlinearity) is needed to model learning/modification and evolution. Any set of ordinary DEs can be written in canonical form as dy ¼ Gð yðtÞ; f ðtÞÞ dt
ðA7:1Þ
For example a simple linear and ordinary DE is d y þ aðtÞyðtÞ ¼ f ðtÞ þ wðtÞ dt
ðA17:2Þ
The forcing f(t ) or w(t ) can be deterministic or random. If f(t ) is deterministic and w(t ) is random then we can think of them as external sources of influence. The particular characteristics of the system (i.e. internal characterization) is given by the left-hand-side (lhs) which is 1st
order/degree, and in this case has a time-dependent parameter a(t ). If a(t ) were constant then it too would be a “frozen” accident or parameter. Furthermore if a(t ) also had sources of randomness then we’d be able to represent systems that fluctuated due to randomness. Still further, the initial conditions of the system are not known, and those initial conditions will affect the outcome y(t ) at any time t, so that we have yet one more source of randomness which because of the way language started is yet another frozen accident. In this paper it will be shown that those properties/characteristics of the brain/mind that are allegedly impossible to model in any abstract way can be done quite fruitfully combining some of the understanding of the recent past dealing with nonlinear dynamics. In fact, these simple analogical models are also the type of models we need for an understanding of the basic issues in all intelligent systems including those that are ensembles of intelligent beings such as historico-econo-politico-social-intelligent (HEPSI) systems.
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Linear systems The definition of linearity is easily explained in terms of operator. Normally in mathematics we deal with a set of objects upon which another set of objects (called operators) act thus changing the objects of our attention to other objects. A linear operator L() is then an operator whose behavior is describable as Lðx þ yÞ ¼ LðxÞ þ Lð yÞ
ðA17:3Þ
In other words, it is an operator that commutes with addition. It is much easier to see if we write it as LðAðx; yÞÞ ¼ AðLðx; yÞÞ
ðA17:4Þ
Since we normally deal with binary operators i.e. an operator that operates on only two objects, we could have written x þ y as A(x,y ) or +(x,y ) or even as (x,y )+. The first notation is called infix notation and is what we normally use for addition, the second prefix and last postfix. As examples of linear operators we have the matrix operator that operates on a vector (array) which we could write as y ¼ Ax: This operation is quite common in the social sciences because of correlation-regression analysis. A one-dimensional example of a dynamic behavior of a system can be written as wnþ1 ¼ bwn
ðA7:5Þ
Figure A9. Intelligence function/potential paradox: The linear relationship I ¼ E þ G 2 1 is approximate and valid only around E ¼ G ¼ 1; therefore the paradox is more difficult to achieve or display. If however, the nonlinear version is used, we can make use of the full scale
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By substituting repeatedly we can see that the solution for all time is wn ¼ b n w0
ðA17:6Þ
Since (A7.6) is a linear equation i.e. can be represented as a vector equation, it is easy to see that the solution can also be represented as a vector equation. However, this formalism is also sufficient for representing multi-dimensional dynamics since we can simply write xnþ1 ¼ Axn
ðA7:7Þ
where the evolution of vector x (at time n þ 1 i.e. xðn þ 1Þ) is determined by matrix A. This can be taken to be a very simple model of an input/output system. It is used in economics, however it can also be used as a toy mathematical model of simple behaviorism in the form rðtÞ ¼ AðtÞsðtÞ in which r(t ) is the response at time t, to a stimulus (input) s(t ) at time t. We can allow the characteristics of the system represented by A(t ) to be a function of time to allow for slightly more flexibility that is normally allowed in simplistic behaviorism. We cannot expect intelligent (learning) systems to respond to the same stimulus in exactly the same way each time. The simple behaviorist view which was meant for low intelligence organisms or for particular aspects of the behavior of HEPSIs such as operant conditioning or classical conditioning cannot be expected to hold at such simplistic levels, although there are elements of such behavior. For example, if a lion were fed at exactly noon everyday and it refused to eat any more for five hours, we can approximate his approximate behavior by making the characteristic/evolution matrix A(t ) appropriately dependent on time instead of constant.We have already discussed linear systems and shown their inadequacies. We would now like to ask the question of what more is needed to model HEPSIs. We can already see that we need a model in which the characteristics of the system changes in response to inputs (stimuli). In a differential equation, the characteristics of the system is a function of the operator. The number of derivatives, their signs, the values of the coefficients, and whether the coefficients are functions of time or are random all determine the response of the system to inputs/stimuli (or forcing/source). The differential equations model explicitly takes into account both the characteristics of the system and the environment (inputs). Secondly by providing us with the possibility of making the forcing/environment deterministic or random or a combination of both it gives us a greater freedom in modeling and provides a greater expressivity in modeling complex systems. Furthermore since it can always be reduced to discrete forms which may be thought of as cellular automata, or iterations of the type that produce fractals, or chaos, and since cellular automata are equivalent to Turing machines, it gives us the means to show that Turing machines are capable of producing a richer repertoire of behavior which can easily be sufficient to be a HEPSI system. To see this we can look at an example of a partial differential equations and how it relates to Turing machines and cellular automata. For example the elliptic partial differential equations in two dimensions known as the Poisson equations is given by d2 d2 uðx; yÞ þ 2 uðx; yÞ ¼ f ðx; yÞ dx 2 dy
ðA7:8Þ
where the values at the boundaries are fixed and given. Typically discretization of this equation involves expressing the relationship of a point u(i, j ) in a grid in terms of the values which are neighbors of this grid. The central-difference method (Burden, Faires, Reynolds, 1981: 511) gives 2
2 2 h h ui; j 2 uiþ1; j þ ui21; j þ ui; jþ1 þ ui; j21 ¼ 2h 2 f xi ; yi k k
ðA7:9Þ
where each u(i, j ) is an approximation for u(xi, yj) which is the discrete grid representation of the continuum problem. This can be re-expressed as a vector equation and solved Gauss-Seidel
method. More complex cases can be solved iteratively. Given the relationship between cellular automata and Turing machines we can see that solutions of continuum problems can be approximately solved by Turing machines. Therefore there must be more to the protests against artificial/machine intelligence than simply the analog capability of natural brains. Simple bending of a metal ruler takes much more equation and more mathematical sophistication than that. Do we expect human behavior to be simpler than a deforming ruler? A linear differential equation easily has nonlinear solutions. What this means is that we can change our domain of inquiry. So in the differential domain we can still create such models from which we can obtain nonlinear solutions. This is another attempt at simplification on the one hand (of model creation) and at the same time producing the ability to create more powerful methods which can yield complex solutions. The answer, almost as a cliche, is in nonlinearity. First of all, we should note that the nonlinearity is in the differential equation and not in the solution; that is, linear differential equations almost always have nonlinear solutions. But nonlinear differential equations are almost always not solvable in closed form. Nonlinear differential equations are those in which the dependent variable or its derivatives do not occur to any power greater than one and do not occur in multiplicative form. Almost all such models are linear in some suitable way. It is could be y ¼ Ax where A is matrix and y, x vectors but there are lots of other ways. Statistical correlation regression tests etc. are of this type more or less. Nonlinearity cannot usually be handled because it is too difficult. It is only now with the power of the digital computer that numerical solutions can be found to some nonlinear differential & integral equations. But this is a bare minimum.
Linear DEs, and time-dependent coefficients, and the black box Starting with the simple linear ODE (A7.2) or (25) with wðtÞ ¼ 0 and a(t )¼a ¼ constant we obtain the solution as in Equation (27). The solution is exponential and not too interesting. The meaning given to it is that f(t ) is external/exogeneous and the coefficient a(t ) or a is a system characteristic, which is a structural or endogeneous parameter. The steady state solution has the “black box” formulation since a discretization of the integral is essentially a matrix multiplied by f which is a vector. With the initial condition I.C. yð0Þ ¼ 0 the solution is Z yðt 2 tÞ ¼ f ðtÞe 2bðt2tÞ dt ðA17:10Þ Another example of such a system is the set of equations given by Equation (A7.5). Both coefficients are constant although the internal parameters of the system do affect each other. The easiest way to make the ODE model more powerful is to allow the coefficients to become functions of time, for example as in Equation (A7.2). But it is still linear and f(t ) is still deterministic. In a “learning system” then this endogeneous parameter should somehow be affected by the inputs (over time) f(t ). Suppose the equation is of type d y þ að f ðtÞÞyðtÞ ¼ f ðtÞ dt
ð17:11Þ
now the system parameter is a function of the external/endogeneous variable and we can think of the system as changing its characteristics over time. In addition we know from the solution of the linear first order DE that f(t ) is the endogeneous variable. We could also make that a function of the system variable itself so that there is a feedback loop created. We now have d y þ að f ðtÞÞyðtÞ ¼ f ð yðtÞÞ dt
ðA7:12Þ
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To create a simple example of this kind of a system let að f ðtÞÞ ¼ 2xðtÞ and f ð yðtÞÞ ¼ 2yðtÞ: dy ¼ 2y þ xy dt
ðA17:13Þ
Let another equation of the same type represent another system interacting with this system. dx ¼ x 2 xy dt
426
ðA17:14Þ
These are the Lotka-Volterra differential equations. Note the rhs of the equation for x, Equation (A7.14), can be written as xð1 2 yÞ which is x AND not-y in fuzzy logic. However with the coefficients such a meaning is a stretching of the imagination. The other can be written as yðx þ 1Þ which no longer has a meaning in such fuzzy logic. That is because even in the 1st equation we can have a (2 ) sign which is not expressible in fuzzy logic because of the constraint of having the values in [0,1]. If we switch to values in, say, [ 2 1,+1] we could then attach meaning to negative values. This can be done simply by shifting (i.e. define new variables). But now we still have the problem of derivatives which do not have meanings (yet) in fuzzy logic. But fuzzy logic is sort of between logic and probability theory, and if we stretch our imagination then derivatives do have meanings in probability theory (i.e. FokkerPlanck methods etc). So differential equtions are more expressive and more powerful than logics. Appendix A.8: Role of genes and heritability: a deeper peek Many dynamic physical phenomena can be described via differential, difference or integral equations. Since most differential or integral equation cannot be solved in closed form but must be solved numerically (i.e. using a digital computer to solve for specific cases called simulation) we can also use difference equation (which are really discretized versions of differential equations). For a linear differential equation, the solution provides a mapping from the source to the output. We can think of this as a black box with the source term as the input and the solution as the output. One way in which to classify differential /is; (a) ordinary vs. partial (b) linear vs. nonlinear (c) deterministic vs. random (d) homogeneous vs. inhomogeneous (e) constant vs. time-dependent coefficients Biological determinism is fundamentally a theory about limits. . . Why should human behavioral ranges be so broad, when anatomical ranges are generally narrower?. . .[I] conclude that wide behavioral ranges should arise as consequences of the evolution and structural organization of the brain. Human uniqueness lies in the flexibility of what our brain can do. What is intelligence, if not the ability to face problems in an unprogrammed (or as we often say, creative) manner? Gould (1981):331 The easier of the two has been denoted above in italics. Unfortunately, real processes in the real world happen not to be so simple. It’s an fortunate fact, however, that we can approximate much phenomena of the real physical world with linear equations. But unfortunately for the biologicallife-social sciences even achieving the simplest models requires competence in mathematics that transcends what is required to study physics, engineering or computer science, the so called hard
sciences. It’s really the soft sciences that are hard. Because they are so hard (i.e. difficult) it’s not easy to either create or explain the mathematical models to those in the field and hence the fields stagnate in the verbal or prescientific state, which allows everyone to have a ‘theory’ since any number of words strung together now comprises a theory. Even the discussion of what is a theory is constrained to discussion via classical bivalent logic or at best of statistical methods. One often encounters the terms equilibrium, stability and steady-state in connection with differential equations. In words, stable systems are those that return to their rest positions when the forcing is removed which implies that the response of the system to a disturbance is to return to its original position after some transients. This implies that there are no more changes in the system after a sufficient amount of time has elapsed. For the homogeneous case, the solution goes to zero as the independent variable (usually time) increases without bound. In general an nth order differential equation with constant coefficients can be written as a first order vector differential equation. In this case, a system is stable if the real parts of the eigenvalues of the matrix of coefficients are all negative. In some systems, the solution does not decay to zero; neither does it increase or decrease. In addition the concept of stationarity is usually assumed in connection with many stochastic processes. Stationarity implies that the statistics associated with the process do not change in time. Obviously, this means that the probability density is not an explicit function of time. The language evolution problem certainly doesn’t fit into this mold, since the density that is derivable from the set of differential equations describing the language’s evolution will be a function of time in general. However, there might exist a steady-state or a stationary version of the density. One final concept, that is often implicitly (and maybe unconsciously) used is that of ergodicity. Simply put, it means that the space (ensemble) averages of a stochastic process will be equal to its time averages. This is an additional concept piggybacked onto stationarity for making problems tractable. One of the ways in which random processes can be modeled, especially if there’s a deterministic component which can be expressed as a differential equation, is via the Fokker-Planck method. If model is given by a set of differential equations dxi ¼ f i ðxi ; tÞ þ Gðxi ; tÞwðtÞ i ¼ 1; 2; 3. . . dt
ðA8:1Þ
where x and f are vectors, G is an n £ m matrix and w(t ) is an m-vector zero-mean white Gaussian noise with the autocorrelation function; kwðtÞwðt 2 zÞl ¼ QðtÞdðt 2 zÞ
ðA8:2Þ
where Q(t ) is positive semidefinite matrix and d(t ) is the Dirac delta function, one can obtain an equatin for the first-order and second-order probability density functions as below; X › ›p 1 X ›2 ¼2 ð pf i Þ þ ½pGQG ij ›t ›xj xi ›xi 2
ðA8:3Þ
For the special case of a single equation the above reduces to
›p › 1 ›2 2 ¼ pðx; tÞ f ðx; tÞ þ ½g ðx; tÞpðx; tÞ
›t ›x 2 ›x 2
ðA8:4Þ
The Equations (A8.3) and (A8.4) are known as the Fokker-Planck equations or the forward Kolmogorov equations for the random differential equations shown. A general ordinary set of differential equations may be written as u0 ¼ Fðu1 ; u2 ; . . .; un ; tÞ
ðA8:5Þ
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where ¼ 1,2,. . .,n and prime indicates derivative. The u may be thought of as the various intensive or extensive parameters describing evolution of a process. In general, the differential equation is a local description of the phenomenon since the effects of the various properties of the system on each other is described in an instant of time in a localized small region of the phase space. In contrast, the solution gives a global description of the behavior of the system. It is for this reason, that we seek solutions to differential equations. Finally, an important reason for this kind of a stochastic model is that it has both deterministic and random components. Even with the simplifications introduced, there are still useful analogies. For example, Gell-Mann, in the discussion of language as a complex adaptive system makes the statement (Hawkins & GellMann, 1989) In every one of these complex adaptive systems, there are fundamental rules, frozen accidents and selection pressures for functional outcomes. As usual, what is effectively a fundamental rule on one time scale may be a result of accident or functional adaptation in one of the complex adaptive systems operating on a longer time scale. We can explain these concepts within the framework of the differential equation model. The Initial Conditions (IC) are one example of the frozen accidents or founder effects. Nonlinear equations exhibit behavior such as catastrophic (fast, discontinuous) changes and bifurcations. Nonlinear equations are also highly sensitive to initial conditions which is a hallmark of chaotic behavior. Since reality is always more complex than our simplifications, it often times behooves us to treat the simplistic deterministic systems as stochastic systems. The stochastic differential equation models are particularly useful since they show a deterministic component as well as a random one piggybacked onto the deterministic component. According to Siski, there are three ways in which randomness can enter into differential equations (Siski-1967; Van Kampen-1976 or Hubey, 1993). (1) Random Initial or Boundary Conditions (2) Random Forcing (3) Random Coefficients The randomness in the initial conditions is the easiest case since the mapping from the initial conditions to anytime which is accomplished by the Green’s function is deterministic and if the source term is also deterministic, this implies simply that the probability density of the solution is a deterministic function of a random variable (the IC) and we can obtain the probability density of the solution (the IC part) via well known methods as can be seen, for example, in Papoulis (1984). The randomness in the source term means that there’s a deterministic mapping (i.e. the Green’s function) of a random function (the random part of the source or the forcing) to produce the final result. We can obtain some statistics about this process in many different ways, and has been extensively discussed in the literature. The third case however is much more complex. A randomness in the coefficients implies that the Green’s function itself is random and that there is now a random mapping (Green’s function) of a random function (source). Thus the output y(t ) is comprised of at least two, possibly independent random processes. In the first case (i.e. randomness in the initial conditions) since the mapping from the set of initial conditions to the solution is deterministic, it essentially involves the calculation of the probability density of the solution from the probability density of the initial conditions using well known transformations. In the second case, the problem revolves around finding the deterministic solution of the equation. If the equation is linear, the steadystate solution can be expressed as an integral with the Green’s function as the kernel of the integral. Thus the solution of the random problem involves a deterministic mapping of the random source and various methods such as mean-square analysis and statistics such as the autocorrelation and the spectral density of the process can be computed. If the equation is
nonlinear, then either linearization or Fokker-Planck methods may be attempted. The random coefficients problem is the most intractable. Methods for solving problems of this type can be found in Van Kampen (1976) among others. Man came first to the realm of minerals, and from them he fell in among plants. . . when he left the plants and joined the animals, he remembered nothing of the vegetative state. . . In the same way he passed from realm to realm, until now he’s intelligent, knowledgeable, and strong. He remembers not his first intellects, and he will leave this present intellect behind. Rumi (Chittick, 1983) As a concrete example of the determinism in evolution, let’s look at a model of random genetic drift in the narrow sense due to Kimura (Kojima, 1970). For equations of this type see also Roughgarden (1979).We assume that mutation, migration, and selection are absent, and that the change of gene frequencies from generation to generation is caused only by the random sampling of gametes in reproduction. We consider a locus in which a pair of alleles A1 and A2 are segregating. Using the Fokker-Planck method the forward Kolmogorov equation for the transition probability density f(x,t ) that the frequency (relative proportion in the population) of A1 lies in between x and x þ dx at time t, given that it is p at the start (i.e. at time t ¼ 0) is given by
›f › 1 ›2 › 1 ›2 2 ¼ f ðx; tÞfðx; tÞ þ M ½ g ðx; tÞ f ðx; tÞ
¼ 2 f ðx; tÞ þ ½V d x fðx; tÞ ðA8:6Þ d x ›t ›x 2 ›x 2 ›x 2 ›x 2 where Vdx and Mdx are respectively the mean and variance of dx, the amount of change in gene frequency x per generation. Now for the specific case in which a pair of alleles A1 and A2 segregating with respective frequencies x and 1 2 x in a random mating population of N monoecious individuals and the mode of reproduction is such that N male and N female gametes are drawn as random samples from the population to form the next generation. Then the mean and variance in the change of gene frequency are Mdx ¼ 0
ðA8:7aÞ
V d x ¼ xð1 2 xÞ=ð2N Þ
ðA8:7bÞ
Then the forward equation becomes (Kojima, 1970, 183)
›f 1 ›2 ðxð1 2 xÞfðx; tÞÞ ¼ · ›t 4N ›x 2
ðA8:8Þ
Now, it’s interesting that this equation is the same equation as the one for the probability density of the stochastic differential equation dx 1 ¼ gðx; tÞwðtÞ ¼ xð1 2 xÞwðtÞ ðA8:9Þ dt 4N where w(t ) is a zero-mean white Gaussian noise. It’s also interesting that Equation (A8.9) without the noise (i.e. wðtÞ ¼ 1) is the famous logistic equation the discrete version of which has now become known as the Feigenbaum Oscillator of chaos fame. What this means is that we have a deterministic process whose coefficients are being multiplied by a random function. The coefficient (1/4N ) is essentially the propensity to segregate in one direction or the other. For a more complicated case in which we consider a reversible mutation rate u
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from A1 to A2 and v in the reverse direction, with x the frequency of A1, we have (Kojima, 1970:186) M dx ¼ ux þ vð1 2 xÞ
ðA8:10aÞ
1 V dx ¼ xð1 2 xÞ Ne
ðA8:10bÞ
From Equation (A8.3) we can see that they describe the stochastic differential equation dx 1 xð1 2 xÞwðtÞÞ ðA8:11Þ ¼ vð1 2 xÞ 2 ux þ dt 2N e Looking at this equation, it’s difficult to use the words deterministic and random in the usual fuzzy sense in which it is used in biology, evolution and psychology. Yes the equation above has both deterministic and random components. And it cannot really be otherwise; no human can give birth to chimps or dogs. It’s as deterministic as 2 þ 2 ¼ 4 from this perspective. However we should note that the coefficients of the equations are considered to be random even if the equation itself without the added noise is a deterministic equation. In this case, there are other methods that can be used to derive solutions in different ways. There is an important result from Van Kampen (1976) that the average of a solution is not the same as the solution of averages. In other words if we have the equations for the evolution of a group of genes which determine a specific trait (i.e. polygenetic trait) then we cannot simply solve the equation and find the average of the solution. This has importance in scaling from the micro viewpoint to the macro viewpoint when the equations are nonlinear and we have more knowledge of the micro behavior (i.e. specific genes) instead of evolution of the whole organism (i.e. macroscopic or polygenic traits). Therefore in order to produce equations for the evolution of a polygenic trait, we need to scale and average upwards from the equations [if we have them] for a set of genes. Similarly we’d have to scale and average upwards again from several polygenic traits toward higher level traits so on until we have some kinds of averaged equations determining evolution of the whole organism. In short, life implies intelligence. The absolute intelligence scale must come from living beings. Intelligence, order, organization, entropy, and probability are related concepts. Hence evolution is probably the best way to define intelligence, and evolution can be thought of as entropy which completes the circle. In truth, the link between information theory and probability theory (and between error and chaos) is probably the best link to enable one to model something as complex as society. In any case, the intelligences of animals (including humans, of course) do seem to fulfill the conditions for scalability of Guttman (see for example, Maranell, 1974). As for an absolute or a ratio scale for intelligence, it remains yet to be seen what can be used except perhaps some kind of a brain-size to body-size ratio. A plot of the DNA nucleotide pairs as a percent of the mammalian DNA content shows a remarkable correlation with what we consider using common sense to be levels of intelligence. (Britten and Davidson, 1969). A log-log plot of the time of origin of the species (years ago) against the number of bits of information encoded in the DNA nucleotide pairs (per haploid cell) (borrowed from Britten and Davidson), and also against the information content of the brain, in Sagan (1977) also shows a correlation with intelligence. Similarly a plot of brain mass against body mass of mammals (and also of all animals) shows an expected ordering according to intelligence (Sagan, 1977). One can see similar patterns or order in the brain masses and also of the parts of the brain associated with higher functions in the primates in Eccles (1989). The overall conclusion is that an absolute scale of intelligence should in some way be associated with comparison of brain mass to body mass. It should probably be of the form
I ¼W
b 2 kB 2 c b2c d ¼W 2k ¼W 2k B B B
ðA8:12Þ
where b ¼ brain mass, B ¼ body mass, and k and c are constants. The reason for subtracting a constant amount from b is that there must be some minimal amount of brain matter necessary to keep the life processes going. The reason for subtracting an amount proportional to the body mass B is that there is probably a need as the body size gets larger for greater or finer control of the voluntary muscles so that an elephant or a whale would seem to need more neurons purely for its existence and not for higher or more abstract levels of cognition. The form of the function W() itself is of great concern but it would seem to be almost arbitrary in the sense that, we’d have to decide with our naked senses that, say we are more intelligent than chimps, and that chimps are more intelligent than dogs, and so on. We might note, for example, that chimps seem to recognize themselves in mirrors. As another example, we might note that a dog or a cat would notice and try to shake off something stuck to its paws whereas a duck might be oblivious to it. We might also indirectly try testing the number sense of various animals. There really is not that much difference in ordering the intelligences of animals according to some minimal types of behavior patterns than noticing that as the temperature gets hotter the liquid in a glass bulb thermometer goes up and not down, or that the things that seem to be heavier stretch a spring more than those that aren’t. We have further hints that the proportion of masses as an indication of intelligence (at least across species) is in the right direction by noting that dimensional analysis which is often used to obtain the forms of the parameters in dealing with complex areas such as fluid dynamics dictates that we use a ratio of the masses so as to make the term dimensionless. The usefulness of dimensional analysis can also be seen in the construction of metric spaces for speech analysis in Hubey (1994). We could, of course, try a logarithmic scale something like the decibel scale using a ratio of human intelligence (or the brain/body ratio as above) to the intelligence of the animal in question. Whatever the case, it is important that an absolute/ratio intelligence scale be used that can pass muster of researchers in artificial intelligence, psychologists, and neurophysicists. And if intelligence is to be broken down into its components (however many) then there should be an operational definition of a single number that is derived from such tests. In such a case we can use a weighted function of the various parts of the brain so that some kind of an absolute intelligence scale should be established with which we can make progress toward machine intelligence instead of simply being forced to deal with unrelated and incoherent ideas such IQ tests using correlation-regression analysis, Turing tests for machine intelligence, expert systems whose knowledge (and intelligence!) defy our imagination.
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The current issue and full text archive of this journal is available at http://www.emeraldinsight.com/0368-492X.htm
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Temporal and simultaneous processing in the brain: a possible cellular basis of cognition Uri Fidelman Department of Education in Technology and Science, Technion, Israel Institute of Technology Haifa, Israel Department of Behavioral Sciences, Emek Izrael College, Israel Keywords Cybernetics, Brain, Cognition, Quantum mechanics Abstract It is suggested that the left hemispheric neurons and the magnocellular visual system are specialized in tasks requiring a relatively small number of large neurons having a fast reaction time due to a high firing rate or many dendritic synapses of the same neuron which are activated simultaneously. On the other hand the right hemispheric neurons and the neurons of the parvocellular visual system are specialized in tasks requiring a relatively larger number of short term memory (STM) Hebbian engrams (neural networks). This larger number of engrams is achieved by a combination of two strategies. The first is evolving a larger number of neurons, which may be smaller and have a lower firing rate. The second is evolving longer and more branching axons and thus producing more engrams, including engrams comprising neurons located at cortical areas distant from each other. This model explains why verbal functions of the brain are related to the left hemisphere, and the division of semantic tasks between the left hemisphere and the right one. This explanation is extended to other cognitive functions like visual search, ontological cognition, the detection of temporal order, and the dual cognitive interpretation of the perceived physical phenomena.
Kybernetes, Vol. 31 No. 3/4, 2002, pp. 432-481. q MCB UP Limited, 0368-492X DOI 10.1108/03684920210422566
Introduction This is a theoretical study. Its purpose is to characterize some fundamental neurological factors which underly several, if not all, of the cognitive functions. We have suggested in previous studies (Fidelman, 1998a; 1999a) two such factors. The first factor is the neurons’ firing rate. Let us assume a metabolic similarity of the plasma of the neurons, e.g. similar density of the ribosomes in the plasma of the neurons. Then the firing rate is influenced by the neurons’ capability of transmitter production, which is determined by the volume of the plasma. Assuming spherical cells, the volume increases as a function of the third power of the radius. That is, the firing rate increases very fast as a function of the neurons’ size. A high firing rate requires also a large rate of receiver production, but this condition is fulfilled automatically. We have suggested that analytic and temporal processing (which may be related to the left hemisphere) require a high rate of neural firing.
A large firing rate of a neural system, however, does not depend merely on the quantities of available transmitter and receiver, which are proportional to the volume of the neuron, in particular the volume of the soma. These quantities depend also on other factors. Such factors are the re-uptake of the transmitter and the type of the transmitter. However, the volume of the plasma is one of the factors determining the firing rate. Therefore, large neurons are expected to have, on the average, a larger firing rate than small neurons. An important factor influencing the firing rate is the number of synapses per neuron. If a neuron has a large volume and therefore a large quantity of available transmitter, and it has also a large number of post neural synapses, the quantity of transmitter per synapse may be small, resulting in a small firing rate. The same is true regarding the number of pre-neural dendritic synapses and the quantity of available receiver. This may explain the phenomenon that small interneurons, which have a small number of synapses, may have a faster firing rate than large neurons having many synapses. High firing rate is not the only way by which a large size of the neurons of a neural system can induce a fast reaction in this system. A large soma can sustain both a large number of axons’ branches and a large number of dendrites. Therefore, in a neural system having large neurons there is a relatively high probability that many branches of axons of a presynaptic neuron create synapses with the dendrites of a certain post synaptic neuron. In this case the summation of the dendritic post-synaptic potentials over the large number of synapses common to both large neurons may cause action potential rapidly. This occurs even if the firing rate of the presynaptic neuron is relatively low. Thus an entire neural system consisting of large neuron may react fast, even if the firing rate of the neurons is low. This argument is an improvement of the early works. This consideration may be extended to many axons of a neural network many axons of which have common synapses with a large number of dendrites of another neural network comprising large neurons. The network which has many dendrites can be activated rapidly. Another factor suggested in the earlier studies (Fidelman, 1998a; 1999a) is the probability of neural transmission errors at individual synapses. If the quantity of available transmitter is too large, there may be an increase in the post synaptic dendritic potential even if the presynaptic neuron did not fire (for example, due to remains of the transmitter). This may influence both the cognitive style and the efficiency of the neural processing. This type of transmission errors may characterize large neurons with a high rate of transmitter production and a high firing rate. If the quantity of available transmitter is too low, then post synaptic dendritic potentials may not increase, in spite of the firing of the presynaptic neuron. This too may influence the cognitive style and the efficiency of the neural processing, but in a different way. This type of transmission errors may
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characterize small neuron which have a low rate of transmitter production and a low firing rate. An innovation of this study is that a third neural factor is suggested, namely, the number of neurons. It is suggested that the larger is this number, the larger is the number of individual data which can be processed simultaneously in the brain. This simultaneous processing is related to the right hemisphere, as follows from the following consideration. Miller (1956) suggested that the brain can process a certain maximal number of data simultaneously. This maximal number is, averagely, seven, and its range is between 5 and 9. When the number of the processed data is larger than this maximal number, the brain integrates chunks of data, each of which is processed analytically as an individual item of data. Papcun et al. (1974) found a left ear (right hemispheric) advantage in the processing of Morse code (by subjects not knowing this code) when the number of dots and dashes was no more than seven. However, when the number of dots and dashes was larger than seven, there was a left ear (right hemispheric) advantage. The findings of Papcun et al. may be interpreted as indicating that Miller’s chunking is performed by the right hemisphere, which includes more cells and more engrams (at least in male subjects) for representing the larger number of items. An extended discussion is presented in Fidelman (1999a). Since the metabolic resources of the brain (glucose and oxygen) are limited, neural systems have two extreme alternative modes of specialization. One mode is to evolve a small number of large neurons having a high firing rate (or, alternatively, many synapses). The other mode is to evolve a larger number of smaller neurons having a lower firing rate (or less synapse). One purpose of this study is to search in the literature and to see whether such a specialization really occurs. Various cognitive functions are analyzed and a relation is suggested between them and the two types of neural systems. This is an interdisciplinary work, covering many fields of cognitive psychology and neuropsychology. Therefore, it cannot be a review of all these fields, due to space limitations. Only several examples of references of evidence for each of the hypotheses on which this study is established are presented. The role of the left hemisphere in temporal processing It is well known in the field of laterality research that the left hemisphere is specialized in fast temporal processing. Efron (1963a, b) presented two successive tactile sensory stimuli directly to the left and the right hemisphere. He achieved this by an application of the fact that tactile information from each hand arrives at the contralateral cerebral hemisphere. Efron presented tactile stimuli to a finger of the left hand and to a finger of the right hand, and asked the subjects whether the stimuli were presented simultaneously. Efron repeated the trials a large number of times and changed the temporal differences
between the stimuli. He found that, on the average, the subjects reported that the stimuli were presented simultaneously when the stimulus presented to the left hand (which sends its neural output to the right hemisphere) had been presented five to seven milliseconds (ms) before the stimulus presented to the right hand (which sends its output to the left hemisphere). This is the time required for a neural pulse to pass from the right hemisphere to the left one. Efron performed a similar experiment in the visual modality by tachistoscopic presentations of stimuli to the left visual field (VF) and to the right VF. The message from each half VF arrives at the contralateral hemisphere. Efron (1963a) concluded that the discrimination whether two tactile or visual stimuli are presented simultaneously, or one after another, is performed in the left hemisphere. A very large number of repetitions of this procedure was needed in order to obtain a significant results. Efron (1963b) found a larger impairment of perceiving simultaneity in subject having a left hemispheric damage who suffered from aphasia than in such subjects without aphasia. He inferred that the left temporal lobe (LTL) is the site of the temporal comparison. Efron suggested that this detection of the temporal order in the left hemisphere underlies the specialization of the left hemisphere in speech comprehension. This suggestion is derived from the observation that damage to the LTL causes both aphasia and the impairment of temporal discrimination. More studies found that the left hemisphere is specialized in the determination of the temporal order of events. According to Swisher and Hirsh (1972) patients with a left hemispheric damage were inferior to patients with a right hemispheric damage, and to normal control, in the determination of the order of two visual stimuli and of two auditory stimuli. Carmon and Nachshon (1971) found that patients with a left hemispheric damage had a difficulty, relatively to patients having a right hemispheric damage, in the determination of the order of three to five stimuli, some of which were auditory, and the others were visual. According to Goldman et al. (1968) patients with a left hemispheric damage have inferior critical flicker fusion (CFF) than patients with a right hemispheric damage. Many more studies, applying various research paradigms, have been performed since these early studies. The relation between the left hemisphere and the temporal analysis of stimuli presented rapidly one after another is now well established. The theory of this study is based on the hemispheric paradigm, which is controversial recently. The main reason of this controversy is the book of Efron (1990). However, Fidelman (1998b) contradicted all the specific arguments of Efron (1990). Moreover, Efron himself (Efron and Yund, 1996) abandoned the theory of Efron (1990).
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Visual search, semantical access, the cerebral hemispheres and ontology Hebb (1949) suggested first that short term memory (STM) is stored in web-like circular neural networks where the neural signal reverberates until it fades due to neural fatigue. These neural networks are called “engrams.” Long term memory (LTM) is related to permanent non-active engrams which are created due to permanent changes in the synaptic efficacies due to repeated transmission of the neural message. Activation of these permanent non-active engrams is related to the elucidating of the memories. Visual search provides us with examples of the functioning of both STM and LTM engrams. In experiments of visual search a target and several distractors are presented to the subjects. The subjects are asked whether the target is presented among the distractors, and the reaction times (RTs) are analyzed. According to the guided search model (Cave and Wolfe, 1990; Wolfe, 1994) the search includes two stages. The first stage is preattentive. The features of the stimuli are stored in retinotopic feature-maps due to bottom-up sensory activation. Then the features of the target are activated further by top-down activation. The superposition of all these activations is the activation map, which represents the stimuli and their locations. The second stage is an attentional serial search for the target. The stimuli are scanned serially one after another. The order of the search is according to the strength of the stimuli in the activation map. The top-down activation of the features of the target in the feature-maps increases the probability that the target will be one of the first stimuli which are scanned. When a stimulus is scanned it is located at the focus of attention, and it is integrated as an object out of its features. When this object is identified as the target, the search terminates. We have suggested that the preattentive stage of visual search is performed by the right hemispheric mechanism, and the feature-maps are located at STM engrams of this mechanism. On the other hand, the second attentional stage is performed by the left hemispheric mechanism, which performs the objectintegration and the serial scanning (Fidelman, 1995a; 1998b; 1999b). The integration of the object out of its features may be performed according to the guided search model (Wolfe, 1994) as follows. The activation map is created as a superposition of the feature-maps. An activated area in the activation-map represents all the features of an object. This activated area may represent the “new whole” which the right hemisphere synthesizes out of the details and the features of the object (Ben Dov and Carmon, 1976). Then the activated areas initiate, according to the order of the strength of their activation, the LTM engrams which designate the objects having the features of the activated area.
It maybe that the activation-map has not a physical existence in the right hemisphere. The feature-engrams may activate the LTM engram representing the object having these features directly. The synthesis of the new whole by the right hemisphere may be merely the simultaneous firing of the right hemispheric engrams designating the features of the object. These right hemispheric engrams may have axons having common synapses with neurons of the non-active left hemispheric LTM engram which represents the object. When all these right hemispheric engrams cause depolarization to neurons of the left hemispheric engram simultaneously, they may cause the appearance of action potential, and the activation of this engram. This activation is the left hemispheric object-integration. This suggestion may relate visual search to ontology as follows. Nominalism believes that only concrete objects exist. The integration of the concrete objects and their processing as concrete objects (i.e. serial scanning) is performed, according to the above model, by the left hemispheric mechanism during the attentional stage of visual search. On the other hand, Platonism believes that features exist. According to the above model features are represented and processed in the right hemispheric feature-maps. Thus nominalism and Platonism are related to the left- and right-hemispheric mechanism, respectively. This suggestion is in line with the experimental findings of Fidelman (1989; 1990). It was found there that subjects whose left hemispheric mechanism is dominant over the right one preferred a nominalist approach to both mathematics and the physical world. On the other hand, subjects whose right hemispheric mechanism is dominant over the left one preferred a Platonist approaches. We have developed (Fidelman, 1998a, b) ideas originated in Posner and Raichle (1994) that there is a relationship between the process of visual search and that of semantic access. Semantic access is the association of one word with another word. Such associations can be enhanced by semantic priming, i.e. presentation of a word having a feature of the target before the presentation of the target. Priming increases the efficiency of the processing of the target. It was suggested (Fidelman, 1998a, b) that the semantic information is stored in permanent LTM non-active engrams. The priming activates the features of the prime in the “semantic feature maps.” The activation of these features activates, in its turn, LTM engrams representing “object-like” semantic entities having these features of the prime. Then the engrams of the “object-like” entities are scanned serially according to the order of their strength. This strength increases together with the number of common features shared by the prime and the “object-like” entity. This scanning is analogous to the serial scanning in visual search, which is ordered according to the stimulus-strength of the visual stimuli. In analogy to visual search, the strongest “object-like”
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semantic entities are the first candidates for the performance of the required semantic task. We have presented evidence (Fidelman, 1998a, b) that the semantic features, which are relatively weak, are represented in the right hemisphere rather than in the left one. On the other hand, the “object-like” semantic entities are stronger (since their strength is supposed to be a superposition of the strengths of their features) and they are represented in the left hemisphere rather than in the right one. Moreover, the serial scanning of the “object-like” semantic entities is performed by the left hemisphere, like the serial scanning of the visual search. This analogy between visual search and semantic access may extend the above discussion regarding the relation of Platonism and nominalism to the right- and left-hemisphere respectively, to semantic Platonic features, like “being an animal” are represented by weak engrams in the right hemisphere, while “object-like” nominalist entities, like “dog” or “cat,” are represent by strong engrams in the left hemisphere. This suggestion is in line with more recent findings. Thus Kiefer et al. (1998) found by the technique of event related potentials (ERP) that direct associations are processed by both hemispheres with a dominance of the left, while indirect remote associations are processed by right hemispheric regions. Examples of indirectly associated words applied by Kiefer et al. (1998) are: “lemon – sweet” and “flower – thorn.” The associations are through the opposing taste-features (sweet and sour), and the feature “being a part of a plant,” which relate each of these pairs of words indirectly. Examples of directly associated words are “hen – egg” and “king – crown,” which designate concrete object which are directly related to each other. Another more recent study is that of Collins and Coney (1998) who applied the technique of tachistoscopic presentation of words to the two hemispheres. They concluded that the left hemisphere comprises a complex network representing concrete and abstract words, “while the right hemisphere operates as a subsidiary word processor of associative connections between concrete, imageable words.” These associative connections are, in fact, features common to the “concrete imageable words.” According to Collins and Coney (1998) these words are represented by the left hemispheric networks (engrams).
The magnocellular and parvocellular visual systems According to Livingstone (1993) there are at least two major processing pathways in the visual system. One pathway is the magnocellular (M) pathway which comprises large cells. The other is the parvocellular (P) pathway, which comprises smaller cells. These two anatomical pathways are identical to two functionally different visual systems which are known as the transient system and the sustained system (Livingstone, 1993).
According to Lovegrove et al. (1986) and Hill and Lovegrove (1993) the transient system (which is, in fact, the M pathway) is fast, it has higher temporal resolution, it is sensitive to rapidly moving and flickering stimuli. It responds only at the appearance and disappearance of stimuli. It is also sensitive to high contrast and to low spatial frequencies, that is, to the entire outer shape on an object and not to its fine details. On the other hand, the sustained system has low temporal resolution, it is most sensitive to stationary stimuli or to stimuli which move slowly. It is most responsive to low contrast and to high spatial frequencies, and to fine details inside an object rather than to the entire object. The transient system is fast, while the sustained system is slow. The transient system is color blind, while the sustained system is sensitive to colors. According to Livingstone (1993) the M system is related to stereoscopic depth perception, and to the perception of movement and of transient situation, to which the P system is not sensitive. There is similarity between the functions ascribed to the M system and the list of features used by the Gestalt psychologists to discriminate objects from each other and from the background (Livingstone, 1993). These features are: common movement, common stereoptic depth, collinearity (i.e. the occlusion of a continuous contour by another object does not prevent it from being perceived as one object), and common color or brightness. Livingstone (1993) inferred from this observation that “Magno functions may include deciding which visual element, such as edges and discontinuities, belong to and define individual object in the scene.” The role of the M visual system in the perception of objects may be related to the object-integration and the attentional perception of object in visual search, which according to Fidelman (1995a; 1999a) are performed by the left hemisphere. That is, the M system may have a role in the suggested left hemispheric mechanism which is related to nominalist cognition. On the other hand, the P system is sensitive to some features, at least to colors (the M system is sensitive only to the existence of a common color to the entire object). The P system is sensitive to details of the presented object, but not to the entire distinct object. Ben Dov and Carmon (1976) developed the analytic synthetic dichotomy of the left- and right-hemisphere, respectively, into a comprehensive model. According to this model the right hemisphere synthesizes several simultaneously presented details into a new whole. Then this new whole is treated analytically by the left hemisphere as a new distinct object. This model is in line with the above empirical evidence, provided that the P system is a part of the “right hemispheric” synthetic mechanism, and the M system is a part of the “left hemispheric” analytic mechanism. Thus we may expect that the right hemispheric neurons are, on average, smaller than the left hemispheric ones. The details of an object may be considered as features of the object. For example, a cat may be defined as an object having the following features: being
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an animal, having a moustache, having sharp ears, etc. Therefore, the P system may be a part of the right hemispheric mechanism which is related to the perception of Platonic features. The role of large cells There is evidence that the left hemisphere consumes more energy than the right one. There is anatomical evidence that the blood supply to the left hemisphere is larger than to the right one (Carmon and Gombos, 1970; Harris, 1985). It has been found by the positronic emission tomography (PET) technique that the left hemisphere consumes at rest more glucose than the right one (Rumsey et al., 1985). There is indirect evidence that the left hemispheric neurons may be (on the average) larger than the right hemispheric ones. This evidence follows from the observation that the visual magnocellular neurons of dyslexics are smaller than those of normal subjects (Livingstone, 1993). The auditory system of dyslexics also includes smaller neurons than that of normals (Galaburda et al., 1994). Dyslexics have, in addition to their reading problems, some additional impairment in their lingual ability which is related to the left hemisphere (Mann, 1991; Wolff, 1993). There is also evidence for left temporal dysfunction in dyslexia (Wood et al., 1991). We observed above that temporal processing is performed by the left hemisphere, and there is evidence that a temporal processing deficit is linked with dyslexia (Farmer and Klein, 1995). Therefore, it is reasonable to hypothesize that efficient left hemispheric data processing (which dyslexics lack) is related to the existence of large neurons in the left hemisphere. Assuming that the neural cells are spheric, we obtain that the metabolic requirements of the neurons (which are proportional to their volume) increase proportionally to the third power of their radius. Therefore, large cells consume at rest considerably more glucose and oxygen than smaller neurons. The above mentioned observation that the left hemisphere consumes at rest more glucose than the right one is in line with the hypothesis that the left hemispheric neurons are, on the average, larger; therefore, they have more metabolic requirements at rest. This observation is in line with the suggested relation between the M system, the left hemisphere, and the “nominalist mode of perception,” i.e. the perception of entire objects. A problem which will be discussed below is why nominalist perception of objects requires large cells. The different roles of small and large cells It has been suggested above that the right hemisphere includes smaller cells than the left one. These smaller cells consume less energy than the large left hemispheric cells. Therefore, the metabolic resources of the right hemisphere enable it to comprise a larger number of neurons. It has been suggested that the
right hemisphere is functionally related to the P visual system, which also includes small neurons. Both the P system and the right hemisphere are related to the perception of a large number of details or features, which are later integrated into entire objects. Each of these many details or features may require a separate Hebbian engram for its encoding. Therefore the right hemispheric synthetic mechanism requires a large number of cells in order to construct these engrams. The number of available cells and engrams is more important for the right hemispheric tasks than the following advantages of large cells. Large cells produce more transmitter and therefore may have a fast firing rate, which contributes to fast processing. Large cells can sustain thicker axons with a thick myeline layer and large conductivity. Therefore a system consisting of large neurons is fast. Alternatively, large cells may have more synapses and therefore a lower firing rate. However, these large cells may have more common synapses with other post synaptic large neurons, and activate them faster. This may explain why the P system is slow, and why the P system and the right hemisphere are inferior to the M system and the left hemisphere, respectively, in the processing of rapidly presented temporal stimuli. The P system and the right hemisphere are slow, and therefore unable to react to fast changes. Therefore, it is reasonable to hypothesize that evolution developed two parallel data processing systems. One system comprising large neurons which react to rapid changes, and performs rapid serial data processing (like the attentional serial processing of the visual search). The second data processing system comprises small neurons, but their number is large. It can store a relatively large number of details simultaneously, but its processing is slow. Nevertheless, since this second system processes data simultaneously and not serially, its slowness is not a considerable disadvantage. Now we have an answer to the problem why the “nominalist mode of perception,” i.e. the perception of objects, requires larger neurons than the “Platonist mode of perception,” namely, the perception of features, at least in the process of visual search. The serial attentional scanning of visual search must be very fast, otherwise the scanned STM engrams will decay beyond recognition, due to neural fatigue, before the termination of the scanning. Therefore, evolution preferred the application of larger neurons, which process data faster, for the representation and scanning of objects. On the other hand, the simultaneity of the processing of the features in the right hemispheric mechanism prevents the decay of the engrams before the termination of the processing. This may have caused evolution to prefer the possibility of processing a larger number of features and details slowly over a faster processing of a smaller number of features.
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Evidence for hemispheric differences in the number and size of neurons The following experimental findings (not including the testosterone theory of Galaburda et al., 1987) were not known to this author when the hypotheses that the left hemispheric neurons are larger and fewer, and the right hemispheric neurons are smaller and more numerous, were formulated. These hypotheses were based on logical considerations and indirect evidence. These experimental works were found later by a computerized search. Therefore, these findings may have a certain predictive value, at least for this author personally. Number of neurons It is well known in the field of laterality that human females have, on the average, left hemispheric cognitive skills, rather than right hemispheric ones. On the other hand, human males have right hemispheric cognitive skill, rather than left hemispheric ones. Rabinowicz et al. (1999) compared the number of neurons in the left- and right-hemispheres of males and females. They found a gender related difference in the human cortex. The male brain (which is right hemisoheric) includes 13 per cent more neurons than the female brain (which is left hemispheric) in 60 cortical loci, which is significant statistically. The maleto-female ratio of neuron numbers is significantly larger in the right hemisphere than in the left one. This last finding is in line with the hypothesis that this gender related difference in the number of neurons is, indeed, related to the right hemispheric cognitive stile of males, versus the left hemispheric cognitive stile of females. This finding is in line with the suggestion of Galaburda et al. (1987) that testosterone increases the number of neurons in the right hemisphere. The sex related difference in the number of neurons found by Rabinowicz et al. (1999) may be in line with the the findings of Pakkenberg et al. (1989) who found that the raw number of the neurons in the brain of females is 10 per cent smaller than in the brains of males. However, the authors did not know whether this finding is a result of a smaller body size of the female sample. Moreover, Rabinowicz et al. (1999) found that the mean cortical thickness of males and females were similar. They inferred from this last finding that there are more “processes” (i.e. neural connections) in the female brain. This conclusion is in line with the hypothesis that the female brain may have more common synapses between the same two presynaptic and post-synaptic large neurons, and therefore may have a faster processing. This phenomenon may occur also in the cortex of at least some non-human mammals, therefore, it may be related to the influence of sex-hormones. Thus Reid and Juraska (1992) found a larger number of neurons in the visual cortex of male rates than in that of female rats. Rabinowicz et al. (1999, p. 103) found only a small difference in the absolute neural density of the left- and right-hemispheres in each of the genders.
However, two of their six male subjects were non-dextrals (the brain of which is less lateralized), while the handedness of the third one was not available. The age of this third subject was 12 years 32 weeks. That is, he was before, or at the beginning of puberty, and testosterone had not yet enough time for influencing the relative number of his right- and left-hemispheric neurons. Therefore, it is reasonable to expect in a sample of right handed males, longer after puberty, more neurons in the right than in the left hemisphere. The brains of females are not lateralized, according to the reviews of Rabinowicz et al. (1999) and of McGlone (1980). This observation and the shortage of testosterone may explain the very small difference between the number of neurons in their hemispheres. Pakkenberg (1993) reported that “for the total neocortex the left – right difference in total neuron number was 2 2.7 per cent.” This finding tends towards my prediction. However, no information is given regarding the gender and the handedness of the sample, and we may assume that in a sample comprising of right-handed males this difference may be larger. The size of left- and right-hemispheric neurons We observed that dyslexic subjects have smaller neurons in the magnocellular visual system, and suggested that this observation may apply also to the hemispheres. This suggestion is in line with the the findings of Tenner et al. (1999). They found that non-dyslexic brains have larger neurons in Brodmann’s area 17 of the left hemispheric primary visual cortex than in the right one. Dyslexic subjects do not have this asymmetry, and their left area 17 neurons are as small as the right ones. Both these findings are in line with the theory of this study. The neurons in the left posterior part of normal subjects are larger than in the right one, and the verbal problems of dyslexics, which are related to temporal processing (Efron, 1963b; Farmer and Klein, 1995), are related to their having smaller cortical neurons. Hayes and Lewis (1993, 1995, 1996) found a similar relation in the anterior part of the brain. They found that the largest pyramidal neurons of layer III in Brodmann’s area 45 (inside Broca’s speech area), which are called magnopyramidal neurons, are larger in the left hemisphere than in the right one. This phenomenon occurs in both males and females. However, it occurs only regarding the largest neurons, the magnopyramidal neurons, and not regarding smaller pyramidal neurons. According to the analysis of Hayes and Lewis (1995) area 45 is related to speech, and it is functionally lateralized. Therefore, its including the largest pyramidal neurons, and the larger size of these left hemispheric neurons relatively to the right hemispheric ones, are in line with the theory of this study. Namely, speech is related to temporal processing, and temporal processing is related to large neurons. The smaller pyramidal neurons may have other tasks than temporal processing. An alternative explanation for the existence of small pyramidal cells in Broca’s area will be discussed below (in the section
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Temporal Discrimination). According to this discussion it maybe that the large neurons inhibit STM “parvocellular” engrams in their vicinity as part of the temporal processing. The smaller are the inhibited neurons, the faster is this suggested inhibition, since the quantity of inhibitory transmitter supplied by the large inhibiting neurons is considerably larger than the quantity of excitatory transmitter supplied by the small neurons of the inhibited engram. According to Hayes and Lewis (1995), however, there was no significant hemispheric difference between the mean size of all layer III pyramids in area 45. This may be explained by the suggestion that only left hemispheric neurons which are specialized in temporal processing evolved a large size. Moreover, the mean cross section of the largest pyramids in area 46 was smaller in the left hemisphere than in the right one. Area 46 in a region of the prefrontal association cortex. As far as we know, it is not functionally lateralized, i.e. it is not related to temporal processing. Maybe that this left hemispheric area includes part of the STM engrams required for the performance of speech, therefore it may require a larger number of smaller neurons, relatively to the corresponding area in the right hemisphere. Another possible explanation for the finding that the largest pyramids in area 46 are in the right hemisphere rather in the left one will be presented in the section “Temporal Discrimination” below. Hayes and Lewis (1995) did not find hemispheric differences in the mean cross section of the largest layer III pyramids in Brodmann’s area 4, at the primary motor cortex. This area is lateralized. However, according to this theory, large neurons are not required for all the tasks performed by the left hemisphere, but only for temporal processing. Area 4 may include STM engrams required for the supervision of the motion of the contralateral limbs. Another finding of Hayes and Lewis (1995) is, apparently, more disturbing to this theory. The male neurons measured by them were significantly larger than the female neurons. No interaction was found between gender and hemisphere. Moreover, Jacobs and Scheibel (1993a) found larger mean soma size for males than for females in Wernicke’s area. That is, in both lingual ares, Broca’s area and Wernicke’s area, the neurons of males are larger than those of females. This finding apparently contradicts this theory, since we may expect that the “left hemispheric” female brain has larger neurons than the “right hemispheric” male brain. In order to solve this puzzle we consider the findings of Gur et al. (1999). According to these authors women have a larger percentage of gray matter (GM) in the supertentorial brain, while men have a larger percentage of white matter (WM) and cerebrospinal fluid (CSF). In men the percentage of GM in the left hemisphere was larger, the percentage of the WM was symmetric, and of the CSF was higher in the right. Women showed no asymmetries. Both GM and WM correlated positively with cognitive tests, but the regression of cognitive performance and the volume of the WM was steeper in women. According to
Gur et al. (1999) the GM consists of the somatodendritic tissue of the neurons, and the white matter consists of myelinated connecting axons. We saw above that according to Rabinowicz et al. (1999) women have fewer neurons than males, and according to Hayes and Lewis (1995) and Jacobs and Scheibel (1993a) the somata of these fewer neurons of females are smaller, at least in several cortical areas. However, the percentage of the female GM is larger than that of males. This means that the female GM includes, in addition to the fewer and smaller somata, a relatively large volume of dendritic tissue. Therefore the female neurons have a larger dendritic arborization than the male neurons. On the other hand, males have longer and more myelinated axons. The more dendrites which a large female neuron has enable it to achieve a fast reaction by having a large number of synapses between its dendrites and several axonic branches of one or more presynaptic neurons. The summation of the post synaptic potentials created on all these dendrites causes a fast appearance of action potential in this large neuron. Suppose that the number of common synapses between a pair A of neurons is, say, twice the number of the common synapses between a second pair of neurons which have less dendrites, B, and therefore less common synapses. Suppose that the number of firings by the presynaptic neuron required in order to cause action potential in the post synaptic neuron of the pair A is N. Let us assume that the same quantity of transmitter is required in the two pairs in order to cause action potential in the post synaptic neuron. Then the presynaptic neuron of the pair B should fire 2 N times in order to cause action potential in the post synaptic neuron. That is, even if the two pairs of neurons have the same size, i.e. they produce and apply the same quantity of transmitter, the pair A of neurons transmits its message at least twice as fast as the pair B (assuming identical firing rate. Then the firing of the first A neuron terminates faster). We assume that the transmitter (and receiver) production capability is proportional to the volume of the plasma, i.e. the number of ribosomes (on which the proteins are produced) is proportional to the plasma’s volume. Then the pair of neurons B requires a larger quantity of transmitter and a faster firing rate, and therefore a larger volume, in order to transmit its message as fast as the pair A. This explains why evolution had to evolve larger magnopyramidal cells in the male brain than in the female brain, the neurons of which have more dendrites. The larger male neurons can maintain longer myelinated axons having more branches than the smaller female neurons. This suits the findings of Gur et al. (1999) that males have more WM than females. The thick myelinated axons of the male neurons enable the message to pass faster in these axons. Thus the larger speed of the message in the axons, together with the higher firing rate, are the basis of the male’s faster activation or inhibition of Hebbian STM engrams in the left hemisphere.
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The long myelinated male axons enable cerebral networks in the male brain to activate or inhibit distant neurons. On the other hand, the female brain has less axons and more dendrites. The smaller neurons which sustain shorter axons can activate only near neurons. The smaller quantity of myelinated axons relatively to the large quantity of dendrites in the female neurons causes the quantity of WM to be the “bottle neck” in the efficiency of the processing. This explains the finding of Gur et al. (1999) that the regression of cognitive performance and WM volume was steeper in women than in men. Hayes and Lewis (1995) concluded in their Discussion that there is more dendritic arborization in the left hemispheric area 45 than in the homotopic region of the right hemisphere. This conclusion is in line with the well known fact that the female brain is “left hemispheric,” and with the above conclusion that the female brain has a larger dendritic arborization. This conclusion regarding the difference between the female “left-hemispheric” brain and the male “right-hemispheric” one is line also with the view of Goldberg and Costa (1981) and of Semrud-Clikeman and Hynd (1990) that the left hemisphere has more intraregional communication and the right hemisphere has more interregional connections. According to Hayes and Lewis (1996) the mean somal size was greater in the left Brodmann area 45 than in the right. They found also that the total dendritic length was positively correlated with somal size in the left hemisphere, but not in the right. This finding is in line with the theory of this article, according to which the purpose of the more dendrites in the magnopyramidal neurons of the left hemispheric Broca’s area (Brodmann’s area 45) is to provide more synapses common to post synaptic and presynaptic neurons, and thus enable a faster activation of the post synaptic action-potential. This faster activation is required, according to this theory, for the temporal processing in the left hemisphere. There are more empirical findings indicating the existence of larger neural soma at various locations of the left hemisphere. Zaidel et al. (1997) found larger neurons in the left than in the right CA2 subfield of the hippocampus. Anderson and Rutledge (1996), (Table I) found that the small supragranular pyramidal neurons of the posterior superior temporal gyrus of men had larger soma in the left hemisphere than in the right. This finding is not significant statistically. The relation between this last finding and the location of Efron’s temporal comparison mechanism will be discussed in the section Temporal Discrimination below. No other relevant experiment, published in the last few years, has been found by this author in a computerized search. The conclusion is that at most cortical areas the left hemispheric somata are larger than the right hemispheric ones.
Long axon engrams We observed in the previous subsection that males have more WM than females, i.e. longer, and more branching axons. On the other hand, males have larger neurons than females, though the preliminary theory expected them to have more and smaller neurons, since they have a right hemispheric brain. The right hemispheric brain of males is a generally accepted fact. It implies a larger Miller’s chunk, i.e. more STM engrams in the right hemisphere. Yet, the male neurons are not smaller than the female ones. This may mean that the male brain applies a different strategy of creating STM engrams than the parvocellular visual system. The above data indicate that this strategy of the male right hemispheric brain includes two components. (1) A larger number of neurons, which can comprise more engrams. (2) Larger neurons which sustain longer and more branching axons than smaller neurons. These longer axons arrive at far areas of the brain and create intraregional engrams, while the shorter female axons can create mainly interregional engrams. (Goldberg and Costa, 1981; SemrudClikeman and Hynd, 1990). We observe that the male brain has more options than the female brain to combine neurons into STM engrams. First, it has more neurons, therefore, more combinations of neurons which create STM engrams are possible. Second, the longer axons can connect a larger variety of neurons to each other, and create more engrams. This strategy of the male brain to create STM engrams is an improvement of the strategy applied by the parvocellular visual system, and it is different. However, since the right hemispheric neurons are smaller than the left hemispheric neurons (even in the male brain, which has larger neurons than the female brain), we will apply the term “parvocellular neural system” or “parvocellular brain” as an abbreviation designating a brain having an efficient unification of the parvocellular visual system and the right hemispheric mechanism.
Spatial frequencies We observed that the M system is sensitive to low spatial frequencies, i.e. to entire objects, and the P system is sensitive to high spatial frequencies, i.e. to the internal details of the object. We also observed that the M system and the P system may be related functionally to the left- and right-hemisphere, respectively. These observations apparently contradict the findings that the right visual field (which sends input to the left hemisphere) is sensitive to high spatial frequencies, while the left visual field (which sends input to the right hemisphere) is sensitive to low spatial frequencies (Sergent, 1982; Christman, 1989; Kitterle et al., 1990; Kitterle and Selig, 1991; Christman et al., 1991).
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In order to understand this apparent contradiction we will analyze the nature of the spatial frequencies perceived by the M- and P-systems on one hand, and the hemispheric mechanisms on the other hand. The sensitivity of the M system to low spatial frequencies is, in fact, its ability to distinguish a large object from its background. We have hypothesized that the large size of the neurons of the M system implies that their number is relatively small. Therefore these neurons can construct only a small number of engrams representing large objects. This small number of large objects constitutes a “low spatial frequency.” Thus the “sensitivity of the M system to low spatial frequency” in explained. On the other hand, the P system is sensitive to many internal details of an object. According to our hypothesis the small size of the neurons of the P system enables it to allocate metabolic resources to a large number of neurons which constitute a relatively large number of engrams. Thus there are many engrams for representing the many internal details of an object. These many details constitute “high spatial frequency.” Thus the sensitivity of the P system to high spatial frequency is explained. The left hemisphere shares with the M system its ability to distinguish between discrete objects and their background. However, while the number of objects detected simultaneously by the M system is limited by the small numbers of engrams included in it, the left hemisphere can process a large number of stimuli analytically, one after another. That is, the left hemisphere, and not the M system by itself, performs analytical processing of visual stimuli. The ability of the left hemisphere to process a large number of details analytically is the basis of its ability to process high spatial frequencies. The right hemisphere can process a certain number of stimuli simultaneously. However, this number is limited by the number of available engrams. We saw above some evidence that this number is the maximal number of elements in Miller’s (1956) chunk, i.e. seven plus or minus two. The analytic left hemisphere does not have this limitation. Therefore the spatial frequencies that the right hemisphere can process are lower than those that the left one can process. However, the processing of the right hemisphere is simultaneous, and therefore faster. A discussion of possible reasons for the evolution of the mechanisms which perceive spatial frequencies is presented at the Appendix. Quantum mechanics and visual search Penrose (1989, pp. 231 –236) described the “archetypical quantum mechanical experiment” with a beam of electrons, or of monochromatic light, which consists of minimal units called “photons.” In this experiment the beam is fired through a single slit, or through a pair of slits, towards a screen or a photographic film. When there are two slits, the photons are scatteded over the screen in a pattern of bright strips and of dark strips.
This phenomenon is typical to interference of waves. The bright strips represent the locations where the crest of the wave originating in one slit strengthens the crest of the wave originating in the second slit. The dark strips are the locations were the trough of one of these waves abolish the crest of the other of these waves. However, the bright strips (or dark strips on photographic film) consist of small individual spots, each of which represents a single photon. When there is only one slit, the pattern of interference strips does not appear, but the spots representing the photons do appear. The intensity of the beam can be reduced, until a single photon (or a single electron) is emitted. Penrose analyzed the behavior of such a single photon. If the two slits are open, it should behave like a wave, i.e. its two parts, which pass through the two slits, interfere with each other. Now let us put a detector for photons at one of the two slits. If this detector detects the photon, then no spot appears on the photographic film. If the detector does not detect the photon, then a spot appears on the film, due to the passage of the photon through the other slit. This behavior is typical to a particle. That is, the photon behaves sometimes as a wave and sometimes as a particle. The behavior of a photon (or an electron) as a particle occurs after an instrument gives us some information about the location of the photon. This phenomenon, the behavior of a photon (or an electron) as a particle, in known as “the collapse of the wave function.” Similar phenomena in which the presentation of entities as discrete objects depends on their having a location are known in the research of visual search. According to Treisman and Schmidt (1982) prior to the object integration, i.e. in the preattentional stage of the visual search, features are “free floating,” namely, have not a location. This idea in based on two kinds of experimental evidence: (1) Subjects may have knowledge about the existence of a feature without knowing to which object this feature belong, or the object’s location. (2) Sometimes there is a phenomenon of illusory conjunction of objects’ features. For example, if a red hat and a blue umbrella are presented to a subject, the subject may report the presence of a red umbrella. According to the guided search model (Cave and Wolfe, 1990; Wolfe, 1994) information about the location of an object is part of the process of the object’s integration from its features. This follows the suggestion that all the features of an object are represented at corresponding locations in the feature-maps, and the object is represented in the activation-map as a superposition of all the activations of the features located at the same location in the feature-maps. This model is not in line with Treisman’s findings that the features are free floating before the object’s integration, and this contradiction worried Wolfe (1994).
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This apparent contradiction can be resolved by the findings of Pohl et al. (1972) who found that there are two independent spatial perceptions. One is right hemispheric and it locates objects relatively to an external frame of reference. The other is left hemispheric, and it locates objects relatively to the body of the subject. The serial scanning of the objects is performed necessarily by the left hemisphere, within the left hemispheric space, while the right hemispheric feature maps are part of the right hemispheric space. Therefore, the right hemispheric features may be free floating relatively to the left hemispheric space of the objects. Moreover, according to Posner and Raichle (1994) there are two different neural paths, the “what path” and the “where path.” The object’s features are encoded in the “what path,” while its location is encoded in the “where path.” That is, the features are encoded independently of their location. According to the theory of this article the features of an object are represented in the visual system by several right hemispheric engrams which may consist of small neurons. The large number of details of the photon’s wave-pattern, its crests and its troughs, are also encoded by several right hemispheric engrams consisting of small neurons. On the other hand, the object integrated from these feature-engrams is represented by a single left hemispheric engram consisting of large neurons. According to Posner and Raichle (1994, p. 97) visual imagery involves the top-down activation of visual areas participating in the actual seeing of the imagined objects. It was suggested by Fidelman (2000) that the conceptions of the wave model and the particle model of the photon are related to the preattentive and the attentional stages of the object-integration, respectively. This may be true regardless of what happens in the actual outer world. When the light beam passes through two slits, the experimenter perceives the interference pattern comprising several details, i.e. strips. When the theoretical physicist conceptualizes the wave-model then he (or she) visualizes a wave-pattern. Therefore, according to Posner and Raichle (1994, p. 94) the brain of the theoretical physicist necessarily applies a top down activation of the parvocellular visual system where the wave pattern is represented. Since the location of the photon is not known, it cannot be integrated into a particle which is represented by the “magnocellular” system. Only when information about the location of the photon is available by the detector at one of the slits (or a spot on the screen, when the detector is present), the physicist can integrate the photon into an object. This integration is performed in the left hemispheric visual cortex, where the visualization of the photon as an object is performed by a top down activation of an engram consisting of large neurons, since we observed above that the magnocellular system perceives discrete objects. Thus, regardless whether the “collapse of the wave function” represents anything in an objective world outer to us, it may be related to the replacing of a “parvocellular” cognitive model of a phenomenon perceived by our senses, by
a discrete object-like “magnocellular” model. That is, the collapse of the wave function is related to a change of the ontological interpretation of the phenomenon from a kind of Platonism into nominalism. The last usage of the term “Platonism” requires clarification. Monochromatic light has the feature “color,”, e.g. red. “Redness” is interpreted by the physicists as having a wavelength of 6560 Angstrom for the emitted electromagnetic waves. That is, physicists identify the feature “redness” and the feature “having a wavelength of 6560 Angstrom” with each other. Therefore, the Platonic feature “red” and the physical wavelength are, in fact, the same entity. Therefore, they are naturally encoded by right hemispheric engrams (the wave requires several right hemispheric engrams for encoding its details). When the photon is intergated as an object out of its features, the feature “having a high frequency of the light-wave” is translated as the photon’s having a large energy. According to Einstein’s relativity theory a large energy means a large mass, or a large size. Thus the different Platonic features, known as “colors” of the light, are intergrated into nominalist corpuscular photons having different sizes. Ethnic differences and implications to education I have suggested (Fidelman, 1992; 1997) that a hot climate causes the evolution of a right hemispheric brain, while a cold climate causes the evolution of a left hemispheric brain. One of the reasons for this view was evidence that the brain produces much more heat than other tissues. The human brain has only 2 per cent of the body mass, and it consumes at rest 18 per cent of the oxygen consumed by the entire body (William and Herrup, 1988, p. 430). Moreover, the brain is extremely sensitive to heat. The right hemisphere produces less heat than the left one. Therefore, a right hemispheric brain has an evolutionary advantage in a hot climate, while a left hemispheric brain has an advantage in a very cold climate. The theory presented in this article provides new arguments for this view. Humanity evolved in the hot climate of Africa, where it evolved a large brain which produces a lot of heat. The first stage of removing heat from the brain is its removal out of the neurons. The volume of a spherical neuron is proportional to the third power of its radius, while its surface is proportional to the second power of the radius. Therefore, a large number of small neurons have a smaller surface than a small number of large neurons having the same total volume. A large surface of the neurons enables a faster removal of the heat out of the neurons and later out of the brain. Therefore, we may expect that ethnic groups which evolved in a hot climate, like in Africa, may have a “parvocellular” brain rather than a “magnocellular” brain. This consideration implies that members of ethnic groups which evolved in a hot climate may have, on the average, a cognitive style different from that of ethnic groups which evolved in a cold climate and have a “magnocellular”
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brain. The brains of the ethnic groups which evolved in a hot climate is expected to have a larger number of neurons, though these neurons may be smaller. Therefore, they may have a larger number of STM engrams, which gives them a cognitive advantage in some cognitive tasks. On the other hand, their smaller neurons may cause them a relative disadvantage on tasks of temporal processing. Some indirect evidence for this hypothesis is presented in Fidelman (1992; 1997). For example, Jensen (1985) measured the achievements of four groups of male subjects in the temporal processing test of WISC-R Digit Span. In this test a small number of digits are presented one digit after another temporally (in the auditory modality) and the subject repeats them in the order of their presentation or in the reversed order. The groups were: low socio-economical status (SES) Afro-Americans, high SES Afro-Americans, low SES whites and high SES whites. The lowest scores on this test were obtained by the low SES Afro-Americans. However, Backman (1972) described a study in which another STM memory test was presented to eight groups of male subjects. The groups were: Low SES and high SES Afro-Americans, Low SES and high SES Jewish whites, low SES and high SES non-Jewish whites, and low SES and high SES orientals. In this memory test, unlike the Digit-Span test, the number of the memorized items was large. Up to 40 items were presented to the subjects. These items were lingual. However, they were not presented one after another temporally in the auditory modality, like the Digit-Span test, but were presented on paper. The finding was that the low SES Afro-Americans achieved the best scores on this memory test. The observation that the same group which achieved the lowest scores on the temporal test of Digit-Span, achieved the highest scores on the many items memory test, may indicate that two different cognitive skills, which compensate for each other, are involved. That is, the low SES AfroAmericans may have the smallest neurons, but they have the largest number of STM engrams, which may be due to their having the largest number of neurons, and/or to their having longer myelinated axons. The observation that these opposing achievements were obtained by a low SES group has been interpreted by Fidelman (1992) as indicating that something is wrong in the American (and more generally, the western) educational system. This system does not apply the cognitive advantage of the subjects who have many engrams consisting of many small neurons, and it is based on temporal processing. For example, reading can be taught one letter after another temporally. Alternatively, reading can be taught globally, a word consisting of several details (letters) as one picture. The first method is suitable for pupils having a small number of large neurons, while the other is suitable for pupils having a large number of STM engrams (consisting of small neurons), which encode the
letters. It seems that the American educational system is adapted to the brains of the first group. Thus the pupils of the second group (like dyslexics and male Afro-Americans) become victims of the lack of knowledge in the educational system, which pushes them to a low SES. The existence of an additional environmental factor makes the situation worst. A suggestion of a possible correction of this damage is presented in Fidelman (1992). Eysenck’s model of the brain’s functioning H.J. Eysenck presented a theory relating both intelligence and the speed of processing data by the human CNS to the probability of transmission errors in the brain (Eysenck and Eysenck, 1985). We have extended Eysenck’s theory into a more general theory of the brain’s functioning (Fidelman, 1998a, b). According to my theory the data processing in the CNS is performed in STM engrams in the sense of Hebb (1949), i.e. in circular neural networks where the neural message reverberates, until it decays due to neural fatigue. The data processing is performed in these engrams by excitation and inhibition of the engrams by other parts of the CNS. According to Eysenck’s theory a neural message is not sent once, but repeatedly, until the message in delivered identically a certain number of times, and only then it is accepted. A small probability of transmission errors causes a faster delivery of the message identically in the required number of times. It was suggested (Fidelman, 1998a, b) that during each cycle of the neural pulse through the engram, axons of this engram which have common synapses with the dendrites of another engram, deliver a message from the first engram to the other one, which we call the target-engram. This repeated sending of the message is in line with Eysenck’s suggestion. When the message causes a large enough depolarization on the dendrites of the target-engram’s neurons, action potentials appear on them, and the target-engram is activated. A small probability of transmission errors in the first engram fastens the identical activation of the target-engram’s dendrites, according to Eysenck’s theory. This smaller probability of transmission errors in the first (delivering) engram causes the message to pass accurately through the same path in this engram. Therefore the delivering engram terminates faster due to neural fatigue. Thus a smaller probability of transmission errors decreases the energy consumption of the relevant neural system. Eysenck’s theory is an extension of the theory of Jensen (1982), according to which intelligence is the outcome of a fast transmission of the neural pulses through the axons. Jensen (1982) found a negative correlation between RT and psychometric g (i.e. the efficiency of the brain), and a larger negative correlation between the variance of RT and psychometric g. Eysenck and Eysenck (1985) based Eysenck’s theory of intelligence on these findings. Unlike Jensen, they suggested that the negative correlation between IQ and RT is due to the large number of transmission errors in the brains of less intelligent subjects, which
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cause a larger number of repetitions of the message before the required number of identical transmissions is obtained. They also suggested that the larger negative correlation between IQ and the variance of RT is due to the random nature of the transmission errors, and therefore of the number of the message’s repetitions. If the neural system is efficient, then it is both faster and has a smaller variance. We have suggested (Fidelman, 1998) an additional factor. Large capability of transmitter production in the neurons of the first engram, which is a feature of large neurons, increases the firing rate and also causes a faster activation of the target engram. Larger firing rate causes also faster neural fatigue. The larger transmitter-production of the larger neurons does not prevent the faster fatigue, since the conduction of the transmitter to the axons and the synapses requires time. The empirical basis of Eysenck’s theory of intelligence The discussion in a previous section about the relation between the lack of knowledge in the educational system and the pushing of entire groups of the population to a low SES may explain some of the difficulties in the empirical verification of Eysenck’s theory. Thus we may learn how to test this theory empirically and avoid methodological traps. Eysenck’s theory that a high intelligence is related to a small probability of errors in the synaptic transmission originated in the work of Hendrickson and Hendrickson (1982). In this work the averaged evoked potentials (AEP) method was applied. A pure sensory stimulus (a flash of light or a noise) was presented to the subject many times, without any cognitive task. The EEG waves were recorded after each presentation and were summed, and the graph of their superposition was created. A significant positive correlation was found between the IQ of the subjects and the AEP measures Amplitude and String (the length of the AEP graph). A significant negative correlation was found between the Variance of the underlying EEG waves and IQ. This experiment has been repeated several times by several researchers (see references in Eysenck and Eysenck, 1985; Fidelman, 1996). The Hendricksons explained the findings as indicating that high IQ is related to a small probability of transmission errors in the synapses. Barrett and Eysenck (1992) performed a similar experiment and found negative correlations between IQ and the Amplitude and String AEP measures, contrary to the Hendrickson paradigm. Nevertheless, they found a negative correlation between the variability measure and IQ. We have compared the samples of the experiments, which were in line with the Hendrickson paradigm, with the sample of Barrett and Eysenck (1992). All the samples in which the Hendrickson paradigm was approved included mostly subjects which are expected to have a left hemisphere more efficient than the right one (Fidelman, 1996). On the other hand the sample of Barrett and Eysenck (1992) included
mostly subjects which are expected to have a right hemisphere more efficient than the left one. Namely, most of them were males who were recruited at the government unemployment office, or by advertisement in the local newspapers, and were paid. Therefore, we may assume that they belonged to a low SES. We observed above that the low SES includes right hemispheric persons who were pushed to the low SES by the educational system. We may conclude that it is reasonable to assume that the Hendrickson paradigm is obtained in samples most of the subjects of which have a left hemispheric cognitive style. On the other hand, we may assume that the findings of Barrett and Eysenck (1992) are obtained in a sample the subjects of which have a right hemispheric cognitive style. The common consensus now is that the EEG fields are created by synaptic activity that changes the polarity of the dendritic trees of pyramidal cells, and not from the action potentials at the axons. According to the above analysis the left hemisphere includes more large pyramidal cells, which have larger dendritic trees, than the right one. The large right hemispheric neurons, in particular those of males, have larger axons (which do not contribute to the EEG), i.e. white matter, and less dendrites (gray matter). Therefore we may expect that the left hemisphere contributes more to the EEG and to the AEP Amplitude and String measures than the right one. Hayes and Lewis (1995) found that there is more dendritic arborization in the left hemispheric area 45 than in the right. According to Jacobs et al. (1993b) the same is true for Wernicke’s area. Therefore, there are stronger EEG field at least in these left hemispheric areas. Hayes and Lewis (1996) found that the total dendritic length was positively correlated with somal size in the left hemisphere, but not in the right. That is, the efficiency of the left hemispheric “magnocellular” mechanism, but not of the right hemispheric “parvocellular” mechanism, is positively correlated with the strength of the dendritic EEG fields. Moreover, we have suggested that the IQ tests, like the educational system, are more suitable to left hemispheric brains (i.e. brains with a left hemisphere more efficient than the right one) than to right hemispheric brains (Fidelman, 1992; 1997). Therefore, in a left hemispheric sample, the subjects of which apply temporal processing for solving IQ tests, the larger are the subject’s pyramidal neurons, the better are the scores of the subject on IQ tests, and the stronger is the EEG field of the subject (and therefore also the Amplitude and String AEP measures). That is, in a left hemispheric sample we may expect positive correlations between these two AEP measures and IQ, i.e. the findings expected by the Hendrickson paradigm. On the other hand, in a right hemispheric sample the subjects solve IQ tests applying their many STM engrams (unlike the subjects of a left hemispheric sample who apply their better temporal differentiation and better sequential processing). The negative correlation between the efficiency of the two
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hemispheres implies that the right hemispheric subjects who score more on IQ tests (and therefore should have more STM engrams) have smaller pyramidal neurons, and therefore smaller EEG fields. This explains the negative correlations between IQ and the Amplitude and String measures found by Barrett and Eysenck (1992). Barrett and Eysenck (1994) repeated their experiment with a sample which has been recruited similarly, i.e. by advertisement in the local newspapers. Therefore it is reasonable to assume that this sample too included low SES subjects. However, this time most of the subject were females. This time no significant correlations were obtained between IQ and the Amplitude and String measures. However, Barrett and Eysenck (1994) found experimentally that when they selected their subjects so that the P180 peak of the AEP was clearly defined, and the variability of the underlying EEG was not too large, the findings of Barrett and Eysenck (1992) were repeated. Barrett kindly sent me data about the selected sample, some of them not included in Barrett and Eysenck (1994). According to these data all but two of the selected subjects were females. Their verbal IQ was about the same as that of the other (non-selected) females. However, their performance IQ was significantly lower. That is, the phenomenon discovered by Barrett and Eysenck (1992) regarding subjects assumed to be right hemispheric is different from the phenomenon discovered by Barrett and Eysenck (1994), since the later sample seems to be left hemispheric. The clear definition of the P180 peak may mean that the selected subjects have larger pyramidal neurons than the non-selected subjects, i.e. the selected subjects are more left hemispheric. Their smaller variability may mean that both their hemispheres operate in synchrony. The smaller performance IQ may mean that these subjects do not have a large number of STM engrams in their right hemisphere. All these observations may be explained by the assumption that the selected subjects are females whose brain is not lateralized, who have speech centers in both hemispheres, and who may be non-dextral. The brain of females is less lateralized than that of males (McGlone, 1980). McGlone and Davison (1973) found that non-dextral females have a smaller visuo-spatial ability even than other females. Their non-dextral female subjects could enumerate (subitize) simultaneously presented dots less efficiently than dextral female subjects, who, in their turn, could enumerate dots less efficiently than males. This means that non-dextral female subjects have a very small number of STM engrams, since their right hemisphere, and not only their left hemispherevf, includes large neurons instead of many STM engrams. This conclusion is in line with the findings of Willerman et al. (1992). They found that the difference between the volumes of the left- and the righthemisphere, and the difference between the Verbal IQ and Performance IQ, correlated +.44 for males and 2 .55 for females. That is, the less lateralized the female brain is, the larger is its right hemisphere due to its having more large
neurons and less small neurons, and the smaller is the Performance IQ, due to the existence of a smaller number of STM “parvocellular” engrams in the right hemisphere. That is, the bottle-neck of the selected subject’s ability to solve IQ tests is their small number of STM engrams. This number is negatively correlated with the size of the pyramidal neurons, which determines the strength of the EEG fields, since in females these neurons have many dendrites, which induce EEG. This explains the negative correlation between the IQ of the selected subjects and the AEP string and variability measures. The findings of the two experiments of Barrett and Eysenck demonstrate how the shortcoming of the educational system may push entire groups of the population to a low SES. We assumed that the samples of both studies belong to a low SES. The teaching of males who have an extremely right hemispheric brain should apply their larger number of STM engrams. Since this is not done, due to lack of knowledge in the educational system, a sample of low SES males may include many right hemispheric subjects, who could not overcome the educational system. According to Barrett and Eysenck (1994) their sample comprised 25 males and 61 females, and there were 26 selected subjects. According to Barrett (private communication) the selected subjects comprised 2 males and 24 females. We assumed that the selected subjects were non-dextrals. The male selected subjects were 8 per cent of the entire male subject, which is reasonable according to the percentage of non-dextral in the entire population. Indeed, according to Oedfield (1971) 25 per cent of the males and 16 per cent of the females claimed that they had some tendency to left-handedness. We assumed that the male subjects belong to a low SES due to their being right hemispheric. Therefore, we do not expect that the sample included many male subjects who are non-dextral due to their having “two left hemispheres.” We may assume that many of the left-handed male subjects, included in the sample, are right-hemispheric subject with a tendency to dyslexia, the left hemisphere of whom is impaired (Geschwind and Galaburda, 1985). These subjects are not included among the selected subjects. This is in line with the observation that the percentage of male selected subject of Barrett and Eysenck (1994) is smaller than the percentage of the male subjects having a tendency to left handedness. However, the percentage of the female selected subjects among the entire female population was 39 per cent. This percentage is considerably larger than expected according to the percentage of non-dextral female in the entire female population according to Oedfield (1971). We assumed that the sample included low SES subjects, and this is the only known peculiarity of this sample. Therefore we may assume that the lower IQ of the selected subjects, in particular their lower Performance IQ, caused them educational problems, which pushed them to a lower SES.
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The teaching of non-dextral females, who have a small number of STM engrams and a larger analytic ability, should avoid using memorizing too many details simultaneously, and apply their relatively better analytic ability. That is, they should apply a larger number of smaller chunks of Miller (1956), and process them analytically. For example, in the learning of reading it is not sufficient to apply the analytic reading of one letter after another. When the words are long, the shorter words comprising the long words should be noted., e.g. the word “symmetry” includes eight letters. The size of Miller’s chunk of non-dextral females is usually smaller than eight. Therefore, it should be explained to them that “symmetry” is a combination of the two Greek words “sym” and “metry.” The meanings of these two Greek words and of their combination should be explained. Thus small Miller’s chunks will be processed efficiently by their brain, one short word after another. Presented Evidence was presented elsewhere (Fidelman, 1992; 1997) that subjects having a small maximal Miller’s chunk and a higher analytical ability, like nondextral females, have more efficient interhemispheric connections relatively to other subjects. Therefore, the larger number of interhemispheric transmission of data, due to the smaller number of elements in the chunks, and therefore a larger number of chunks (which are transmitted to the left hemisphere for analytical processing, and then transmitted to the right one for integration into chunks of chunks), is performed by the nondextral females efficiently. It is predicted that if the experiments of Barrett and Eysenck (1992; 1994) will be repeated with a sample comprising left hemispheric subjects, which does not include non-dextral subjects, the findings will be in line with the Hendrickson paradigm. If the sample will comprise right hemispheric subjects, the findings of Barrett and Eysenck (1992) will be repeated. If the sample will comprise non-dextral females, the findings of Barrett and Eysenck (1994) will be repeated. References of recommended hemispheric tests are presented in the Discussion section. The role of neural transmission errors in hemispheric specialization As discussed in the previous section, the probability of neural transmission errors is one of the factors determining the efficiency of a neural system (see Fidelman, 1998a, b). Now we will discuss the interaction between this factor and the number of small neurons in the right hemisphere on one hand, and the speed of processing in the left hemisphere on the other hand. First, we observe that testosterone may impair the functioning of the left hemisphere (Geschwind and Galaburda, 1985) and increase the relative number of neurons in the right one through a trophic effect (Galaburda et al., 1987). On the other hand, female hormones impair the right hemispheric cognitive functions and enhance left hemispheric ones (Kimura, 1989; Hampson, 1990). During the part of the menstrual cycle when female hormones are active the
women’s cognitive style is more left hemispheric, and during the other part – their cognitive style is more right hemispheric. Therefore, we may hypothesize the existence of a negative correlation between the efficiencies of the two hemispheres. At least in the case of variations in the cognitive style of females, sex hormones cannot change the size and number of existing neurons during the menstrual cycle, thus the negative correlation between the efficiencies of the hemispheres is not likely to originate in the transmitter production rate. Therefore we may hypothesize that the sex hormones influence the probability of transmission errors in the synapses, and there is a negative correlation between these probabilities in the two hemispheres. We have suggested that the size of the brain is related to the probability of transmission errors in the CNS through the a trophic effect. That is, neurons which do not communicate with other neurons die (e.g., Fidelman, 1993). Thus a large probability of transmission errors in the CNS reduces the communication between the neurons, and reduces their number. That is, a large brain is the outcome of high intelligence, and not the opposite (Fidelman, 1993). Therefore, the findings of Galaburda et al. (1987) that testosterone may increase the number of neurons in the right hemisphere may be the outcome of its inducing a smaller probability of transmission errors in this hemisphere. This smaller probability of transmission errors causes also a smaller energy consumption in engrams of the right hemisphere (as discussed in the section Eysenck’s Theory of the Brain’s Functioning). Thus the energy allocated to the right hemisphere of males may be sufficient for more cells which may construct more engrams. Thus the smaller is the probability of transmission errors in the right hemisphere, the larger may be its ability to process a large number of details. A small probability of transmission errors in the right hemisphere is related to a high probability of transmission errors in the left one, which causes two outcomes: (1) Death of left hemispheric neurons, due to trophic effect. This outcome is in line with the findings of Geschwind and Galaburda (1985). (2) A smaller speed of transmission of neural messages in the left hemisphere, since according to Eysenck’s theory a large number of transmission errors causes the repetition of the message more times before its acceptance. The role of neural transmission errors in semantic We observed above that the right hemisphere is related to weak associations, while the left hemisphere is related to strong ones. The right hemispheric semantic features do not necessarily always activate left hemispheric engrams
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related to “object-like” entities. The reason is that a single right hemispheric engram representing a semantic feature may not be strong enough for activating the left hemispheric engram related to the semantic “object” having this feature. Often more than one feature is required in order to activate the semantic object-engram. The empirical evidence indicates, however, that a right hemispheric semantic feature-engram activates other such engrams, which represent similar or related features. Kiefer et al. (1998) presented an example of a week association: lemon – sweet. In this example the left hemispheric object-engram representing “lemon” causes the activation of the engram representing one of its most characteristic features, “sour.” This last engram activates the related feature-engram representing “sweet.” It maybe that the feature-engrams representing “sour” and “sweet” share common neurons, or have synaptic connections, related to their being tastes. This may explain why the activation of one of these engrams activates the other one, thus causing an association. On the other hand, the left hemisphere is related to strong associations like dog – cat. According to Kiefer et al. (1998) strong associations are processed by both hemispheres with a dominance of the left. This means that strong associations cannot be explained only by the sharing of left hemispheric neurons or synapses by the two engrams, as suggested for weak associations. The findings of Kiefer et al. may mean that the activation of a left hemispheric semantic object-engram is related to the activation of its right hemispheric feature-engrams, which, in their turn, activate another left hemispheric objectengram sharing these features. This procedure is similar to the effect of priming. “Dog” activates the right hemispheric feature-engrams representing “being a carnivorous animal,” “being a pet,” and “being rivals.” These feature-engrams activate the left hemispheric object-engram “cat.” This model is in line with the findings of Kiefer et al. (1998), who applied the ERP research method, that both hemispheres participate in this association. The dominance of the left hemisphere (in the ERP experiment) is explained by its larger pyramidal neurons, which have more dendrites and therefore cause larger EEG fields. We have participated in a study (Thimor and Fidelman, 1995) which may provide more experimental evidence for the above analysis of the functioning of the semantic engrams. In this study subjects learned to construct bottom-up and top-down concept-maps. Then they were asked to construct such conceptmaps according to their own preferences. A bottom-up concept construction begins with elementary entities, like “pupil,” and a more comprehensive concept, like “class,” is constructed out of them. A top-down concept-creation begins with the more comprehensive entity, like “class,” and then its elements are specified. Thus the sentence “a pupil is a member of a class” is a bottom-up concept-mapping, while the sentence “a class includes a pupil” is a top-down concept-mapping.
The subjects were tested by hemispheric tests. The right hemispheric test measured the number of details perceived by the subject simultaneously (the enumeration of dots or forms flashed simultaneously, known as subitizing). The left hemispheric test measured the temporal differentiation and the temporal analysis of the subjects (the enumeration of dots flashed during a very brief time one after another with very brief temporal intermissions). References presenting the empirical relation between these tests and the hemispheres are presented in Thimor and Fidelman (1995). The subjects were divided into two groups. One comprised subjects who performed top down concept-mapping, and the other comprised subjects who performed bottom-up concept-mapping. The means of the scores on the leftand right hemispheric tests, and the differences between these scores for individual subjects, were compared by T-test. In addition, for each group the scores of the subjects on the correctness of the concept-mapping were correlated with their scores on the hemispheric tests and the differences between them. The statistical analysis showed that the construction of top-down conceptmaps by a subject is related to the subject’s having an efficient right hemispheric mechanism. On the other hand, the construction of bottom-up concept-maps by a subject is related to the subject’s having an efficient left hemispheric mechanism. Concepts are, in fact, features of objects. Therefore, concept-mapping and semantic associations involve similar mechanisms. Therefore, the existence of two cognitive styles of concept-mapping, may indicate that there are also two styles of semantic associations. If the right hemisphere of the subject is efficient, i.e. if there is a small probability of transmission errors in it, then few right hemispheric semantic feature-engrams may be able to activate a left hemispheric semantic object-engram. The negative correlation between the efficiencies of the two hemispheres may imply that when the right hemisphere is efficient, there may be a difficulty for a left hemispheric object-engram to activate right hemispheric feature-engrams of this object’s features. That is, the associations are “top-down.” Otherwise, the probability of transmission errors in the right hemisphere is large, and the probability of transmission errors in the left hemisphere is small. Then the efficient left hemispheric semantic object-engram activates one or several feature-engrams which represent the features of the presented “object.” This is a “bottom-up” activation. If the association is a feature of the presented object, then this is a “bottom-up” association. Temporal discrimination Introduction In the visual modality an inter stimuli interval (ISI) of almost 20 ms is required in order to determine that two stimuli were presented one after another
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temporally (Hirsh and Sherrick, 1961). An ISI of 50 –55 ms is required in the visual modality for determining a temporal gap between two stimuli (Di Lollo et al., 1982). However, in the auditory modality only an ISI of 2–3 ms is required for determining that two stimuli were presented one after another temporally, while an ISI of 5 – 16 ms is required for detecting a temporal gap between two auditory stimuli (Farmer and Klein, 1995, p. 465). A model is suggested in order to explain these empirical observations. In this following model it is assumed that there are neural networks which represent the beginning and the termination of the sensory stimuli. This assumption is more established for the visual modality, since the M visual system is specialized in detecting both the beginning and the termination of visual stimuli. We assume also that as long as the visual stimulus endures it is represented by an engram of the sustained (parvocellular) visual system and by a cerebral engram comprising small neurons. It should be stated that the following model may be only one of several possible models. It is proposed only in order to show that the relation between fast neural reaction and temporal discrimination, which exists empirically, may be explained by the assumptions of this study. The magnitudes of ISIs presented in the former paragraph depend on a variety of variables. Nevertheless, the possible variety of these numbers does not change the possibility of the model. Indeed, an example of a possible explanation of the discrepancy between the findings of two experiments performed in different conditions is presented. Namely, the discrepancy between the findings presented above in this section and the findings of Efron (1963a). We observed that Efron (1963a, b) succeeded to discover the site where simultaneity of two visual stimuli is determined, by presenting stimuli to the right hemisphere 5 ms earlier than to the left hemisphere. This observation apparently contradicts the observation of Hirsh and Sherrick (1961) that subjects report the simultaneity of two stimuli for ISIs as large as 20 ms. We can remove this apparent contradiction by explaining the phenomena according to Eysenck’s model. Before justifying the possibility of Efron’s (1963a, b) findings a possible model of temporal comparison in the brain is suggested.
Temporal order without a temporal gap According to the above assumptions when two visual stimuli are presented each of them is represented by three cortical engrams. The first stimulus is represented by the transient “magnocellular” STM engram related to its initiation (designated by I1, Initiation), by the sustained “parvocellular” STM engram which is active as long as the visual stimulus is received (designated by S1, Stimulus), and by the transient “magnocellular” STM engram related to the termination of the stimulus, designated by T1 (Termination). The second
(later) stimulus is represented similarly by three engrams I2, S2, and T2, respectively. According to Efron (1963a, b) the temporal comparison is performed at the left hemisphere, and the LTL participates in it. The left hemisphere includes larger neurons, therefore its networks may react faster than those of the right hemisphere. We assume that these fast-reacting left hemispheric networks participate in the temporal processing. The following model takes into account all the above mentioned assumptions and information, and it is at least a possible model for temporal processing. As far as it is known each cognitive datum is related to the operation of an engram. The termination of the activity of an engram does not represent a cognitive datum. On the contrary, it is related to the disappearance of cognitive information. Therefore, the knowledge that there is a temporal order between two stimuli is related to the activity of a STM engram. Similarly, the knowledge that there is a temporal gap between two stimuli is related to the activity of another STM engram. Therefore, the engram representing the existence of a temporal order between two stimuli, as well as the engram representing the existence of a temporal gap between them, are activated as a result of an interaction between the engrams S1 and T1 on one hand, and S2 and I2 on the other hand. A possible cue about the nature of this interaction may be the finding of Hayes and Lewis (1995), described above, that there was no significant hemispheric difference between the mean size of all layer III pyramidals in area 45. These authors found also that the magnopyramidal neurons of layer III in Brodmann’s area 45 are larger in the left hemisphere than in the right one (Hayes and Lewis 1993; 1995; 1996). This can be true only if the small pyramidal cells of area 45 in this sample are smaller in the left- than in the right-hemisphere. It should be noted that according to Hayes and Lewis (1996) the mean somal size was greater in the left Brodmann area 45 than in the right. However, we may interpret the combined findings of Hayes and Lewis (1995; 1996) as indicating that the difference between the small and the large neurons is larger in the left hemisphere than in the right one, even if the mean size of the left hemispheric neurons is larger due to the considerably large size of the left hemispheric magnopyramidal neurons. This interpretation may indicate that the temporal differentiation process in the left hemisphere includes an inhibiting of a “parvocellular” engram by a “magnocellular” one. A large difference between the size of the inhibiting neurons and the inhibited ones means that there is a considerably larger quantity of inhibitory transmitter than of excitatory transmitter, and therefore the inhibition is quicker. That is, the process of detecting a temporal order may include the inhibition of the “parvocellular” engram S1 (and not of the “magnocellular” engram T1) by the “magnocellular” engram
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I2 (and not by the “parvocellular” engram S2). The inhibition of the weak S1 by the strong I2 is faster than its inhibition by the weak S2, and has evolutionary advantage. Now we have to answer the question: Why should S1 be inhibited? The cognition of the existence of a temporal order between the two stimuli is related to the activation of a STM engram, which we designate by TO (temporal order). The activation of TO is related to the beginning of the second sensory stimulus after (or together with) the end of the first stimulus. The continuation of the first sensory stimulus is represented in the central neural system (CNS) by the STM engram S1. S1 is sustained by the sensory stimulus as long as the stimulus endures. However, S1 continues to endure for some time after the termination of the first sensory stimulus, until it decays. In order that TO will represent the existence of a temporal order between the two stimuli, it should be activated after the decay, or inhibition, of S1 (provided that S1 is not inhibited before the termination of the first sensory stimulus) and as soon as possible after the activation of I2. This can be achieved if the following assumptions are true. (1) As long as S1 endures it inhibits the neural pathway through which TO is activated. (2) I2 activates TO and inhibits S1. (3) S1 cannot be inhibited as long as the first sensory stimulus is presented. Since S1 is sustained by the first sensory stimulus, it is not easily inhibited as long as this stimulus continues. However, its inhibition by I2 can shorten the duration of the endurance of S1 after the termination of the first stimulus. Since S1 inhibits the pathway through which TO is activated by I2, the inhibition of S1 by I2 fastens the activation of TO. According to these assumptions the ISI required for the detection of a temporal order is the time required for S1 to decay (without inhibition) to the degree in which I2 can inhibit S1 and activate TO. We may consider TO to be “an outcome of a temporal comparison” engram, i.e. it represents a temporal order. Temporal gap The first assumption of the extending of the model to the detection of a temporal gap is that the ISI periods presented above can be used as cues for the nature of the process. These periods depend on the conditions of the experiment. Nevertheless, the proportions of the ISIs required for detecting a temporal order and a temporal gap in the visual and auditory modalities are about similar, which may mean that they are due to a common feature of the neural process. In the visual modality the ISI required for detecting a temporal gap (50 – 55 ms) is more than twice (but less than three times) the ISI required
for detecting a temporal order (20 ms). The same is true regarding the corresponding minimal ISIs in the auditory modality: 2 –3 ms are required for detecting a temporal order, while 5 – 16 ms are required for detecting a temporal gap. My suggestion to take as a cue for the neural process of detecting a temporal gap only the minimal ISIs in the auditory modality requires a justification. There is a similarity between reaction time (RT) and ISI. Both depend on the rate of neural processing. We observed above that Jensen (1982) had found a negative correlation between RT and psychometric g (i.e. the efficiency of the brain), and a larger negative correlation between the variance of RT and psychometric g. If the neural system is efficient, then it is both faster and has a smaller variance. This means that subjects who have a small ISI also have a small variability. That is, the minimal ISI cannot vary much. On the other hand, subjects having a large ISI have also a large variability, and the maximal ISI is increased considerably by the random variability. This means that the minimal ISIs are more reliable data for establishing the model. In the visual modality an ISI of at least 50 ms is required in order to detect a temporal gap. This time is beyond the time required for the decay of S1 without being inhibited by I2, which is at least 20 ms. That is, at lest 20 ms of the ISI are required in order that S1 will decay without inhibition, so that TO will not be activated (and the subject will not cognize merely a temporal order a temporal gap). We may assume that the process of activating a STM engram representing a temporal gap (designated by TG) is similar to the process of activating TO, with the engram T1 replacing S1. That is, T1 inhibits the neural pathway through which I2 activates TG, and I2 (which is activated this time after the decay of S1) activates TG and inhibits T1. The extra time which is required for the minimal time of this process (beyond the time required for the initiation of TO) is due to: (1) The longer time required for the decay of S1 without inhibition. (2) The longer time required for the inhibition of the strong “magnocellular” engram T1 by I2, which is more than the time required for the inhibition of the weaker “parvocellular” engram S1 by I2. The selection of which engram is activated by I2, TO or TG, depends on the engrams which are active when I2 is initiated. If S1 is active, then I2 activates TO after the termination of S1. If T1 is active alone, then I2 activates TG after the termination of T1. This model is more established for the visual modality than for other modalities, since we know that the magnocellular visual system is sensitive to the beginning and the end of visual stimuli. It is an assumption of the model that a similar phenomenon occurs also in the other modalities.
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Wernicke’s area According to Efron (1963a, b) the detection of the temporal order is performed at the left hemisphere, at least partly at the LTL. Efron (1963b) suggested that the location of both lingual tasks and temporal processing at the LTL is due to the temporal nature of the lingual functions. The anatomical evidence presented above, which is in line with the model, is regarding Broca’s area and not regarding the temporal lobe. However, Broca’s area too has lingual functions which require temporal processing. The model suggested for Broca’s area may be valid for the temporal lobe too, as may follow from the following empirical data. According to Jacobs et al. (1993a, b) the mean of the dendritic measures (for their entire sample) in Wernicke’s left hemispheric area was larger than in the homotopic right hemispheric area, though the difference was not large. They also found that females exhibited slightly greater dendritic values than males. Since larger dendritic arborization requires larger somata, we may infer that the neurons in Wernicke’s area are larger than those of the homotopic right hemispheric area, and they react faster due to the activation of more dendritic synapses simultaneously. There are indications that the detection of temporal differences at the LTL (Efron, 1963a, b) is not performed by the strategy of high firing rate. According to Ojemann and Schoenfeld-McNeil (1998, p. 321) low frequency of activity is typical of recording from temporal cortical neurons of humans. This may mean that the brain applies for temporal discrimination at the LTL the faster strategy of activating many dendrites of the same neuron simultaneously. This strategy reduces the firing rate necessarily, since it reduces the quantity of transmitter per synapse. We observed above that according to Gur et al. (1999) the left hemisphere has larger dendritic arborization than the right one, and females, who have a “left hemispheric” brain, have larger arborization. Now we observe that this is true also for Wernicke’s area. This may mean that in Wernicke’s area many axons of an engram can activate, or inhibit, more dendrites of the same neuron, or of another engram, simultaneously. That is, both activation and inhibition of engrams may be faster in Wernicke’s area than in the homotopic right hemispheric area. This indicates that the model of detecting a temporal order suggested above may suit Wernicke’s area and the temporal lobe too. Why the findings of Efron are possible? Now we can see how the findings of Efron (1963a, b) regarding the visual modality are in line with this model, in spite of the apparent contradiction due to the detection of temporal order when the ISI is merely 5 ms, which is less than 20 ms. The initiation of the outcome of the comparison engram TO depends on a series of previous engrams, beginning with the STM engram
which represents the visual icon in the ganglion of the retina (in the case of tactile modality, spinal or subcortical engrams). According to Eysenck’s theory the timing of the activation of each of these engrams depends on the probability of transmission errors in the previous engram. Thus the timing of the initiation of the engram TO may be influenced by the accumulation of the temporal deviations in the activation of the previous engrams. Therefore, even if the stimulus presented at the left VF (LVF) is presented 5 ms earlier than the stimulus presented at the right VF (RVF), they do not always arrive at the temporal comparison mechanism simultaneously. Each of the two stimuli may arrive at the temporal comparison mechanism somewhat earlier or later. Moreover, all the CNS neurons of the path form the RVF to the LTL are left hemispheric and react faster, while the path from the LVF to the LTL includes also right hemispheric neurons, which react slower. Therefore, the stimulus presented at the RVF (left hemisphere) requires less time per unit of distance for arriving at the LTL, and the temporal difference between the arrival of the two stimuli at the LTL increases. Thus the temporal difference between the arrival of the two stimuli at the LTL may, sometimes, be sufficient for the activation of TO, even if the ISI is smaller than 20 ms. Suppose that the ISI of the visual stimuli presented to the left- and right-VF in a certain experiment is limited to be below 20 ms (or, rather, below 15 ms, when the first stimulus is presented at the right VF), and the number of trials is very large. Then the engram TO, which designates a temporal order, will be activated only by few pairs of stimuli, as a result of the random temporal deviations in operation of their engrams. The probability that this activation will occur when the presentation to the LVF is 5 ms earlier than to the RVF is the smallest, since then the stimuli arrive at the temporal comparison mechanism, on the average, simultaneously. Therefore the subjects will report more times that the stimuli were presented simultaneously when the stimulus at the LVF is presented 5 ms earlier than the one at the RVF, than for any other temporal difference. Thus the effect described by Efron (1963a, b) can be detected. Why fast temporal processing is performed in the left hemisphere? We have observed that for all the sensory modalities, visual, tactile and probably auditory, the determination of temporal order is performed in the left hemisphere. In particular, the existence of an ISI is determined in the LTL. However, there is an advantage of the right hemisphere in some types of temporal processing. Kimura (1964) found by the technique of dichotic listening a left ear (right hemispheric) advantage in the recognition of melodies by subjects who were not professional musicians. We observed in the Introduction that Papcun et al. (1974) found a left ear (right hemispheric) advantage in the processing of Morse code (by subjects not knowing this code) when the number of dots and dashes was no more than
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seven. However, when the number of dots and dashes was larger than seven, there was a left ear (right hemispheric) advantage. Ben Dov (1978) and Ben Dov and Carmon (1984) found a similar phenomenon with visual rhythms comprising temporal intervals separated by flashes of light presented to the left- and right-hemisphere. They found that increasing the number of intervals in a rhythm resulted in a shift in cerebral dominance from left to right. When the number of intervals was one or two there was a left hemispheric advantage. However, when the number of intervals was three or four, there was a right hemispheric advantage. In the experiment of Papcun et al. (1974) the subjects were instructed to attend only to auditory stimuli (dots and dashes). In the experiment of Ben Dov (Ben Dov, 1978; Ben Dov and Carmon, 1984) the subjects were instructed to attend to rhythms consisting of temporal intervals between visual flashes. A rhythm consisting of one temporal interval comprises three elements: two separating flashes and the temporal interval. A rhythm consisting of two intervals comprises five elements: three separating flashes and two intervals. According to Miller (1956) most subjects can process this number of items analytically. However, when the number of temporal intervals is three and four, the number of processed items is seven and nine, respectively. Then most subjects must process them by chunking, applying the right hemisphere, which includes more cells and therefore more engrams. That is, the temporal processing by the left hemisphere is related to its analytic ability. It can be done solely by the left hemisphere when the number of items is small. Another finding of Ben Dov (1978), which is not included in Ben Dov and Carmon (1984), contradicts the model of Efron (1963a) regarding the determination of simultaneity. Ben Dov (1978) applied the apparatus of Ben Dov and Carmon (1984). She found that when two flashes of light are presented to the right VF (left hemisphere) simultaneity is determined more accurately than when two flashes of light are presented to the left VF (right hemisphere). According to the model of Efron (1963a) the passages of the two stimuli, presented to the left VF (right hemisphere), through the corpus callossum to the left temporal lobe (where simultaneity is determined) requires the same time. Similarly, the two flashes of light presented to the right VF (left hemisphere) require identical durations of time for arriving at the left temporal lobe. Therefore, there should be no difference in the accuracy of the determination of simultaneity of each of the two pairs of stimuli. Thus the better determination of simultaneity of stimuli presented to the right VF (left hemisphere) contradicts Efron’s model. Ben Dov (1978) explained this finding by the “quantisization” of time, with a larger “temporal quantum” in the right hemisphere than in the left one. If two non-simultaneous stimuli are presented within the same “temporal quantum” they are perceived as occurring simultaneously. The findings and the explanation of Ben Dov (1978) are, to a certain extent, in line with the model
presented in this study. The “temporal quantum” is, in fact, the ISI required in order to determine temporal order. It is the time required by the engram S1, after the termination of the sensory stimulus sustaining S1, to decay to a degree that I2 can inhibit it. The inhibition of S1 is a prior condition for the activation of TO by I2. Ben Dov (1978) did not relate the beginning of the temporal quantum to a certain temporal instant. In fact, she assumed that time is divided into temporal quanta, regardless of the presented stimuli. According to my model the temporal quantum begins at the end of the first sensory stimulus. This “temporal quantum” is different for the various modalities. The model of Ben Dov (1978) implies that there are two temporal comparison mechanisms, one in each hemisphere, and the mechanism lateralized to the left is more efficient due to a faster reaction at the left hemispheric temporal comparison mechanism. The findings of Efron (1963a) are explained by the observation that the left hemispheric temporal comparison mechanism is both faster and more efficient. Therefore there is an evolutionary advantage to the acceptance of the judgment of the left hemispheric mechanism. The only exception is when the two visual stimuli are presented to the right hemisphere. This exception may be due to the observation that sending both stimuli to the left hemisphere through the corpus callossum involves both extra time and more transmission errors. The transmission errors cause, according to Eysenck’s theory, larger temporal delay. Moreover, the delays in the delivery of the two stimuli may vary due to a different number of transmission errors, thus increasing the probability of a wrong decision regarding the simultaneity of the two stimuli. Therefore evolution favored in this case the decision of the right hemispheric temporal comparison mechanism. Nevertheless, it maybe that even two stimuli presented at the LVF (right hemisphere) are compared for simultaneity at the LTL. In the previous paragraph we observed that according to Eysenck’s theory there is a disadvantage to the processing of a temporal comparison of two visual stimuli, presented at the LVF, in the LTL (relatively to two stimuli presented at the RVF). This disadvantage by itself may explain the findings of Ben Dov (1978). It is not decided in this work which of these two models is correct, though arguments are presented in favor of both. It maybe that the brain applies both of these models, and the first outcome of the two processings is cognized. We saw above that Anderson and Rutledge (1996, Table I) found larger somata in small pyramidal neurons of the LTL than in the RTL, but the difference was not significant statistically. Since we hypothesized above that the detection of temporal order requires the inhibition of an engram consisting of small neurons, the finding that the small pyramidal neurons are smaller in the right hemisphere, is in line with the suggestion of Ben Dov (1978) that there is a temporal comparison also at the RTL. The relative advantage of this right hemispheric mechanism may be its smaller “parvocellular” neurons.
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The finding of Hayes and Lewis (1995) that the mean cross section of the largest pyramids in area 46 was smaller in the left hemisphere than in the right one may also be in line with the suggestion of Ben Dov (1978). Maybe that the right hemispheric temporal differentiation mechanism includes inhibiting “magnocellular” engrams at area 46, and inhibited “parvocellular” engrams at the temporal lobe (where the small neurons are smaller than in the LTL). Similarly, the left hemispheric temporal differentiation mechanism may have inhibiting and inhibited engrams at different left hemispheric areas, and may be that only one of them is located at the LTL. Probably, area 45 may be the site of the inhibiting “magnocellular” engrams. It is also possible that a temporal order is detected at the LTL, while a temporal gap is detected at Broca’s area. The temporal comparison mechanisms for different modalities We observed that according to Efron (1963a, b) the detection of simultaneity is performed at the LTL for both tactile and visual modalities. Therefore, we may hypothesize that a crucial stage of the temporal comparison is performed at the LTL. However, the considerable differences in the ISIs, required for the auditory and the visual modalities, mean that different engrams are compared for different modalities. A possible model is that the engrams involved in the detection of a temporal order and of a temporal gap are located at far extending regions of the brain. These regions probably include the visual or the auditory areas, depending on the modality of the compared sensory stimuli. Part of these engrams, at least the engrams related to detecting temporal order, include neurons of the LTL. The considerable temporal difference between the ISIs of the auditory and the visual modalities may be due to differences in the structure of the compared engrams related to stimuli in the two modalities. The auditory engram may consist of neurons having a faster reaction, probably due to a larger size and larger dendritic arborization, which have many common synapses with previous neurons of the engram or of the activating engram. The larger neurons can sustain thicker axons with a thicker layer of myeline, and therefore they can conduct the neural signal faster. The auditory engram may be shorter, since the auditory cortical areas, at least in the left hemisphere, are closer to the temporal comparison mechanism than the visual areas. Thus, according to our assumptions, the engram representing the auditory stimulus is both shorter and has a faster pulse. That is, the neural pulse passes through it more times per a unit of time. Therefore, it is terminated faster, due to neural fatigue, after the end of the auditory stimulus. That is, the inhibition of the S1 and T1 engrams of the auditory stimulus are faster than in the visual modality, and the activation of the TO and TG engrams, respectively, by I1 is faster. This consideration may explain the shorter ISIs in the auditory modality.
Discussion Conclusions This study comprises two parts. The first part is searching the literature (taking into account space limitations) in order to see whether the basic assumptions are verified empirically. The first basic assumption is that the left hemisphere, like the magnocellular visual system, includes large cells which produce large quantities of neurotransmitter and neuroreceiver. These large quantities cause a fast reaction of the left hemispheric networks due to either high firing rate or to many synapses common to two neurons. The second assumption is that the right hemisphere includes a larger number of smaller neurons comprising a larger number of Hebbian STM engrams. The empirical data confirms the existence of larger magnopyramidal neurons in the left hemispheric lingual area of Broca than in the homotopic right hemispheric area. Similarly, it was found that left hemispheric neurons were larger in several left hemispheric areas than in the homotopic right hemispheric areas. This is in line with the first assumption. However, Anderson and Rutledge (1996) found only regarding the small pyramidal neurons in a part of the temporal lobe that they are larger in the left hemisphere than in the right one, and this finding was not significant statistically. This finding is explained by the suggestion of Ben Dov (1978) that there are temporal differentiation mechanisms at both hemispheres, and the hypothesis that the efficiency of these mechanisms is proportional to the difference between the size of the large neurons of an inhibiting engram, and the small neurons of an inhibited engram. Thus the efficiency of the right hemispheric temporal differentiation mechanism may be due to the smaller size of the neurons of the inhibited engram. The empirical data also show that the male brain, which is expected to be right hemispheric, has more neurons than the female brain. This finding is in line with the second hypothesis. However, the male neurons were found to be larger than the female ones. Two conclusions are derived from these empirical findings and the findings that females have more GM than males, and males have more WM than females. The first conclusion is that the female brain achieves a fast reaction (in the left hemisphere) by the activation of many synapses of a neuron simultaneously, while the male brain achieves this task by a higher firing rate and a faster passage of the neural message through the highly myelinated axons. The second conclusion is that the male brain creates a larger number of STM engrams not only by having more neurons, but also by applying the longer and highly myelinated axons for creating engrams comprising neurons distant from each other. There is also some direct evidence, though not conclusive, that there are more neurons in the right- than in the left hemisphere. The second part of this work is the characterization of the similarity in the functioning of the magnocellular visual system and the left hemisphere on one
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hand, and of the parvocellular visual system and the right hemisphere on the other hand. The existence of these similarities enabled us to consider each of these two pairs of neurals systems as an integrated neural system, having a combined perceptual and cognitive function. Thus the neural system is virtually divided into two distinct functional components, which were described above. The first component is characterized by larger and fewer neurons having a high firing rate, or, alternatively, many synapses common to a pair of neurons (or to an activating engram and an activated neuron). It includes the left hemisphere and the magnocellular visual system. The second component is characterized by a large number STM Hebbian engrams. It may comprise smaller neurons having a low firing rate and/or long highly myelinated axons, which construct engrams out of neurons distant from each other. It includes the right hemisphere and the parvocellular visual system. The “left hemispheric” component of the CNS applies its faster reaction in order to achieve a better temporal resolution. It is the site of the engram representing the integrated object in visual search, therefore its fast reaction enables it to scan these integrated objects one after another in time very rapidly. Therefore the nominalist ontology, which accepts the existence of concrete objects, is related to this neural mechanism. The larger temporal resolution of this mechanism gives it an advantage in the analysis of temporally presented auditory speech, as well as in other temporally organized lingual functions. The “right hemispheric” component includes a larger number of engrams (which may comprise smaller neurons), and therefore it can be the site where the numerous features of an object are represented in order to be integrated into a new whole or an object. Therefore the existence of features, i.e. Platonic ontology, is related to the “right hemispheric” component. Thus nominalist and Platonist ontologies are related to the “left hemispheric” and to the “right hemispheric” component of the CNS, respectively. In semantic access the “left hemispheric” component of the CNS includes engrams denoting objects which are strongly associated with other objects. The “right hemispheric” component includes engrams denoting features which are weakly associated with objects and with each other. The strong associations in the left hemisphere, and the weak associations in the right one, may be related to the faster reaction in the left hemisphere, and the slower reaction at the right one (which may be related to a lower firing rate, and/or to a smaller number of synapses common to two neurons or to two engrams), respectively. A left hemispheric object-engrams may activate another such engram not only directly (if a direct activation is possible), but also through the activation of their common feature-engrams in the right hemisphere. It is suggested that the abstract semantic objects are integrated from abstract semantic features in a mode analogous to the integration of the concrete objects from their sensory features in visual search. Thus this
suggestion relates semantic to ontology. The difference is only in the locations of these integrations. In visual search the integration occurs at the visual area in the posterior part of the cerebrum, while the integration of the semantic objects occurs at the anterior part of the brain (see Fidelman, 1998a, b). Another suggestion is that the duality in physics, like visual search and semantic, is related to the duality in human ontology. That is, the particle cognitive model of the photon is related to nominalism and the left hemisphere. On the other hand, the wave cognitive model of the photon of light is related to the Platonic features of light and to the right hemisphere. Suggested experiments This study is theoretical, therefore some experiments which may test its predictions are suggested. One suggested experiment is to teach a class, including both normal readers and dyslexic subjects, the Platonist and nominalist ontological approaches, during several lessons. After the teaching the ontological preferences of the subjects will be compared. It is predicted that the dyslexic subjects, who have smaller neurons in some neural structures, will tend more to accept Platonism than the normal readers. Another prediction is that dyslexic subjects will be impaired, relatively to normal readers, in determining whether stimuli were presented one after another temporally, and in detecting a temporal gap between stimuli. This prediction is not theoretically new. However, it is suggested to repeat the experiment of Ben Dov (1978), described above, with a sample including both normal readers and dyslexics. It is predicted that the difference between the efficiency of the left hemisphere and the right one in detecting simultaneity will be larger in the sample of normal readers than is the sample of dyslexic subjects. This prediction is independent of the problem whether the larger “temporal quantum” for stimuli presented at the left VF originates in a comparison mechanism at the right hemisphere, or in longer engrams, compared at the left one. This prediction is inferred from the finding of Jenner et al. (1999) that there is a smaller difference between the size of neurons in the left- and right-hemisphere of dyslexics than in the hemispheres of normal readers. It is also suggested that there are two cognitive styles of semantic access, related to the relative efficiencies of the hemispheric mechanisms. We suggest an experiment with semantic access which includes testing the subjects by hemispheric tests to perform is performed. It is predicted that subjects having an efficient right hemisphere will have, more often than other subjects, associations of object-words which are aroused by a feature of these object. The hypothesis regarding the relation between the duality in physics and the two neural subsystems can be tested experimentally. We suggest that two classes of students are taught optics in two modes. The first mode is to teach the laws of refraction and of reflection of light explained by the assumption
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that light is a phenomenon of waves. The second mode is to teach the same laws explained by the assumption that light comprises particles. Each class should learn the two presentations of optics in the opposite order. The students will be tested also by hemispheric tests. Their scores on the understanding of the lessons and the explanation preferred by them for the nature of light will be correlated with their scores on the hemispheric tests. It is predicted that the subjects who have a more efficient left hemisphere will score more, relatively to themselves, on the corpuscular explanation of light, and will prefer this model over the waves model of light. On the other hand, a student having a more efficient right hemisphere will score more, relatively to themselves, on the waves model of light, and will prefer this model over the corpuscular model. We have performed a somewhat similar experiment (Fidelman, 1989). In this experiment students were asked to prefer between two explanations to Zeno’s paradoxes. One explanation is the atomist explanation (of Democritos), which considers the physical world to consist of indivisible particles, called atoms. This explanation is nominalist. The second explanation is the Eleatic one (of Parmenides), which considers the world to be a spatial structure of logical concepts, and it is a precursor of Platonism. The findings were that the subjects who have a more efficient left hemisphere preferred the atomist explanation, while the subjects with a more efficient right hemisphere preferred the Eleatic explanation. This finding was significant at p , 0:05 in a two tailed test. This experimental method suits also an experiment regarding the duality in physics. Interested readers may find details of the experiment and of the hemispheric tests in Fidelman (1989). The experiments suggested here and additional experiments, can be applied in order to test this theory experimentally. One of the purposes of this paper is to suggest the performance of these experiments. An implied philosophical issue This study is an attempt to explain cognitive psychology using neurological methods. Neurology, in turn, may be presented from a biochemical point of view. Finally, biochemistry can be considered, in a sense, as a branch of physics. In this study, however, we have suggested that physical cognition itself, namely, the dual physical models of particles versus waves, can be explained by neuropsychology. This observation implies the existence of a “vicious cycle” in the core of Integrated Science, which includes both physics and psychology. This means that Integrated Science is a self-referential system. Self-reference is a “Goedelian situation.” That is, science cannot prove its own consistency. Indeed, Penrose (1989) proved that Goedel’s incompleteness theorem implies that no formalizable theory can represent the physical universe, which includes the human brain. I have extended this proof (contrary to Penrose’s Platonic
philosophy) to include any consistent theory which can be presented lingually (Fidelman, 1999c). That is, no conceivable consistent Integrated Scientific theory, which can be communicated to other persons, can modelize the totality of empirical phenomena. Therefore, “hard science” has no philosophical advantage over psychology. Nevertheless, the establishing of a psychological theory on “hard science,” and vice versa, increases the probability that the predictions of this theory will be approved empirically, since this theory is in line with a wider range of empirical phenomena.
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Appendix: Possible evolution of spatial frequency perception Bryden and Underwood (1990) suggested that the processing of stimuli presented in the upper visual field (VF) and the lower VF is similar to the processing of stimuli presented in the right VF and the left VF, respectively. That is, they predicted that high spatial frequencies are processed better than low spatial frequencies in the upper VF, while low spatial frequencies are processed better when presented in the lower VF. Christman (1993) confirmed this prediction experimentally. Moreover, Efron and Yund (1996) found that the stimulus-strength of stimuli presented at the right VF is stronger than that of stimuli presented at the left VF, and stimulus-strength of stimuli presented at the upper VF is stronger than that of stimuli presented at the lower VF.
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The stronger stimulus-strength at the right VF and at the upper VF are in line with the hypothesis that these VFs are more related to the M system (which has larger neurons and therefore a higher rate of transmitter-production) than to the P system. However, the findings regarding the spatial frequencies should be explained. We have suggested (Fidelman, 1999b, Appendix) that there is an evolutionary pressure encouraging the better processing of high spatial frequencies at the upper VF, and the processing of low spatial frequencies at the lower VF. Spatial frequencies, like stripes, are used by many animals as camouflage. When the animals are distant the stripes are perceived at a small angle as high spatial frequencies, and the animals are perceived at the upper VF. Close animals are perceived in the lower VF at a larger angle, and their stripes are perceived as low spatial frequencies. We can perceive at the higher VF several distinct objects, like a pack of wolves, and prepare to confront them. This task too is defined as perception of a spatial frequency. The perception of a flock of animals is performed analytically, one animal after another. When the potentially dangerous flock is distant and perceived at the upper VF, there is enough time for analytic processing, one animal after another, by the left hemispheric mechanism. This mode of analytic scanning has an evolutionary advantage, since it provides the information that we encounter a pack of wolves and not, say, a herd of sheep. When a distant tiger is perceived, necessarily at the upper VF, the stripes may be processed analytically by the left hemisphere, one stripe at a time, as distinct objects. Thus when the danger is a distant tiger, no damage is done by the analytic processing of the stripes, since there is enough time for preparations. The analytic processing in the upper VF is best performed by the M visual system which is specialized in the perception of distinct objects. The M system is also specialized in the perception of a contrast, and may distinguished better between individual stripes and process them analytically. This strategy of “high spatial frequency processing” at the upper VF comprises the identification of individual objects by the M visual system and processing them analytically by the left hemispheric mechanism. We may conclude that the sensitivity of the M system to low spatial frequencies is merely its ability to discriminate between large individual objects and their background. The M system may be able to do this also with small objects, provided that they are perceived as individual objects, and not as details or features of individual objects. This strategy has evolutionary advantage for distant objects perceived at the upper VF. However, when the tiger is close, there is no time for the analytical processing of the stripes one after another by the M system and the left hemispheric mechanism. That is, an alternative faster strategy of processing is required. When a tiger is near, its strips are perceived as a lower spatial frequency at the lower VF. Thus the perception of low spatial frequency at the lower VF requires the fastest data processing in order to avoid an immediate danger. The fastest possible processing is the integration of, say, the entire tiger from the stripes and the other details presented at the lower VF. These details are presented, according to this theory, as STM engrams consisting of small neurons of the right hemispheric mechanism and the P visual system. These details and stripes comprise the low spatial frequency. The observation that the right hemispheric mechanism and the P system can process more details (high spatial frequencies) does not prevent it from a fast processing of low spatial frequencies as well. This processing comprises two stages: (1) Simultaneous representation of the details comprising the (low) spatial frequency (e.g., the stripes of the near tiger) as right hemispheric engrams. (2) These right hemispheric engrams activate the left hemispheric engram related to the object “tiger.”
Since the left VF and the lower VF are related to a similar data processing, the above discussion explains how the P system which is sensitive to “high spatial frequency” can be related to the operation of the left VF which is sensitive to low spatial frequencies. The “high spatial frequency,” to which the P system is sensitive, may be no more than the ability to process relatively more details and features simultaneously, due to the existence of more engrams in this system. However, the P system can process also a small number of details. The sensitivity of the left VF and the lower VF to low frequencies may be the evolutionary outcome of the necessity to process low spatial frequencies as fast as possible, that is, integrating the object from the presented details at once, without the application of serial scanning of the details. The application of this strategy of processing for data presented at the lower VF is understandable. The application of this strategy of processing data presented at the left VF may follow from the following consideration. Low spatial frequency may mean a danger even when it is presented at the upper VF, e.g. a tiger on a near tree. Since the left VF sends input to the right hemisphere, this input can be processed right away by an immediate integration of the object related to the low spatial frequency. If the near tiger on the tree is presented at the upper right VF, the low spatial frequency indicates that the the number of details is not large. Therefore their analytic processing is relatively short. Similarly, high spatial frequency may mean a danger even if it is presented at the lower VF, e.g. a near small snake camouflaged by high spatial frequency spots. The processing of the details of the snake is performed, is this case, by the P system and the right hemispheric mechanism, applying their ability to represent a large number of details as STM engrams comprising small neurons. Then axons originating in these right hemispheric STM engrams cause the integration of the snake as an object by activating simultaneously many dendrites of the neurons of the left hemispheric engram which represents this object. If the small snake is on a near tree, its detection is performed either simultaneously in the upper left VF (right hemisphere), or analytically in the right VF (left hemisphere). Since a near small snake up on the tree is a less immediate danger than a near tiger, its processing at the upper right VF by the analytic data processing mechanism is not a serious evolutionary disadvantage. In conclusion, the so called “sensitivity of the M system to low spatial frequencies” may be no more than its ability to detect large individual objects and distinguish between them and the background. The so called “sensitivity of the P system to high spatial frequencies” may be no more than the its ability to represent a large number of details as STM engrams prior to the object-integration.
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PART II
Research studies with interdisciplinary methodologies
The current issue and full text archive of this journal is available at http://www.emeraldinsight.com/0368-492X.htm
and Direct and indirect causality in indirectDirect causality in living systems living systems
Miguel Lloret-Climent University of Alicante, Spain Keywords Cybernetics, Living systems Abstract Not all cells are equal: each tissue and organ has its own type of cell. Although the nucleus of each cell in a living system has the same genetic information, each one dispenses of the lion’s share of that information and only those genes that are necessary for carrying out the function of the particular organ or tissue to which they belong remain active. Despite the fact that in specific scientific fields, such as ecosystem studies, it is possible to measure the relationships between different variables and to compare the various direct and indirect effects they may have on one another, there has been no such development in the wider context of a General Systems Theory. This paper sets out to address the question of cellular change by interpreting processes such as direct and indirect causality, cellular meiosis and mutation of cells.
485 Received March 2001 Revised March 2001 Accepted August 2001
Introduction The cells resulting from the first cellular divisions of the ovum until the moment an embryo is formed are all identical. These cells are called mother cells and can be converted into any type of cell and can manufacture all types of tissues. As a result, specialised cells which are characteristic to a particular tissue or organ are formed. Another important difference between mother cells and specialised cells is that mother cells can be multiplied and divided indefinitely while specialised cells can, at most, be divided and multiplied a finite number of times (approximately 50 times). Another type of cell that can be divided and multiplied indefinitely, in the same way as the mother cells, are tumour cells. The reason for this is that the Telomerasa gene remains active. We aim to formalise these problems within the framework of the systems theory. Notation The lifespan of a cell x will be represented by tE(x ), such that tE ðxÞ ¼ ½t 0 ; t f where t0 is the instant when the cell is formed, for example by cellular meiosis, and where tf is the instant when the cell ceases to exist, for example, as a result of cellular meiosis or death. The lifespan of a mother cell would be [0,+1[, the same as a tumour cell. The state in which a cell x is found at instant t will be determined by the set of all the behaviours of the genes of the cell at that particular moment, which will be denoted xE(t ), thus giving us xE ðtÞ ¼ g 1 ðtÞ; g 2 ðtÞ; . . .; g l ðtÞ}. xE(t ) is the behaviour and/or value of the behaviour of cell x in time t (present). x[t0,tf] is the
Kybernetes, Vol. 31 No. 3/4, 2002, pp. 485-495. q MCB UP Limited, 0368-492X DOI 10.1108/03684920210422575
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behaviour of cell x at any instant of the interval ½t0 ; tf and is the value of the behaviour of cell x at any instant of the interval ½t 0 ; t f (duration of the behaviour). x t is the behaviour and/or value of the behaviour of cell x at any instant prior to t, that is, in ½t0 ; t½ (past). xt is the behaviour and/or value of the behaviour of cell x at any instant later than t, that is to say t; t f (future). Similarly, ;r [ R½t0 ; tf where r is a relationship. r½t 0 ; tf is the relationship at any instant of the interval ½t 0 ; t f . r(t ) is the relationship in time t (present). r t is the relationship at any instant prior to t (past) and rt is the relationship at any instant later than t (future). If ðxðtÞ; yt Þ [ r t or (x(t ))rt(yt), the behaviour and/or value of the behaviour of cell x in t, when influenced by the relationship r at any instant later than t, produces a behaviour and/or value of the behaviour yt. This will not be the only behaviour and/or value of the behaviour yt, since there are other relationships and other behaviours that may affect the behaviour of y. If ðx t ; yðtÞÞ [ r t or (x t)r t(y(t )), the behaviour and/or value of the behaviour of cell x at any instant prior to t, when influenced by the relationship r at any instant prior to t, produces a behaviour and/or value of the behaviour y in t. Definition 1 (Lloret et al., 1998). A system-linkage S ¼ ðM ; RÞ is the pair formed by a set object M and a set of binary relations R, in such a way that R , PðMxM Þ ¼ PðM 2 Þ. That is to say: ;r [ R=r , MxM where r ¼ ðx1 ; y1 Þ; ðx2 ; y2 Þ; . . .; ðxl ; yl Þ=ðxi ; yi Þ [ MxM}. The concept of system-linkage is developed on living systems.
Living cellular systems Definition 2 (Lloret, 1999a). These are system-linkages which have the form S½t0 ; tf ¼ ðM ½t 0 ; t f ; R½t0 ; tf Þ where M ½t 0 ; t f is the object set of cells and R½t0 ; tf is the set of relations, where R½t0 ; tf , M ½t 0 ; t f xM ½t0 ; tf . The set M ½t0 ; tf ¼ M R ½t0 ; tf < M M ½t 0 ; t f < M S ½t 0 ; tf < M T ½t 0 ; t f where: M R ½t0 ; tf ¼{the male or female reproductive cells that only contain half of the complete genetic part} M M ½t0 ; t f ¼{mother cells, which can be converted into any type of cell} M S ½t0 ; tf ¼{specialised cells located in any tissue or organ resulting from the growth of the egg} M T ½t0 ; tf ¼{tumour cells obtained from the mutation of specialised cells} R½t0 ; tf ¼{cellular meiosis, mutation, etc.}
Cellular formalisation Each cell can be identified as a vector of n components formed by the four bases adenine (A), cytosine (C), guanine (G) and thiamine (T) through the genome function defined in the following way:
f:{cells} ! {A,C,G,T}n such that each cell x determined by the set of genes Direct and {g1, g2,. . .gl} is identified with a vector of n components formed by a succession indirect causality of four nucleotides A,C,G and T. in living systems For example, f ðxðtÞÞ ¼ f ðg1 ðtÞ; g 2 ðtÞ; . . .; gl ðtÞÞ ¼ ðA; C; C; G; .nÞ. .; G; AÞ so that gene g1 at instant t is identified by a vector formed by the l1 first positions of the succession, gene g2 at instant t corresponds to the subsuccesion of 487 nucleotides of size l2 that starts once g1 ends. This process continues until gene gl at instant t corresponds to the subsuccesion of ln nucleotides, which begins once gl2 1 ends. Evidently, l 1 þ l 2 þ . . . þ l l ¼ n as a result of the construction carried out. Furthermore, the function f for each cell x can be decomposed in l functions f 1 ; f 2 ; . . .; f l such that f 1 ðg1 ðtÞÞ ¼ A; C; G; T}l1 , f 2 ðg 2 ðtÞÞ ¼ A; C; G; T}l2 ; . . .; f n ðgn ðtÞÞ ¼ A; C; G; T}ll provided that we have the succession of nucleotides that corresponds to each of the genes, an issue that as yet has not been completely resolved. In still general terms, we will suppose that gene g1 corresponds to the Telomerasa gene. If the Telomerasa gene is active, it is of maximum length. This enables the cell to be reproduced indefinitely. Although an individual’s cells are all equal as regards their genetic makeup, they are not equal as regards the behaviour of each of their genes. For this reason, we provide the following definitions: Definition 3. Reproductive or mother cells: These are the cells resulting from the first divisions of the ovum until the moment an embryo is formed. Their genome function indicates that, except for the Telomerasa gene, their genes are not active. Definition 4. Specialised cells: These are cells which correspond to a particular tissue or organ. They are characterised by the fact that their genome function indicates that the Telomerasa gene is passive. This means that the specialised cells can, at most, be divided and multiplied a finite number of times (approximately 50 times). Evidently, all the specialised cells which correspond to one particular tissue or organ will have the same genome function, whereas the specialised cells which correspond to different tissues or organs will have different genome functions. The identification of this genome function is currently being studied in the field of cellular biology. Definition 5. Tumour cells: Like mother cells, these are also immortal. Their genome function also indicates that the Telomerasa gene is active and that reproduction is indefinite. This change is either due to cellular mutation at some point during the cell’s lifecycle or to cellular division. Proposition 1. Mother cells can be divided and multiplied indefinitely. Moreover, the number of descendants that may be obtained from a mother cell in the n-th division is 2nþ1 2 1 ;n [ N < {0} Proof. By mathematical induction: A For n ¼ 0, 20þ1 2 1 ¼ 1, which is the mother cell x.
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For n ¼ 1, when applying cellular meiosis to cell x: ’{y,z} cells/DðxÞ ¼ {y; z}, then in total we would have cells {x, y, z}, that is to say 3 ¼ 22 2 1 Let us suppose that for i: the number of descendant cells is 2iþ1 2 1 of which i 2 are reproduced by cellular division, where 2.2i new cells are obtained. The total number of cells would be: 2iþ1 2 1 þ 2:2i ¼ 2iþ1 2 1 þ 2iþ1 ¼ 2:2iþ1 2 1 ¼ 2iþ2 2 1: Proposition 2. Tumour cells can be divided and multiplied indefinitely. Moreover, the number of descendants obtained from a tumour cell in the n-th division is 2nþ1 2 1 ;n [ N < {0}, as demonstrated in the previous proposal. Proposition 3. Specialised cells can be divided and multiplied a finite number of times (approximately 50). Consequence 1. The Telomerasa gene is only active in mother cells and tumour cells.
Cellular meiosis for mother cells Definition 6. Let [ti,t ] and [t,tj] be the lifespans of the mother cells x and y respectively, where x½ti ; t [ M M ½t0 ; tf , y½t; t j [ M M ½t 0 ; t f . We shall say that cell x[ti,t ] is transformed by meiosis into cell y[t,tj] and we shall denote this by ðxðtÞ; yt Þ [ r t where r is the relation of cellular meiosis if, as a result of a biological process (which we shall take to be a function in the mathematical sense), we obtain cell y[t,tj] from cell x[ti,t ]. Then: if tE ðxÞ ¼ ½ti ; t and t E ðyÞ ¼ ½t; t j , ðxðtÞÞrt ðyt Þ or ðxðtÞ; yt Þ [ r t , {’D : M ½t0 ; tf ! PðM ½t0 ; t f Þ function } ^ {’!z½t; tk [ M M ½t0 ; tf z is a mother cell / z½t; tk – y½t; tj ; tE ðzÞ ¼ ½t; tk } ^ Dðx½ti ; tÞ ¼ y½t; tj ; z½t; tk } [ PðM ½t 0 ; t f Þ For the unique cell z[t,tk] of the definition, we shall also say that cell x[ti,t ] is transformed by division into cell z[t,tk], which will be denoted by (x(t ))rt(zt) or ðxðtÞ; zt Þ [ r t In the first steps of the cellular division process for mother cells, f ðxðtÞÞ ¼ f ðyt Þ ¼ f ðzt Þ. This indicates that the same type of mother cells are obtained through this process. Furthermore, at a later stage in the cellular division process and for as yet unknown reasons, f(x(t )) is different to f(yt) or to f(zt). It is at this moment that specialised cells begin to be obtained. The formalisation of this situation is: If f ðxðtÞÞ ¼ f ðyt Þ ¼ f ðzt Þ: mother cells have been obtained during the cellular division process. If f ðxðtÞÞ – f ðyt Þ and f ðxðtÞÞ ¼ f ðzt Þ: during the cellular division process, cell zt remains invariable and cell yt is a cellular mutation of cell x(t ). Therefore cell yt will be a specialised cell of any tissue or organ if the Telomerasa gene remains passive; or a tumour cell if the Telomerasa gene remains active with maximum length.
If f ðxðtÞÞ ¼ f ðyt Þ and f ðxðtÞÞ – f ðzt Þ: during the cellular division process, Direct and cell yt remains invariable and cell zt is a cellular mutation of cell x(t ). Therefore, indirect causality cell zt will be a specialised cell of any tissue or organ if the Telomerasa gene in living systems remains passive; or a tumour cell if the Telomerasa gene remains active with maximum length. If f ðxðtÞÞ – f ðyt Þ and f ðxðtÞÞ – f ðzt Þ, and furthermore f ðyt Þ ¼ f ðzt Þ: during 489 the cellular division process, cells yt and zt are cellular mutations of cell x(t ). Therefore, cells yt and zt are specialised cells of the same tissue or organ if the Telomerasa gene remains passive; or tumour cells if the Telomerasa gene remains active with maximum length. If f ðxðtÞÞ – f ðyt Þ and f ðxðtÞÞ – f ðzt Þ, and furthermore f ðyt Þ – f ðzt Þ: during the cellular division process, cells yt and zt can either be specialised cells that do not belong to the same particular tissue or organ if the Telomerasa gene remains passive in the two cells; or one of them may be a specialised cell and the other a tumour cell if the Telomerasa gene of either of them remains active with maximum length. Cellular meiosis for specialised cells Definition 7. Let [ti,t ] and [t,tj] be the lifespans of the specialised cells x and y respectively, where x½ti ; t [ M S ½t0 ; t f ; y½t; t j [ M S ½t0 ; tf . We shall say that cell x[ti,t ] is transformed by meiosis into cell y[t,tj] and we shall denote this by ðxðtÞ; yt Þ [ r t where r is the relation of cellular meiosis if, as a result of a biological process (which we shall take to be a function in the mathematical sense) we obtain cell y[t,tj] from cell x[ti,t ]. Then: if tE ðxÞ ¼ ½ti ; t and t E ðyÞ ¼ ½t; t j , ðxðtÞÞrt ðyt Þ or ðxðtÞ; yt Þ [ r t , {’D : M ½t0 ; tf ! PðM½t0 ; tf Þ function } ^ {’!z½t; t k [ M S ½t0 ; tf z is a specialised cell / z½t; t k – y½t; t j ; t E ðzÞ ¼ ½t; t k } ^ Dðx½t i ; tÞ ¼ {y½t; t j ; z½t; t k } [ PðM ½t 0 ; t f Þ: For the unique cell z[t,tk] of the definition, we shall also say that cell x[ti,t ] is transformed by division into cell z[t,tk], which will be denoted by (x(t ))rt(zt) or ðxðtÞ; zt Þ [ r t : In the first steps of the cellular division process for the specialised cells, f ðxðtÞÞ ¼ f ðyt Þ ¼ f ðzt Þ. This indicates that the same type of specialised cells are obtained through this process. Furthermore, at a later stage during the cellular division process these cells may cease to be reproduced. This occurs when the Telomerasa gene is less than a certain length and is when the decaying process of a human being begins. It is possible that, for environmental, genetic or simply unknown reasons, that a mutation may occur in a specialised cell during its lifecycle, or that the mutation may appear at the point when this cell undergoes cellular division. Some of these changes may produce tumour cells. The formalisation of this situation is:
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If f ðxðtÞÞ ¼ f ðyt Þ ¼ f ðzt Þ: specialised cells which are of the same type as cell x(t ) have been obtained during the cellular division process. If f ðxðtÞÞ – f ðyt Þ and f ðxðtÞÞ ¼ f ðzt Þ: during the cellular division process, cell zt remains invariable and cell yt is a cellular mutation of cell x(t ). Therefore, cell yt will be a specialised cell of any tissue or organ, if the Telomerasa gene remains passive; or a tumour cell if the Telomerasa gene remains active with maximum length. If f ðxðtÞÞ ¼ f ðyt Þ and f ðxðtÞÞ – f ðzt Þ: during the cellular division process, cell yt remains invariable and cell zt is a cellular mutation of cell x(t ). Therefore, cell zt will be a specialised cell of any tissue or organ, if the Telomerasa gene remains passive; or a tumour cell if the Telomerasa gene remains active with maximum length. If f ðxðtÞÞ – f ðyt Þ and f ðxðtÞÞ – f ðzt Þ, and furthermore f ðyt Þ ¼ f ðzt Þ: during the cellular division process, cells yt and zt are cellular mutations of cell x(t ). Therefore, cells yt and zt are specialised cells of the same tissue or organ, if the Telomerasa gene remains passive; or tumour cells if the Telomerasa gene remains active with maximum length. If f ðxðtÞÞ – f ðyt Þ and f ðxðtÞÞ – f ðzt Þ, and furthermore f ðyt Þ – f ðzt Þ: during the cellular division process, cells yt and zt can either be specialised cells that do not belong to the same particular tissue or organ if the Telomerasa gene remains passive in the two cells; or one of them may be a specialised cell and the other a tumour cell if the Telomerasa gene of either of them remains active with maximum length. Mutations in cellular meiosis The DNA-alphabet has four letters (four nucleotides) and the protein-alphabet contains 20 letters (20 amino acids). A sequence of three specific letters (triplet) of the DNA-alphabet (called codons) forms one amino acid and a specific sequence of amino acids forms a protein. These biological phrases do not have any other characteristics such as punctuation, spaces, commas, accents, etc. Mutations are changes in the sequence of nucleotides forming DNAdeletions of one or more nucleotides, insertions on one or more nucleotides, transpositions of one or more segments of nucleotides from one place to another, and duplications of one or more segments of nucleotides. Definition 8. Given the specialised cell y[t,tj] of the living system S½t0 ; tf ¼ ðM ½t0 ; tf ; R½t0 ; tf Þ obtained by the cellular division of the specialised cell x[ti,t ], we will say that cell y[t,tj] is a mutation by suppression of cell x[ti,t ] if: f ðxðtÞÞ ¼ f ðg1 ðtÞ; g2 ðtÞ; . . .; gl ðtÞÞ ¼ ¼ {ðA; C; G; TÞl 1 ; ðA; C; G; TÞl 2 . . .; ðA; C; G; TÞl l } ¼ UVW where U,V and W correspond to consecutive sections of nucleotides such that the length of U þ V þ W ¼ n
f ðyðtÞÞ ¼ f ðg1 ðtÞ; g2 ðtÞ; . . .; gm ðtÞÞ ¼ ¼ {ðA; C; G; TÞm1 ; ðA; C; G; TÞm2 . . .; ðA; C; G; TÞmm } ¼ UW where U and W correspond to consecutive sections of nucleotides such that the length of U þ W ¼ p being p , n: Definition 9. Given the specialised cell y[t,tj] of the living system S½t0 ; tf ¼ ðM ½t 0 ; t f ; R½t0 ; tf Þ obtained by the cellular division of the specialised cell x[ti,t ], we will say that the cell y[t,tj] is a mutation by inversion of cell x[ti,t ] if: f ðxðtÞÞ ¼ f ðg1 ðtÞ; g2 ðtÞ; . . .; gl ðtÞÞ ¼ ¼ {ðA; C; G; TÞl 1 ; ðA; C; G; TÞl 2 . . .; ðA; C; G; TÞl l } ¼ UVW where U,V and W correspond to consecutive sections of nucleotides such that the length of U þ V þ W ¼ n f ðyðtÞÞ ¼ f ðg1 ðtÞ; g2 ðtÞ; . . .; gm ðtÞÞ ¼ ¼ {ðA; C; G; TÞm1 ; ðA; C; G; TÞm2 . . .; ðA; C; G; TÞmm } ¼ UV R W where U, V R and W correspond to consecutive sections of nucleotides such that the length of U þ V R þ W ¼ n, and furthermore that the nucleotides of V and V R are the same but where the nucleotides appear the other way round. Definition 10. Given the specialised cell y[t,tj] of the living system S½t0 ; tf ¼ ðM ½t 0 ; t f ; R½t0 ; tf Þ obtained by the cellular division of the specialised cell x[ti,t ], we will say that cell y[t,tj] is a mutation by transposition of cell x[ti,t ] if: f ðxðtÞÞ ¼ f ðg1 ðtÞ; g2 ðtÞ; . . .; gl ðtÞÞ ¼ ¼ {ðA; C; G; TÞl 1 ; ðA; C; G; TÞl 2 . . .; ðA; C; G; TÞl l } ¼ UVWZ where U,V,W and Z correspond to consecutive sections of nucleotides such that the length of U þ V þ W þ Z ¼ n f ðyðtÞÞ ¼ f ðg 1 ðtÞ; g2 ðtÞ; . . .; gm ðtÞÞ ¼ ¼ {ðA; C; G; TÞm1 ; ðA; C; G; TÞm2 . . .; ðA; C; G; TÞmm } ¼ UWVZ where U,W,V and Z correspond to consecutive sections of nucleotides such that the length of U þ W þ V þ Z ¼ n x evolves in y transposing a segment to a new position
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Definition 11. Given the specialised cell y[t,tj] of the living system S½t0 ; tf ¼ ðM ½t 0 ; t f ; R½t0 ; tf Þ obtained by the cellular division of the specialised cell x[ti,t ], we will say that cell y[t,tj] is a mutation by duplication of cell x[ti,t ] if: f ðxðtÞÞ ¼ f ðg1 ðtÞ; g2 ðtÞ; . . .; gl ðtÞÞ ¼ ¼ {ðA; C; G; TÞl 1 ; ðA; C; G; TÞl 2 . . .; ðA; C; G; TÞl l } ¼ UVW where U,V and W correspond to consecutive sections of nucleotides such that the length of U þ V þ W ¼ n f ðyðtÞÞ ¼ f ðg1 ðtÞ; g2 ðtÞ; . . .; gm ðtÞÞ ¼ ¼ {ðA; C; G; TÞm1 ; ðA; C; G; TÞm2 . . .; ðA; C; G; TÞmm } ¼ UVVZ where U,V,V and Z correspond to consecutive sections of nucleotides such that the length of U þ V þ V þ Z ¼ p being p . n:
Causality in living systems Direct and indirect causality in ecosystems have been dealt with extensively by authors such as Higashi and Patten (1989), Patten (1985, 2000), Patten et al. (1976, 1982) Higashi and Bums (1991), Higashi and Nakajima (1995), etc. in the study of different types of interaction (relationship) such as depredation, competition, etc. Definition 12. Given the living system S½t 0 ; t f ¼ ðM½t0 ; tf ; R½t0 ; tf Þ, we will say that cell x½t i ; t [ M ½t 0 ; tf has a direct influence on cell y½t; t j [ M ½t 0 ; t f as regards both route and time, at instant t [ ½t 0 ; t f if (1) ’r½t 0 ; t f [ R½t 0 ; t f where r may be the relation of cellular division, some mutation, environmental conditions, etc., such that ðx½t i ; t; y½t; t j Þ [ r½t 0 ; tf (2) Cell x[ti,t ] at instant t [ ½t0 ; tf and through the relationship r produces an effect on the attribute or behaviour of cell y[t,tj] in time t þ dt [ ½t0 ; t f , which we will represent by r t ðx t Þ ¼ yt Definition 13. Given the living system S½t 0 ; t f ¼ ðM½t0 ; tf ; R½t0 ; tf Þ, we will say that cell x½ti ; t [ M ½t 0 ; tf has an indirect and direct influence, as regards route and time respectively, on cell y½t; t j [ M ½t0 ; tf at instant t [ ½t 0 ; t f if: (1) Cells x1 ½tiþ1 ; t; x2 ½t iþ2 ; t; . . .; xn ½tiþn ; t [ M ½t 0 ; t f exist where ti , tiþ1 , tiþ2 , . . . , tiþn , t and relationships r1 ½t0 ; tf ;
r2 ½t0 ; tf ; . . .; rn ½t0 ; tf [ R½t0 ; tf exist, which may be equal or different, Direct and such that indirect causality ðx½ti ; t; x1 ½t iþ1 ; tÞ [ r 1 ½t0 ; t f ; ðx1 ½tiþ1 ; t; x2 ½t iþ2 ; tÞ [ r 2 ½t0 ; t f ; . . .;
in living systems
; . . .; ðxn21 ½tiþn21 ; t; xn ½tiþn ; tÞ [ r nþ1 ½t0 ; tf (2) Cell x[ti,t ], at instant ti+1 and through the relationship r1, produces an effect on the attribute or behaviour of cell x1[ti+1,t ] in time t iþ1 þ dt [ ½t 0 ; t f denoted by rt1iþ1 ðx tiþ1 Þ ¼ ðx1 Þtiþ1 . Cell x1[ti+1,t ] at instant ti+2 and through the relationship r2 produces an effect on the attribute or behaviour of cell x2[ti+2,t ] at instant t iþ2 þ dt [ ½t0 ; tf denoted by t t r2iþ2 ðx1iþ2 Þ ¼ ðx2 Þtiþ2 and so on, until finally cell xn[ti+n,t ] at instant tiþnþ1 ¼ t and through the relationship rn+1 produces an effect on the attribute or behaviour of cell y[t,tj] in time t þ dt [ ½t0 ; tf denoted by rtnþ1 ðxtn Þ ¼ yt Definition 14. Given the living system S½t 0 ; t f ¼ ðM½t0 ; tf ; R½t0 ; tf Þ, we will say that cell x½t i ; t [ M ½t0 ; tf has a direct and indirect influence, as regards route and time respectively, on cell y½t; t j [ M ½t0 ; tf at instant t [ ½t 0 ; t f if: (1) ’r½t 0 ; t f [ R½t 0 ; t f where r may be the relation of cellular division, some mutation, environmental conditions, etc., such that ðx½t i ; t; y½t; t j Þ [ r½t 0 ; tf (2) Cell x[ti,t ], at instant t [ ½t0 ; tf and through the relationship r, produces an effect on the attribute or behaviour of the cell y[t,tj] in time t þ mdt [ ½t 0 ; t f , where m is a positive integer m . 1 that we shall represent by r t ðx t Þ ¼ ytþðm21Þdt . This indicates a certain delay in the produced effect. Definition 15. Given the living system S½t 0 ; t f ¼ ðM½t0 ; tf ; R½t0 ; tf Þ, we will say that cell x½ti ; t [ M ½t0 ; tf has an indirect influence as regards both route and time on cell y½t; tj [ M ½t 0 ; t f , at instant t [ ½t 0 ; tf if: (1) Cells x1 ½tiþ1 ; t; x2 ½t iþ2 ; t; . . .; xn ½tiþn ; t [ M ½t 0 ; t f exist where ti , tiþ1 , tiþ2 , . . . , t iþn , t and relationships r 1 ½t0 ; tf ; r 2 ½t 0 ; t f ; . . .; r n ½t 0 ; t f [ R½t 0 ; t f exist, which may be equal or different, such that ðx½ti ; t; x1 ½t iþ1 ; tÞ [ r 1 ½t0 ; t f ; ðx1 ½tiþ1 ; t; x2 ½t iþ2 ; tÞ [ r 2 ½t0 ; t f ; . . .; ; . . .; ðxn21 ½tiþn21 ; t; xn ½tiþn ; tÞ [ r nþ1 ½t0 ; tf (2) Cell x[ti,t ], at instant ti+1 and through the relationship r1, produces an effect on the attribute or behaviour of cell x1[ti+1,t ] in time t iþ1 þ mdt [ ½t 0 ; t f m . 1 denoted by r t1iþ1 ðx tiþ1 Þ ¼ ðx1 Þtiþ1þðm21Þdt . Cell x1[ti+1,t ], at
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instant ti+2 and through the relationship r2, produces an effect on the attribute or behaviour of cell x2[ti+2,t ] in time tiþ2 þ mdt [ ½t 0 ; t f t t denoted by r 2iþ2 ðx1iþ2 Þ ¼ ðx2 Þtiþ2þðm21Þdt and so on, until finally cell xn[ti+n,t ], at instant tiþnþ1 ¼ t and through the relationship rn+1, produces an effect on the attribute or behaviour of cell y[t,tj] in time t þ mdt [ ½t 0 ; t f m . 1 denoted by rtnþ1 ðxtn Þ ¼ ytþðm21Þdt . Conclusion Authors such as Higashi and Nakajima, (1995) have developed a quantitative method (The Chain Rule Approach) for analysing indirect effects on ecological interaction networks. In this way, they have discovered the expression of the indirect effect transmitted through an individual path to be the products of the direct effects associated with the connections which constitute the path. As in an ecosystem, it is known that the indirect effects can be more powerful than the direct effects, Patten (2000). In the particular case of living systems, this issue is still an unresolved problem. The aim of this paper was to examine direct and indirect causality in living systems by studying cells and analysing relationships such as cellular meiosis and mutation. References Higashi, M. and Burns, T.P. (1991), Theoretical studies of ecosystems. The network perspective, Cambridge University press. Higashi, M. and Nakajima, H. (1995), “Indirect Effects in Ecological Interaction Networks I The Chain Rule Approach”, Mathematical Biosciences, Vol. 130, pp. 99-128. Higashi, M. and Patten, B.C. (1989), “Dominance of indirect causality in ecosystems”, The American Naturalist, Vol. 133, pp. 288-302. Lloret-Climent, M. (1999a), “Cellular meiosis: A system-linkage theoretic approach”, Cybernetics and systems: An international Journal, Vol. 30, pp. 1-8. Lloret-Climent, M. (1999b), “Systems approach to the concept of mutation”, Cybernetics and systems: An International Journal, Vol. 30, pp. 249-59. Lloret-Climent, M. (1999c), “Measures of cellular change in systems theory”, Kybernetes, Vol. 28, No. 9, pp. 1016-26. Lloret-Climent, M., Villacampa, Y. and Uso, J.L. (1998), “System-Linkage: Structural functions and hierarchies”, Cybernetics and systems: An International Journal, Vol. 29, pp. 29-39. Mesarovic, M.D. and Takahara, Y. (1975), General Systems Theory: Mathematical Foundations, Academic Press, New York. Mesarovic, M.D. and Takahara, Y. (1989), “Abstract Systems Theory“, in, Lecture notes in Control and Information Science, Thomas, M., Wyner, A., (Eds) Springer-Verlag, Berlin. Patten, B.C. (1978), “Systems Approach to the concept of Environment”, Ohio J. Sci., Vol. 78, No. 4, pp. 206-22. Patten, B.C. (1982), “Environs: Relativistic Elementary Particles for Ecology”, The American Naturalist, Vol. 119, pp. 179-219.
Patten, B.C. (1985), “Further developments toward a theory of the quantitative importance of indirect effects in ecosystems”, Contributions in Systems Ecology, Vol. 65, pp. 271-84. Patten, B.C. (2000), Holoecology. Manuscript. Patten, B.C., Bosserman, R.W., Finn, J.T. and Cale, W.G. (1976), “Propagation of cause in ecosystems“, in, Systems Analysis and simulation in Ecology, Patten, B.C. (Eds), Academic Press, New York Vol. 4, pp. 457-579. Patten, B.C., Richardson, T.H. and Barberm, G. (1982), “Path analysis of a reservoir ecosystem model”, Canadian Water Resources Journal, Vol. 7, No. 1, pp. 252-82.
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The current issue and full text archive of this journal is available at http://www.emeraldinsight.com/0368-492X.htm
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Physical system theory: fundamentals, recent developments and relationships with system dynamics Sushil Department of Management Studies, Indian Institute of Technology, Hauz Khas, New Delhi, India Keywords Cybernetics, Systems theory, Modelling Abstract Two important methodologies having some common grounds, but based on differing contexts and paradigms are Physical System Theory (PST) and System Dynamics (SD). The developments in both the fields have taken place almost independently, and attempts have been made to integrate the two to complement their strengths and limitations. This paper provides an overview of PST in terms of its foundations, philosophy, fundamental postulates, recent developments on its simplification and enlargement, and applications to socio-economic and managerial systems. A comparison of PST is made with SD on different fronts so as to understand their similarities and differences for carving out their place in modelling of managerial and socio-economic systems and integrating the two more meaningfully and flexibly. The paper is concluded emphasizing the need for a ‘Flexible System Theory’ which can relate many such systems based approaches and techniques on the whole continuum from hard to soft systems thinking to cater the whole spectrum of problem situations effectively.
Kybernetes, Vol. 31 No. 3/4, 2002, pp. 496-528. q MCB UP Limited, 0368-492X DOI 10.1108/03684920210422584
1. Introduction A large class of real world problems can be easily comprehended by taking a physical view of systems associated with them. These systems could be real physical systems such as electrical, mechanical, hydraulic, or other engineering systems; or conceptual physical systems such as managerial or socio-economic systems. The concrete and real physical systems exist in space and time, made of matter and energy and organized by information. The relationships among things or units in these systems are measurable. However, in case of conceptual physical systems, which are usually the living systems, the relationships such as love, hate, prestige, richness etc. are of the kind which involve meaning as interpreted by an observer, as discussed by Miller (1978). Thus, in case of the conceptual physical systems or the managerial or socio-economic systems the mechanistic metaphor gives only one viewpoint through which we can conceptualize the real world. Depending upon the problem situation, the system needs to be conceptualized using either the mechanistic, organismic, cultural or political metaphors (Flood and Jackson, 1991) or a combination of these. Thus,
a theory of physical systems can help us in developing a mechanistic view of Physical system the systems to begin with, which can then be developed further by using other theory approaches along with the physical system theory. According to Etizioni (1968), a physical system theory is easy to understand. The world is usually seen as a collection of concrete objects in space-time, though relationships are less obvious. Conceptualization of biological and 497 social aspects of life and behaviour in terms of concrete system in space-time enables us to benefit from a life time of experience in thinking that way. Though a physical view of the real world has its own limitations of not being able to capture many complexities in terms of growth, reproduction, multiple perspectives, and conflict, it has its own merits of a tangible assessment and clearer conception. In a variety of systems, such a physical view may be either sufficient or provides a sound basis, which can then be refined and enriched by subtler understanding of the real world. Thus, a theory of physical systems can help a long way in assisting the analysis and design of various systems of interest. In this paper, fundamentals and developments of the linear graph theoretic physical system theory (PST) are discussed, which is then compared and interfaced with a more organismic methodology, i.e. system dynamics (SD) to generate flexibility in modelling. The historical development of PST can be traced into three phases, viz. developments prior to the formulation of the theory; formulation of PST principles and concepts; and development towards its generalization to the application to managerial and socio-economic systems. The principle of conservation of energy offers a sound basis for the development of PST. The theory of electrical networks developed by Kirchhoff in 1847 laid the foundation of PST. The Kirchhoff’s nodal law states that the charge can neither be stored or lost at a node was converted into two complementary laws viz., current law, and voltage law. This law is analogous to Newtons law that the sum of all the forces acting on a body at rest is equal to zero (Evans and Disxhoorn, 1974). Another set of important developments took place in the area of developing electrical analogs of non-electrical systems. For example, oscillographic solutions of electro-mechanical systems by Nickle (1925), dynamic analogies by Olson (1943), and engineering analogies by Murphy et al. (1963). While developing the analogies, Firestone (1933) introduced the distinction between two types of physical variables; across variables and through variables. The across variables are measurable across two spatially different points, such as voltage, velocity, pressure, temperature, etc. The through variables are measurable at a single point such as current, force, fluid flow, heat flow, etc. On the other hand, developments in the area of linear/control system theory developed a sound basis for the modelling of physical systems. The third set of developments took place in mathematical graph theory and these have been utilized by Trent (1955) to provide isomorphism between
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oriented linear graphs and lumped physical systems. On the basis of this, many more developments took place, which culminated into the development of a new field of physical system engineering through the electro-mechanical system theory of Koenig and Blackwell (1961). Here the concepts of graph theory were applied to develop free body models and system linear graph was used to write interconnection constraints. This has shifted the emphasis from developing analogies to linear graphs. The work was further developed by Koenig et al. (1967) to develop a discipline of analyzing discrete physical systems in the form of PST, which was applied to electrical, mechanical, and hydraulic systems. Contemporarily another development was made by Roe (1966) to use graph theoretic techniques for analysis of physical systems. Further developments in generalization based on the use of graph theoretic concepts resulted into the development of PST as a theory of general systems by Satsangi and Ellis (1971a). This has provided general principles of modelling to include conceptual physical systems (socio-economic) along with the real physical system (engineering). Koenig et al. (1972) have then applied PST to engineering – economic and ecological systems and discussed the models of basic processes in an ecosystem, which is then used as a basis for its further application in managerial and socio-economic systems. In the context of managerial and socio-economic systems, it is simplified by Sushil (1992a) and the paradigm is enlarged to generalize the variables by Sushil (1990a). PST is interfaced with many other modelling methodologies to suit the requirements of the problem situation and systems in a flexible systems methodological framework (Sushil, 1994b). In this paper, it has been critically compared with SD and the schemes of its integration with SD are outlined.
2. Foundations In order to appreciate the PST methodology and the modelling of systems using this methodology, it would be desirable to develop understanding about its roots. It initially was developed as a general theory of physical –real systems, which is later on generalized to model and analyze physicalconceptual systems also. Thus, the foundations were more led with the principles of physical –real systems, with inputs from mathematics to facilitate abstraction as a means of making it a general system theory. This fact is amply reflected in the nature of PST and its roots can be traced to linear graph theory, electrical network theory, and linear system theory that facilitate the PST modelling process. The mathematical tools such as matrix algebra and differential equations are then used for analyzing and studying the system behaviour. These foundations and their provision to PST are shown in Figure 1.
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2.1 Linear graph theory PST is primarily developed to model and analyze discrete physical systems. In case of the physical real-discrete systems, the structure is transparent, components are discrete, and interfaces are finite. Thus, these systems can be conveniently represented in the form of abstract graphs having nodes or vertices and directed edges connecting these nodes. The graphic representation abstracts the real system in terms of its structure, which is the focus of study in system theory. Thus, it can act as a means through which the systems can be generalized transcending the disciplinary boundaries. The PST develops on the assumption whether it is an electrical, mechanical, hydraulic, managerial or socio-economic system, that its structure can be represented in the form of a linear graph (Harary et al., 1965). This requires a well-defined system structure, which is true for the physical real systems and also for a class or physicalconceptual systems (managerial and socio-economic) such as material flow systems, energy flow systems, money flow systems etc. As PST is primarily interested in flow modelling, the directed graphs are used to represented the structure. The beginning point is, thus, qualitative definition of the system/process to be modelled. In case of managerial and socio-economic systems, the graphic representation acts as a means of system conceptualization and communication of system structure. Though theoretically it demands a transparent structure, its practice helps in clearing the ambiguity and making systems more transparent to the users. The use of graph theoretic consideration in PST not only helps in abstraction of the system, it facilitates the system comprehension, and at the same time provides a framework through which the power of graph theoretic tools and techniques can be effectively used for systems modelling and analysis.
Figure 1. Foundations of physical system theory
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2.2 Electrical network theory The graphic representation acts as a basis of system conceptualization, but for the purposes of analytical treatment it needs to be converted into a mathematical model. The graphic representation facilitates the development of mathematical equations representing the internal behaviour of the components of the systems as well as their interconnections. The mathematical model makes use of Electrical Network Theory (Evans and Dixhoorn, 1974; Seshu and Reed, 1961; Roe, 1966) to portray the mathematical representation and principles of conservation in the system in representing the components and their interconnections. The principle of conservation holds good in most of the physical real systems, but in case of physical conceptual systems it may not be true in all cases and for all parts of the system. Thus, while extending the application of PST to physical conceptual systems certain adaptations/modifications are to be performed to close the system model. The principle of conservation helps in developing mechanistic models of the system, and thus, delimits the applications of PST in managerial and socio-economic systems where it is neither always possible nor always desirable to have a mechanistic view of the systems. 2.3 Linear system theory According to Glass (1976), “Although the physical laws which govern the dynamic behaviour of ocean waves differ from those which control a transistor amplifier, the processes by which we analyze them are quite similar. The physical laws are distilled into mathematical models . . .”. A mathematical model for a dynamic system is a set of differential equations which predicts its behaviour under typical inputs. Linear system theory provides basis for the development of mathematically tractable models, whereas in case of nonlinear systems little general theory is available. The state-space approach has come to play a central role in the theories of optimal control of constrained as well as unconstrained systems (Zadeh and Desoer, 1963). This provides a strong basis for mathematical analysis. The mathematical model so developed can be manipulated analytically to study the system behaviour by using mathematical tools such as matrix algebra, and differential equations. The use of graph theory in PST facilitates the model building process as well as it unbolts the power of these analytical mathematical tools for analysis purposes. Thus, it can be seen that PST is developed on strong theoretical background and uses powerful tools and techniques which increase its utility. 3. Philosophy PST is very clearly governed by the underlying systems philosophy that the system behaviour is dependent upon the structure of the components as well as
their interrelationships. There are two basic philosophies that help in Physical system developing such structures viz., theory (1) Causal philosophy; and (2) Chance philosophy. According to the causal philosophy, there is always a cause behind an effect 501 seen. The Indian philosophy is essentially causal, which is deterministic in nature. The philosophy of ‘karma’ is causal in nature, according to which the fruits or effects obtained are an outcome of the ‘karma’ caused by an individual. Thus, every happening can be traced back in the causal chain and, in principle, some cause can be assigned for the happening of any event in the world. On the other hand, the chance philosophy treats the events to be as chance happenings. The outcome of an event is not certain or deterministic rather probabilistic. Same action might result into many possible happenings according to some probability distribution. Thus, any event in real life can be assessed by the probability of its happening. PST is based on a philosophy, which is essentially causal in nature in an implicit sense. The PST models, thus, in a way represent the causal relationships in the system. The fundamental premises of PST are discussed as follows: i. Fundamental premise I “Systems behaviour is primarily determined by the structure of its components and their interrelationships.” This premise directs to view the structure of the system in terms of its components and interactions to study the system behaviour. A corollary of this dictates that the system behaviour can be controlled by intervening with the system structure. That is, a system can be designed to give a desired behaviour and this can be achieved by changing the components or interactions or both. The other two premises state about the underlying philosophy in developing the component structure, and interactions respectively. ii. Fundamental premise II “In a discrete system (having discrete components united at a finite number of interfaces), a component/sub-system can be modelled in isolation to characterize its behaviour as an independent entity irrespective of its interactions with other components and the environment.” This means that the components are independent so far their internal behaviour is concerned. The components can be taken out either physically or conceptually to develop their models representing their internal behaviour and then they can be refitted. The component models can be aggregated to get the model of the overall system. This is essentially a mechanistic view point of the system and is a strong theoretical assumption so far as the managerial and socio-economic systems are concerned. However, in practice this assumption is
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relaxed to a great extent in PST modelling of these systems, as usually the components in the managerial and social systems are defined uniquely by their placement in the system, and the number of their interfaces depend on their interrelationship with other components. This is unlike the physical real components such as mass, spring, resistor, capacitor, triode, etc. which have well defined terminals irrespective of the system and their interrelationship in a particular system. Whereas, the number of terminals for a transformation process depend on its linkages with other processes.
iii. Fundamental premise III “The components of a system interact in a manner such that the principle of conservation holds good within the system for all of its variables.” This takes a view of the system which has strict continuity and compatibility in its relationships. Again this conforms to a mechanistic view point and though true for mechanistic systems it may not always hold good for all managerial and socio-economic systems. The principle of conservation can be better assumed in these systems dealing with more tangible variables, e.g. material flow systems, energy flow systems, and money flow systems. The information flow systems are essentially non-conserved in nature. In view of the above, it can be said that philosophically PST conforms of the “mechanistic” metaphor of the systems and it can be used to analyze even the managerial and social systems from a mechanistic view point, which is also desired in a wide variety of situations. Further, as it provides a mechanistic representation of the systems, which can be used as good basis for “optimization” and system design, PST is philosophically inclined towards “hard systems thinking” (Checkland, 1981).
4. Fundamental postulates As discussed in the previous section, PST is based on the hard and analytic systems philosophy. It follows the basic analysis-synthesis cycle of systems approach, i.e. dividing whole into parts and resynthesizing back into the whole to infer about the whole more rationally, systematically and objectively The three fundamental postulates of PST (i) component postulate, (ii) system postulate and (iii) interconnection postulate, conform to this analysis-synthesis cycle as shown in Figure 2. PST (Koenig et al., 1967) is based on the notion of a linear graph. The first notion of the theory is that a single oriented line segment represents the measurement of two complementary variables (an “across” variable and a “through” variable). The “across” variable normally exhibits different values at the terminals, whereas the “through” variable exhibits the same value at both the terminals. The fundamental postulates of PST are as follows:
i. Component postulate Physical system “The pertinent performance and behavioural characteristics of every n-terminal theory component in an identified system structure are completely and uniquely defined by a component terminal graph of ðn 2 1Þ edges and a set of ðn 2 1Þ equations relating ðn 2 1Þ ““across” and ðn 2 1Þ “through” variables.” In order to develop PST model of any system, the components are first 503 defined and the component models are developed for each component separately. The component models are composed of component terminal graphs or free body diagrams, and component terminal equations. The component terminal graphs are dependent upon the number of terminals (interfaces) of the component. The component terminal graph ideally has a tree structure, i.e. it interconnects all the terminals without forming a loop. Thus, n terminals of a component can be connected by a component terminal graph of ðn 2 1Þ edges. Every edge is associated with an “across” variable and a “through” variable, and thus, there are 2ðn 2 1Þ terminal variables for component having n terminals. In general, for actually establishing the component model after selecting a terminal graph, it is required to select one set S1 of ðn 2 1Þ terminal variables as independent variable functions of time, and the remaining set S2 of ðn 2 1Þ terminal variables as dependent variable functions of time. The only requirement on sets S1 and S2 is that each contains ðn 2 1Þ terminal variables. The component model consists of a set of ðn 2 1Þ relations or, more generally, a mapping showing the variable functions of time in S2 as a function of the variable function of time in set S1 (Koenig et al., 1967). ii. System postulate “On uniting the component terminal graphs in a one-to-one correspondence with actual interconnections or interrelations of the components in the system, the system graph is obtained.” The system graph is characterized as having ‘v’ vertices corresponding to interfaces between the identified components in the system, and ‘e’ edges corresponding to the total number of edges in the component terminal graphs
Figure 2. Analysis-synthesis cycle of physical system theory
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and dummy edges used to close the graph (if any). Thus, the total number of variables associated with the system graph are “2e”. Usually, some variables are exogenously specified as stimulus to drive the system and are called as drivers. Let nx be the number of specified across variables or across drivers, and ny be the number of specified through variables or through drivers, then the total number of unknown variables are ð2e 2 nx 2 ny Þ; which required ð2e 2 nx 2 ny Þ number of equations to be solved. Out of these, already ðe 2 nx 2 ny Þ component equations are developed to specify the performance characteristics of the components; the remaining e liner homogeneous algebraic equations are obtained as structural constraints by the interconnection pattern of the components. The total structure of the system model is shown in Figure 3. iii. Interconnection postulates The interconnection postulates synthesize the different components into the system framework on the basis of the ability of the system graph to permit description of the interconnection pattern of the components. As there are two categories of variables, the principle of conservation operating on each one leads to a separate law or sub-postulate. Cutset subpostulate or continuity law “The law states that the flow is conserved in the system i.e. the algebraic sum of the flow rates into vertices or the algebraic sum of the through variables implied by the oriented edges of any cutset in the system graph is zero.” This law is based and is the generalization of the Kirchhoff’s Current Law and also referred to as GKCL. The total number of continuity or fundamental cutset equations are ðv 2 1Þ; which are determined by the number of branches in the tree identified on the system graph with “v” vertices. This law defines strict continuity of flow through the system. Circuit subpostulate or compatibility law “The ‘across’ variable evaluated of whatever path is followed between any two vertices will be the same, i.e. the algebraic sum of the across variables implied by the oriented edges of any circuit in the system graph is zero.” The law is based on the principle of conservation regarding the across variables, is the generalization of the Kirchhoff’s Voltage Law, and is also referred to as GKVL. The total number of compatibility or fundamental circuit equations are ðe 2 v þ 1Þ which are determined by the number of chords in the co-tree (complementary to tree) in the system graph with “v” vertices and “e” edges. This law defines the compatibility of across variables within the system.
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Figure 3. Structure of a physical system theory model
5. Strengths and limitations In order to appreciate the applicability of PST to model real life systems, it is desirable to examine critically from the viewpoint of its strengths and limitations; some major ones are outlined as follows: Strengths The significant strengths of the PST modelling approach are as follows: (1) It has a graphic input interface, which helps in conceptualizing the system as well as in communicating the system structure. (2) It uses powerful tools of linear graph theory, matrix algebra, and differential equations so as to make the modelling as well as analysis an efficient process. (3) There are well-defined steps of modelling and the model structure is transparent. The modelling process is having a high degree of repeatability and is, thus, having a scientific rigour.
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(4) Since it is based on the principles of linear graph theory, it can be used as a general system theory; the physical (real as well as conceptual) components of different types can be abstracted in the form of a graph, which facilitates generalization of modelling as well as analysis process.
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(5) It inbuilds “across” as well as “through” variables in a single model, and thus, provides a more comprehensive framework than other flow modelling approaches, which primarily consider the through or flow variables. (6) The dynamic model in the framework of state-space modelling is well established in linear system theory and gives a lot of insight about the dynamic behaviour of the system. The strong points of control system theory are inbuilt in such a model and powerful solution procedures can be used. (7) It provides an analytical solution to the system models as well as simulation or numeric analysis. (8) The coefficients are defined for each component separately, and thus, can be easily measured as the attention is to be focused only on one component at a time. It also helps in channelizing the modelling and validation process. (9) These models can be integrated with other mathematical models and can be used at different levels of planning.
Limitations Some major limitations of the PST modelling approach are as follows: (1) It takes a mechanistic view of the system, which is good for the engineering systems, but has a limited utility for a large class of managerial and social systems, which are more organic and cultural in nature. (2) The formulation is very rigid in terms of the definition of variables and the steps of modelling and analysis, which restricts its application to ill defined problems that are very common in managerial and socioeconomic systems. (3) The theory is based on the principle of conservation, and thus, cannot be applied directly in the systems where principle of conservation does not hold good. However, with certain modifications, such as introduction of dummy edges, it can handle non-conserved situations also, but certainly it increases the number of variables and system equations.
(4) It does not model the information system very explicitly and clearly. The Physical system information linkages in terms of across variable interactions are theory modelled implicitly. (5) Mathematical treatment is very rigorous, which restricts the application to softer/ill defined managerial and social systems. (6) It is difficult to apply to degenerate systems. (7) The model basically assumes linear relationships, and thus, has a limited utility in managerial and social systems, which have high degree of nonlinearity. Though it can also handle non-linear relationships with advanced treatment the process becomes complex, which again delimits applicability. (8) Basically the model is deterministic and can handle stochasticity only with advanced treatment, which demands higher degree of expertise as well as more extensive mathematical treatment. (9) The dynamic model is having a rigid structure and demand a lot of mathematical manipulation. Thus, the applicability to policy modelling and analysis is limited. (10) The state-space model may have rate to rate loops, which in managerial and social systems may exhibit false dynamics. (11) Non-availability of user-friendly specialized software is a major factor deterring its application to managerial and socio-economic systems. The control systems theory based software is available for analysis, which does not provide modelling support. (12) For the modelling, the variables in the system have to be categorized into “across” and “through” variables, which at times, are difficult to be defined. Moveover, many variables in the system may not fit in the strict definition of “across” and “through” variables and strong assumptions are to be made to handle those variables. (13) Simultaneous equations are to be solved for analyzing the model, which for a larger system becomes complex and, at times, indeterminate. Many of the above mentioned limitations have been overcome by simplifying the modelling in the context of managerial and socio-economic systems, and by enlarging the paradigm to generalize the variables as discussed in the subsequent sections. Further, it can be made more applicable by suitably interfacing it with System Dynamics (Forrester, 1961) and other modelling methodologies which is presented in the coming sections. This is creating flexibility in its application and enhancing the utility for modelling a vast category of physical systems.
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6. modelling of managerial and socio-economic systems When it is required to have a mechanistic view of the managerial and socioeconomic systems, PST offers a sound modelling framework. As has already been discussed, the application of PST requires the following: (1) The system should be discrete. (2) Variables are measurable and definable in “across” and “through” categories. (3) Structure of the system is clear in terms of components and their interfaces, which are finite in number. (4) Principle of conservation holds good for the system. The managerial and social systems are discrete with finite number of interfaces. In many cases, particularly, when the flow variables are tangible, e.g. material flow, energy flow, money flow, etc., the “across” and “through” variables can be defined and the principle of conservation also remains valid. In such situations, the structure is transparent, e.g. manufacturing systems, financial systems, intersectoral interactions, and the components can be defined at different levels of aggregation depending upon the purpose of the study. If the structure is not well defined, then first of all an exercise should be done to clarify the structure, say by using Interpretive Structural Modelling (Warfield, 1990) and then the PST model can be developed, e.g. in the case of Management Information Systems (Gupta, 1990). Once the variables are defined and components/sub-systems for modelling are identified the first task is to develop component models. In case of engineering (physical real) systems, the mathematical models of different basic components, e.g. resistor, capacitor, inductor, mass, spring, damper etc. are well established, whereas in the managerial and socio-economic systems there are no such well defined components with established models representing their internal behaviour. In many cases, the component models are based on empirical analysis or assumed behaviour. However, the PST modelling of managerial and socio-economic systems is greatly facilitated by developing generalized models of some basic processes that can be identified in these systems. Experience in modelling managerial and socio-economic systems using PST has shown that the generalized process of modelling leads to some redundancies in the PST models of these systems. Thus, owing to the peculiar features of the systems, simplification of modelling as well as analysis can be attempted. In many practical situations, all the variables cannot be classified according to the strict definitions of “across” and “through” variables. The paradigm, thus, can be enlarged by generalizing the definition of variables, facilitating the application of PST to comparatively softer systems.
6.1 Basic processes Physical system Koenig et al. (1972) proposed ecosystem concept to extend the application of theory PST to engineering-economic systems; according to which, different components of the ecosystem are conceptualized to perform any one or a combination of the three basic processes: material transformation process; material transportation process; and material storage process. These processes 509 can be generalized for any resource viz. material, energy, money, manpower, information, etc. in the context of various managerial and socio-economic systems. Though theoretically different components of the system under consideration can be conceptualized to perform any one or a combination of these three basic processes, it is worthwhile to define their variants to suit to practical situation demanding specialized treatment. Thus, for example, aggregation, disaggregation, and inspection processes are specialized transformation processes and separate models for these could be developed. Hence, it is proposed to consider the following basic processes to help the modelling of managerial and socio-economic systems. i. Transformation process The transformation process can be defined as a transformation of resources to achieve a well-defined change in their physical, chemical, technological, biological or functional characteristics. A transformation process can have multiple inputs and multiple outputs. To facilitate the modelling, a multi-inputmulti-output transformation process can be conceptualized as if there is one main output and the remaining outputs are auxiliaries or waste as shown by the free-body diagram of a typical generalized transformation process in Figure 4(a). The process vertex is treated as datum node which acts as reference. The component terminal equations for such a transformation process with n inputs, m wastes and one main output have one cost (across variable) equations for the main output and ‘n+m’ flow equations for the n inputs and m wastes. Some examples of transformation processes in managerial and socio-economic systems are: manufacturing stages, information processing or decision-making stages, managerial position, project activities, energy processing, training of manpower, etc. ii. Transportation process This can be viewed as a special type of transformation process wherein resource is simply moved from one geographic location to another at a cost. This includes collection, translocation and distribution of various kinds of resources with an origin and a destination as shown in Figure 4(b). The component terminal equations for such a process will be exhibiting the internal characteristics of the transportation process, may be the cost-volume relationship. If it is a multi-commodity flow transportation process, then additional equations for aggregating multiple flows into an equivalent flow, and multiple cost equations will be there. Some common examples of transportation
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Figure 4. Free body diagrams, (a) transformation process, (b) transportation process, (c) storage process, (d) aggregation process, (e) dissaggregation process, (f) inspection process
process are: materials handling in a plant, transportation/distribution of goods, Physical system passenger transportation, energy transmission, information communication, theory etc. iii. Storage process This can be viewed as a special type of transformation process in which the input and output are identical in form, and the resources are carried over time. The free body diagram of a generalized storage process is shown in Figure 4(c). In this case, there are two accumulation or level equations; one for the basic resource flowing through the system and the other for the associated valued flow may be energy or capital accumulation. For the output edge, a cost (across variable) equation is written, and for the input edge a flow equation is written depending upon the ordering policy. Some examples of storage processes are: storage of material, information storage, cadre in a manpower planning system, etc. iv. Aggregation process This is similar to transformation process, but there is no change in characteristics. The process simply aggregates multiple inputs into a single output. The free body diagram of a generalized aggregation process is shown in Figure 4(d). There is a one cost (across variable) equation for the aggregated output and n flow equations for n inputs. Some examples of aggregation process are: aggregation of finance from different resources, aggregation of returns from different projects, etc. v. Disaggregation process This is also a special class of transformation process with a single input desegregated into multiple outputs with/without change in characteristics. The free body diagram of a generalized disaggregation process is shown in Figure 4(e). The component terminal equations will involve cost (across variable) and flow equations and can be defined as per the situation. Some examples of disaggregation processes are: classification of a product into different streams; investment of finances to different projects, etc. vi. Inspection process This is a special type of transformation process where the input is classified into two categories viz. conforming and non confirming by inspecting the input. In the process, there could be two kinds of inspection errors, viz. declaring conforming parts as non-conforming, or declaring non-conforming parts as conforming. The free body diagram of a generalized inspection process is shown in Figure 4(f). The component terminal equations involve the definition of these inspection errors, costing of the conforming output, and flow of input and waste (non conforming output). Some examples of inspection
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processes are raw material inspection, stage inspection, finished goods inspection, etc. Standard generalized models of many more specialized processes can be developed through research to facilitate model building for managerial and socio-economic systems. One important process that can be of great importance in business systems could be exchange or transaction process where input and output are different resources.
6.2 Simplifications There are certain peculiar features that can be observed in the PST models of managerial and socio-economic systems, particularly based on the model of the transformation process. Two significant features of these systems are as follows: (1) In most of the interconnections of components, there is one-to-one correspondence, i.e. what is the output of one component, is the input to some other component as shown in Figure 5(a). The branching at the connecting node takes place in very few cases as shown in Figure 5(b). (2) A large class of these systems are forward flow or hierarchical in nature, i.e. the input is processed in stages, one after the other. (3) The first feature results in simplification in modelling, whereas the second feature results in simplification in solution of the model (Sushil, 1992a). 6.2.1 Simplifications in modelling. As PST provides a generalized framework to model a variety of systems, the direct application of the PST approach to the modelling of managerial and socio-economic systems leads to redundancy in the original model owing to the above mentioned peculiarities of these systems. In view of these peculiar features, the PST models can be considerably simplified resulting in reduction of the number of variables and number of system equations. The rules for simplification for such a system are as follows: (1) Eliminate the interconnection node (Figure 5(a)) and merge the links which are in one-to-one correspondence by maintaining the notation of the input links as shown in Figure 5(c). (2) Merge the continuity and compatibility equations in the component models by changing the notation of relevant variables and by removing the negative sign of cost (across variables) of output from the component equation of the previous component. (3) Eliminate all the dummy edges interconnecting the exogenous input/output/waste with the reference node.
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Figure 5. Types of interconnections and simplification, (a) a typical interconnection, (b) an interconnection with branching (c) a simplified Interconnection
The above simplification will result in halving of the number of edges in the system graph (if it has no branching nodes); the number of variables and the number of system equations also reduce to half of the original. It may be worth noting that in the process of simplification, nothing is lost regarding the details of the system. The simplified model will only have the component equations with inbuilt structural constraints; there may be a few structural constraints in the simplified models in case if branching at the interconnection nodes takes place. 6.2.2 Simplifications in analysis. In case of forward flow or hierarchical systems, the solution procedure of a PST model can be considerably simplified by using a step-by-step solution procedure which facilitates the application of PST to large scale managerial and socio-economic systems. The steps for obtaining the values of through variables are as follows: (1) Solve the flow equations for the last process to determine the values of flow of input/waste. (2) Solve the continuity equations to obtain the corresponding values of flows from other processes; only in case of interconnection nodes with branching. (3) Repeat steps (i) and (ii) successively until the value of flow of terminal input/waste is obtained.
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The steps for obtaining the values of “across” variables are as follows: (1) Solve the cost (across variables) equation for the first process having only exogenous inputs/wastes. (2) Solve the compatibility equation to obtain the value of the corresponding across variable going to next process, only in case of interconnection nodes with branching. (3) Solve the cost (across variable) equation of the next process. (4) Repeat steps (ii) and (iii) successively until the final cost (across variable) is calculated. 6.3 Generalization of variables – enlarging the paradigm All the variables in managerial and socio-economic systems usually need not conform to the previously mentioned strict definitions of “across” and “through” variables. For example, the quality variables of a component may take different values at both the terminals for some components, whereas they ma take the same values at both the terminals for certain other components of the same system, depending upon the characteristics of the different components. Further, the quality variables cannot be subject to the strict constraints on either the compatibility or the continuity for all the circuits or interconnecting nodes in the system graph. Thus, it is not possible to model quality or other soft variables by treating them either as “across” variables or as “through” variables, as the nature of the constraints that are imposed on such variables are usually not met by quality and other soft variables in reallife systems. In order to take care of such soft variables, the “across” and “through” variables have been generalized and an enlarged paradigm of PST has been developed by Sushil (1990a). In the extended framework, every edge in the system graph is considered to be associated with an additional generalized variable typified as “quality”, i.e. there are three variables associated with every edge: an “across” variables x, a “through” variable y, and a “quality” variable z. The quality variable is governed by the following considerations. (1) For components, the quality variable can take either the same value at both the terminals of an edge or different values at both the terminals depending upon the characteristics of the component. (2) In the system, the quality variable is governed by the quality transfer constraint which takes care of the perception gap at every interconnection by introducing a quality transfer coefficient or function. The “quality” variable is the generalized variable, whereas the conventional “across” and “through” variables are specific cases of the “quality” variable. The “across” variable is a special case of the quality variable when it takes different values at both the terminals for all the components and quality
transfer coefficients are unity for all the interconnections. The “through” Physical system variable is a special case of quality variable when it takes the same value at theory both the terminals for all the components and the summation of all quality transfer coefficients for an interconnection is unity. Some examples of quality variables are level of service in transport, quality of resources in economy, and managerial effectiveness in organizations. The 515 introduction of quality as a generalized variable in the enlarged paradigm of PST helps in the modelling of soft variables in the managerial and socioeconomic systems and contributes towards the development of a “Flexible Systems Theory”. 6.4 Applications PST has been extensively applied in the context of electrical, mechanical, hydraulic and other engineering systems. Though its application to managerial and socio-economic systems was initiated by Koenig et al. (1972), and Koenig and Tummala (1972) in the context of engineering-economic and ecosystems, the development in this field had been comparatively slow and sporadic. Initially, the applications had been restricted at the macro level on the lines similar to proposed by Koenig et al. (1972) and Satsangi and Ellis (1971a). Gradually, the applications shifted to micro level manufacturing systems and then to a variety of systems with tangible as well as intangible flows. The applications of PST to managerial and socio-economic systems can be classified as per the resource flow (through variable) as shown in Figure 6. i. Material flow applications Since material flow is tangible and follows the principal of conservation, the application of PST was first extended from engineering systems to material flow systems both at the macro and micro levels. At the macro level, the applications have successfully been made for modelling of national economic systems by (Satsangi and Ellis (1971b); Bhargava and Satsangi (1978); Sushil (1989; 1994a)) water resource systems by Agnihotri (1984); and transportation systems by Satsangi (1977) and Mathur (1984). At the micro level, applications have been made to productivity analysis (Sushil et al., 1988), multistage manufacturing systems (Singh and Sushil, 1990a; Sushil, 1992b; Kak, 1993), closed chain production systems (Sushil, 1991), project systems (Kanda, 1986), technology selection (Singh and Sushil, 1990b), and quality planning and analysis (Sushil, 1990a; Sushil and Pankaj 1992). ii. Energy flow applications Another physical resource which follows the principle of conservation is energy, and thus the PST modelling can be easily applied to energy flow systems. It can be applied to do flow modelling of energy along with the material flow (Sushil, 1989 and Singh, 1994), costing of energy, energy planning
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Flow
Flow
Resources
Flow
Flow
Flow
Figure 6. Applications of physical system theory
at the national level (Vij et al., 1988a), as well as the enterprise level and energy audit studies. iii. Money flow applications The associated flow with all physical resources is that of money, which is tangible and follows the principle of conservation. At the macro level, the applications in the area of economic planning deal with the flow of monetary resources as can be seen in Ellis (1965); Satsangi and Ellis (1971b); Sushil (1989); Vij (1990). It has also been applied in the context of central budget by Gupta (1991) to bring a performance orientation in the budgeting process. Budgeting applications have also been made in the area of marketing management to estimate the advertising and sales promotion budget. It has been applied to integrate all the three major financial decisions, viz., financing decision; investment decision; and dividend decision along with mathematical programming by Kulkarni (1993). Applications of PST can certainly be made in most of the cash flow problems. iv. Manpower flow applications Though the manpower related systems are associated with softer and ill defined aspects, there are quite a few areas which are comparatively tangible where the applications of PST have been made, e.g. manpower planning,
costing of training systems, project planning, etc. It can be seen that in the Physical system conventional rigid framework there is a limited scope of PST in this area. theory However, by using the enlarged paradigm with generalized variables, and by suitably interfacing it with System Dynamics, and Fuzzy Set Theory many more applications in human resource management can be successfully made. v. Information flow applications Flow of information is comparatively less tangible and is not conserved. However, at the architectural level with the application of information technology the information flow has become more tangible. Applications of PST to meta analysis of Management Information System architecture with respect to cost, flow and quality of information to aid various decisions such as budgeting, technology selection, and capacity planning can be made as can be seen in Gupta (1990). Since the flow of information is associated with ambiguities, integration of enlarged PST paradigm with System Dynamics and Fuzzy Sets will help a great deal in the analysis of Information Systems. vi. Flow of soft resources Attempts have also been made in the softer areas, such as organizational analysis and design to explore the utility of PST (Sushil, 1990b; Sushil and Raghunathan, 1994). In the organizational context, the organizational structure is comparatively a more defined area and both the mechanistic and organismic considerations are of interest. The parity of authority and responsibility and a balanced allocation of authority and responsibility are of central concern in organizational design, which can be flexibly modelled by PST in conjunction with Fuzzy Sets, and other qualitative analysis techniques. vii. Waste flow applications One significant area which shows the utility of PST in the real life managerial and socio-economic systems, and which could not be effectively handled by other modelling methodologies is of Waste Management. With every resource flow system, there is associated flow of waste and the PST modelling framework can capture this very explicitly. The aspects of waste recycling, rework, and disposal have been modelled by using PST at the macro level (Sushil, 1989), municipal waste management (Jalan et al., 1994), and in the manufacturing management (Singh, 1994). All operations management decisions can incorporate the waste considerations in manufacturing by using PST. 7. Relationships with system dynamics 7.1 Introduction to system dynamics System Dynamics is yet another methodology which is based on the concept of flow modelling like PST. It originated with the pioneering work of Forrester
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(1958) and developed subsequently during 1960s (Forrester, 1961, 1968, 1969, 1971). Till 1980 it has taken the shape of a discipline and then its further developments in the area of qualitative system dynamics (Wolstenhome, 1990), interfaces, use of AI, etc. (Sushil, 1993a, b) have been initiated. It is built upon the foundations laid by three background disciplines, viz., traditional management of social systems, feedback theory or cybernetics, and computer simulation. System Dynamics finds its roots essentially in the causal philosophy, assuming that the behaviour of a system is principally governed by its structure, and flow structure is most effective way of viewing an organization. Its main purpose is “gaining deep insight into the working of a system” and thus, governed by the “learning” or soft systems thinking paradigm (Checkland, 1981). It has more applications at the level of policy analysis and is supported by a set of diagramming aids, and specialized user friendly simulation packages. 7.2 A critical comparison PST and SD are critically compared for their similarities and differences for important facets such as, foundations, philosophy, paradigm, metaphor, system structure, variables, modeling, analysis, software, management level, strengths, limitations and applicability for managerial systems, as shown in Table I. It can be observed through such a critical comparison, that the two modeling methodologies are essentially complementary in nature and more meaningful applications of these methodologies can be made in a “Flexible Systems Methodology” framework (Sushil, 1994b) by suitably interfacing the two as per the requirements of the problem situation. 7.3 Integration or interfacing In view of the complementarity of PST and SD, as is reflected by the critical comparison in the preceding section, the two modelling methodologies can be suitably integrated or interfaced to cater to the differing requirements of problem situation lying on the whole continuum from well structured to unstructured in a Flexible Systems Methodology framework. As can be seen in Figure 7, PST covers more the operational level problems, whereas SD caters more effectively the strategic level problem situations. A suitable integration of the two will help bridging the gap and the problem situations from strategic to tactical management can be analyzed in an integrative and cohesive framework. Three possible schemes of integration of PST and SD are shown in Figure 8, which have been tried in past and exhibited meaningful application of the integrated models. i. In succession As shown in Figure 8(a), PST and SD can be effectively integrated in succession, i.e. developing a model in PST framework, and then converting it
Based on flow modelling. Models are static as well as dynamic. Modelling is governed by mixed formulation: (i) Differentiation in time domain, (ii) Linear time invariant equation in s-domain, and (iii) Integrated differential equations in operation domain. Basically considers linear relationships, and deterministic in nature. The analysis is done analytically using powerful tools of matrix algebra and differential equations. Sensitivity analysis is done through differentiation, which is essentially parametric.
Modelling
Analysis
Variables
Metaphor System structure
Paradigm
Sensitivity as well as policy analysis is done through experimentation including both the changes in parameters and model structure. (continued)
The analysis is done in simulation framework.
Nonlinear and deterministic.
Variables theoretically in modelling are treated as continuous Based on flow modelling. Models are essentially dynamic. Governed by the concept of integration
It is primarily based on management principles and associated disciplines such as traditional management, cybernetics, and computer simulation. The philosophy is explicitly causal. Flow structure is determined in terms of causal feedback chains, which deals with both the physical and information flows explicitly. Lays more emphasis on feedbacks. Essentially focused on learning about systems, and thus, oriented towards soft systems thinking. Governed by an organismic view of the system. Problem situations and systems with ill defined structure can be modelled which makes it more flexible in operation. Applied basically to continuous systems. The basic variables considered are “levels” and “rates”.
It’s roots primarily lie in engineering principles and disciplines such as graph theory, electrical network theory and linear system theory. The philosophy is implicitly causal. Flow structure is determined in terms of components and interrelationships, which deals primarily the physical flow; the information flow is implicitly considered. Lays more emphasis on principle of conservation. Provides a basis for optimization, and thus, oriented towards the hard systems thinking. Governed by a mechanistic view of the system. The underlying philosophy, paradigm and metaphor requires a well defined system structure, which makes it more rigid in operation. Applied basically for the analysis of discrete systems. Basically two types of variables are considered, i.e. “across” and “through”. Variables in modelling are treated as continuous.
Foundations
Philosophy
System dynamics
Physical system theory
Basis of comparison
Physical system theory
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Table I. A critical comparison of physical system theory and system dynamics
Table I.
Strengths
Limitations
Practice
Non-linearity and stochasticity can be handled by advanced treatment. Policy analysis cannot be carried out effectively.
The framework is more rigid and can be applied to well defined situations and systems only. Demands a high degree of mathematical knowledge and expertise by the user. Basically linear and deterministic in nature.
High degree of repeatability in modelling and thus the models can be generated through computer.
Graphic interface helps conceptualization Precision in results due to analytical modelling framework Powerful tools of mathematics can be utilized.
Has scientific rigour.
Applicability
Management level
Specialized software for managerial applications not available. Control system theory based software are used. In view of the attributes of the methodology, it fits, better to analyze the problems and systems at the operational management level (Figure 7) It is applicable very effectivity to the engineering systems, and the managerial and socio-economic systems with tangible flows, e.g. material, energy, money, manpower. Can be adapted to model intangible flows as well, e.g. information, responsibility etc. Individual modeller, demand high degree of expertise, basically applied at quantitative level.
Physical system theory
The repeatability is low; each modeller may have a different model of the same problem situation. Requires more involvement of managers.
Models may incorporate the bias of managers.
Specialized simulation packages are developed, which are user friendly with graphic features. In viewing of basically modelling the policy structures it is best suited at the strategic management level (Figure 7). It is more applicable to managerial and socio-economic systems. It takes both the tangible and intangible flows at one place, and can amalgamate multiple flows together. Group based modelling, participants need not have high degree of expertise, applied both at qualitative and quantitative levels. Can accommodate the modelling of Ill defined situations Diagramming aids are available for conceptualization More flexible in application. The mathematics used is very simple and can be easily comprehended by the manager. It translates the mental models and helps in storing as well as communicating managerial experience and knowledge. Can be used for development of games and decision support systems. Can capture the non-linearity and dynamics. Lacks scientific rigour.
System dynamics
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Software
Basis of comparison
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Figure 7. Applicability of physical system theory and system dynamics as per management levels
into a SD simulation model for simulating the behaviour, as applied by Kumar and Satsangi (1993). The PST model in simplified framework can be easily simulated using DYNAMO simulation package. ii. Submerging one into another As shown in Figure 8(b), the PST model can be made a submodel in a larger SD model so that it can be solved in every time period in a SD simulation framework. In this manner, hierarchical SD models can be developed and the strengths of PST may be used for that part of the problem situation which is well structured. An application of this type of integration is made by Vij et al. (1988b, 1993) in the context of energy policy planning in the Indian economy. It has developed an integrated PST-SD model in which a PST subroutine is developed to model the input-output relationships in an inter-sectoral economy and various policies are modelled in a SD framework. iii. Amalgamation As shown in Figure 8(c), the two techniques can be amalgamated so that within a single model part application of both are made as part requirements leading to a totally new formulation. An attempt in this direction has been made by Kumar et al. (1994) while developing a DSS for unit exchange spares management for Indian Railways. It develops the simplified system graph of various maintenance stages as transformation processes whereas the storage processes and their policies are developed in a SD framework owing to the inherent advantages of SD policy analysis. This has resulted into a simulation model of maintenance management in spreadsheet framework in which “whatif” analysis can be effectively done in a DSS environment.
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Figure 8. Schemes of integration of physical system theory and system dynamics, (a) integration in succession, (b) submerging one into another, (c) amalgamation
8. Flexibility in application In view of the requirements of a variety of systems, the physical system theory can be applied in a flexible manner on the fronts of systems to be analyzed, paradigm, structure, metaphor, level of application, type of model, variables and relationships, and nature of analysis. Flexibility refers to dynamic interplay on continuum exercising the freedom of choice (Sushil 1997; 1999). In order to exercise the freedom of choice, the first task is to identify the continua. Some important continua on which PST can be flexibly applied are shown in Figure 9. The PST can be applied on one hand to physical real systems, whereas, with suitable adaptation it can be applied, on the other hand, to physical conceptual systems exhibiting its adaptability for variety of system applications. Though, primarily PST is inclined towards hard systems paradigm and used more as an approach of system design, it can also be used in a soft systems thinking framework, as a learning aid to clarify the system structure in terms of its components and relationships, as well as the system behaviour under varying conditions. When PST is integrated to SD it operates from hard to soft systems thinking. The basic paradigm of PST applications is analytic, as shown in Figure 2. However, it can be applied synthetically in the context of flexible systems management using situation-actor-process (S-A-P) paradigm (Sushil, 1997).
Physical system theory
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Figure 9. Flexibility in Physical System Theory
First of all, the context of the system under consideration is clarified and the situation in which it is placed is analyzed. Next the actors dealing within the system are identified and in the light of the situation and actor the role of the system is defined which is used as a basis for its analysis under the three postulates. Such a synthetic paradigm provides a basis for reengineering the process/system under consideration. The structure of PST could be traditional or simplified depending upon the type of system under analysis. Further, if the variables are abstract and general the enlarged framework with quality variables can be applied. Usually, PST is based on mechanistic metaphor and applied at the operational level of management. However, by interfacing with SD it can be applied with organismic view and the application can be extended to deal with issues of policy at the strategic level.
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Figure 10. Trends towards developing a flexible system theory
y
The PST model can be developed empirically by field data or can be placed in a simulation framework. On the front of variables, it can use specific “across” or “through” variables or can have all the variables as general variables. The variables could be deterministic, stochastic or combined. Similarly, the variables, as well as relationships can be fuzzified to deal with qualitative or linguistic variables. Traditionally, PST uses linear relationships. However, in case of managerial and socio-economic systems the mathematical models can be framed to incorporate non-linearity in a simplified simulation framework. The PST models can be analyzed analytically or can be simulated in SD or Monte Carlo framework to capture dynamics and stochasticity. PST, thus, should be applied in a more flexible manner by examining the requirements of the situation, and actors so as to adapt to the problem situation more realistically. 9. Concluding remarks It can be observed by the preceding discussion that the applicability of PST to managerial and socio-economic systems can be enhanced by overcoming its
limitations through simplification of modelling as well as analysis; enlarging Physical system the paradigm to include generalized variables and resulting in a less rigid theory formulation; suitably interfacing it with SD and other modelling methodologies to cater to comparatively softer problems, and fuzzifying the qualitative variables. As shown in Figure 10, this trend of development is giving way to a “Flexible System Theory” through which many more systems based 525 techniques (qualitative as well as quantitative) can be integrated with PST in an innovative manner to help develop the models of managerial and socioeconomic problem situations lying on the whole continuum from well structured to unstructured, and enmeshing different system methaphors and perspectives as per the requirements of reality. Some of the system based techniques that can be considered in the Flexible System Theory framework in addition to PST and SD are Interpretive Structural Modelling, Fuzzy Set Theory (Zadeh, 1965), Linear/Goal Programming, Analytic Hierarchy Process (Saaty, 1980), Simulation, Delphi/Scenario building, statistical techniques, Soft Systems Methodology (Checkland, 1981), Interpretive Planning (Ackoff, 1981), Neural Nets, Expert Systems, and so on. It is hoped that such a spectral and integrative framework will help in a synthesis and convergence of the systems theory in tune with the emerging trends. This will aid in the analysis of manufacturing, business and economic processes to prepare a ground for their restructuring and reengineering, and ultimately regenerating innovative strategies in a flexible manner. This will, thus, provide a strong basis for the effective management of physical systems. References Ackoff, R.L. (1981), Creating the Corporate Future, John Wiley and Sons, New York. Agnihotri, G. Dynamic Modelling Studies of Water Resources Systems for Regional Development Applications: The Case of Punjab Region of India Unpublished Ph.D. Thesis IIT, Delhi. Bhargava, D.K. and Satsangi, P.S. (1978), “Economic system: a multidimensional model”, Int. J. of Systems Science, 9 No. 6, pp. 603-12. Checkland, P. (1981), Systems Thinking: Systems Practice, John Wiley and Sons, New York. Ellis, J.B. The Description and Analysis of Socio-Economic Systems by Physical System Techniques Ph.D. Thesis Michigan State University. Etizioni, A. (1968), The Active Society, Free Press, New York. Evans, F.J. and Dixhoorn, J.J. Von (1974), “Towards more physical structure in systems theory”, in Physical Structure in Systems Theory, Dixhoorn, J.J. Von, Evans, F.J. (Eds), Academic Press, London. Firestone, F.A. (1933), “A new analogy between mechanical and electrical systems”, J. Acoust Soc. A., 4, pp. 249-67. Forrester, J.W. (1958), “Industrial dynamics: a major break through for decision makers”, Harvard Business Review, 36 No. 4, pp. 37-66. Forrester, J.W. (1961), Industrial Dynamics, MIT Press, Cambridge, Massachusetts. Forrester, J.W. (1968), Principles of Systems, MIT Press, Cambridge, Massachusetts. Forrester, J.W. (1969), Urban Dynamics, MIT Press, Cambridge, Massachusetts.
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Forrester, J.W. (1971), World Dynamics, MIT Press, Cambridge, Massachusetts. Flood, R.L. and Jackson, M. (1991), Total System Intervention, John Wiley, New York. Glass, C.M. (1976), Linear Systems-with Application and Discrete Analysis, West Publishing Co., New York. Gupta, M.P. (1990), Modelling and Analysis of Management Information System Architecture at National Level, Ph.D. Thesis IIT, Delhi. Gupta, M. (1991), Systems Modelling for Central Budget, M. Tech. Dissertation IIT, Delhi. Harary, F., Normal, R.Z. and Cartwright, D. (1965), Structural Models: An Introduction to the Theory of Directed Graphs, John Wiley and Sons, New York. Kak, A. (1993), Application of Physical System Theory to Operations Management: A Case Study, M.Tech. dissertation IIT, Delhi. Kanda, A. (1986), Project Planning Studies with Renewable and Non-renewable Resources, Unpublished Ph.D. Thesis IIT Delhi. Kulkarni, R.G. (1993), Modelling and Analysis of Financial Decisions Using Physical System Theory and Goal Programming, M. Tech., Dissertation IIT Delhi. Koenig, H.E. and Blackwell, W.A. (1961), Electro-mechanical Systems Theory, McGraw Hill Book Co., New York. Kumar, S. and Satsangi, P.S. (1993), “System dynamics simulation of physical systems with application to Hopfield neural network”, in Systems Dynamics for Management Support, Sushil (Ed.), Wiley Eastern Ltd., New Delhi. Koenig, H.E. and Tummala, R.L. (1972), “Principles of ecosystem design and management”, IEEE Trans. Systems, Man and Cybernetics, 2 No. 4, pp. 449-59. Koenig, H.E., Tokad, Y. and Kesavan, H.K. (1967), Analysis of Discrete Physical Systems, McGraw Hill Book Co., New York. Koenig, H.E., Cooper, W.E. and Falvey, J.M. (1972), “Engineering for ecological, sociological and economic compatibility”, IEEE Trans. Systems, Man and Cybernetics, 2 No. 3, p. 319. Kumar, N., Vrat, P. and Sushil (1994), “A.simulation study of unit exchange spares management of diesel locomotives in the Indian Railways”, International Journal of Production Economics, 33, pp. 225-36. Mathur, M.C. (1984), Physical Systems Theory Research Investigations of Large Scale Transportation System Problems, Unpublished Ph.D. Thesis IIT, Delhi. Miller, J.G. (1978), Living Systems, McGraw Hill, New York. Murphy, G., Shippy, D.J. and Luo, H.L. (1963), Engineering Analysis, Iowa State University, University Press, USA. Nickle, C.A. (1925), “Oscillographic solution of electro-mechanical systems”, Trans AIEE, 44, pp. 844-56. Olson, H.F. (1943), Dynamical Analogies, Von Nostrand, Princeton. Roe, P.H. (1966), Networks and Systems, Addison Wesley, New York. Saaty, T.L. (1980), The Analytic Hierarchy Process, McGraw-Hill, New York. Satsangi, P.S. (1977), “A physical system theory modelling framework for transportation system studies”, IEEE Trans. Systems, Man and Cybernetics, 7 No. 11, pp. 763-78. Singh, T.P. (1994), Analysis and Design of Waste Management System for Engineering Industry, Ph.D. Thesis TIET Patiala. Sushil (1989), Systems Approach to National Planning: A Study in Waste Management, Anmol Publications, New Delhi.
Sushil (1990a), “Modelling of quality in physical system theory: an extended framework”, Int. J. of Systems Science, 21 No. 12, pp. 2489-512. Sushil (1990b), “Modelling and analysis of authority-responsibility relationships in an organizational System-A physical system theory approach”, Systems Research, 7 No. 4, pp. 255-71. Sushil (1991), “Simplified physical system theory model of a closed chain production system”, Int. J. of Systems Science, 22 No. 6, pp. 987-99. Sushil (1992a), “Simplification of physical system theory in the modelling of manufacturing, organizational and other socio-economic systems”, Int. J. of Systems Science, 23 No. 4, pp. 531-43. Sushil (1992b), “Costing and production planning using physical system Theory-A case study of manufacturing of compression springs”, Production Planning and Control, 3 No. 2, pp. 141-50. Sushil (1993a), Systems Dynamics: A Practical Approach for Managerial Problems, Wiley Eastern Ltd., New Delhi. Sushil (1993b), System Dynamics for Management Support, Sushil (Eds), Wiley Eastern, New Delhi. Sushil (1994a), “Application of physical system theory and goal programming to modelling and analysis of waste management in national planning”, Int. J. of Systems Sciences, 24 No. 5A, pp. 957-84. Sushil (1994b), Flexible System Methodology, Systems Practice, 7(6), Also Presented as Key Note Address at the Ninth International Congress on Cybernetics and Systems, New Delhi January, 1993. Sushil (1997), “Flexible systems management: an evolving paradigm”, Systems Research and Behavioral Science, 14 No. 4, pp. 259-75. Sushil (1999), Flexibility in Management, Global Institute of Flexible Systems Management, Vikas Publishing House, New Delhi. Satsangi, P.S. and Ellis, J.B. (1971a), “General systems from network systems: a philosophy of modelling”, Int. J. of Systems Science, 2 No. 1, pp. 1-16. Satsangi, P.S. and Ellis, J.B. (1971b), “A generalized system theoretic framework for modelling large scale national economic systems in dynamic, structural and spatial terms”, Int. J. of Systems Science, 2 No. 2, pp. 213. Seshu S. and Reed M.B. (1961), Linear Graphs and Electrical Network, Addison-Wesley Pub. Co., Reading. Singh, N. and Sushil (1990a), “A.physical system theory framework for modelling manufacturing systems”, Int. J. of Production Research, 28 No. 6, pp. 1067-82. Singh, N. and Sushil (1990b), “Technology selection models for multistage production systems: joint application of physical system theory and mathematical programming”, European Journal of Operational Research, 47, pp. 248-61. Sushil and Pankaj (1992), “Systems modelling of cost and quality: an application of extended physical system theory and simulation”, Systems Research, 9 No. 1, pp. 43-66. Sushil and Raghunathan, B. (1994), “Interactive decision support system for organizational analysis”, Decision Support Systems, 11, pp. 283-98. Sushil, Singh, N. and Jain, B.K. (1988), “A physical system theory approach to material flow and productivity analysis”, Engineering Costs and Production Economics, 13, pp. 200-15.
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Trent, H.M. (1955), “Isomorphisms between oriented linear graphs and lumped physical systems”, J. Acoust Soc. A., 27, pp. 500-27. Vij, A.K. (1990), Energy Policy Modelling and Analysis for the Indian Economy, Ph.D. Thesis IIT, Delhi. Vij, A.K., Sushil and Vrat, P. (1988a), “Energy modelling for national economy-a generalized model based on physical system theory approach”, Int. J. of Systems Science, 19 No. 6, pp. 857-73. Vij, A.K., Vrat, P. and Sushil (1988b), “Energy modelling through integration of system dynamics with physical system theory”, System Dynamics: International Journal of Policy Modelling, 1 No. 1. Vij, A.K., Sushil and Vrat, P. (1993), “Energy policy analysis for the indian economy through integration of system dynamics and physical system theory”, in, System Dynamics for Management Support, Sushil (Eds), Wiley Eastern Ltd., New Delhi pp. 371-83. Warfield, J.N. (1990), A Science of Generic Design: Managing Complexity Through Systems Design, Vol. I and II, Intersystems Publication, USA. Wolstenholme, E.F. (1990), Systems Enquiry: A System Dynamics Approach, John Wiley and Sons, Sussex, U.K. Zadeh, L.A. (1965), “Fuzzy sets”, Information and Control, 8, pp. 338-53. Zadeh, L.A. and Desoer, C.A. (1963), Linear Systems Theory: The State Space Approach, McGraw Hill Book Co., New York. Further reading Satsangi, P.S. (1969), A Physical Systems Theory Modelling Framework Generalized for Large Scale Economic Systems, Ph.D. Thesis University of Waterloo, Canada. Sushil, Jalan, R.K. and Srivastava, V.K. (2000), “The management of municipal solid waste using flexible systems approach”, in, Solid Waste Management, Grover, V.I., Guha, B.K., Hogland, W., McRae, S.G. (Eds), Oxford and IBH Pub. Co., New Delhi.
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Neural network process vision systems for flotation process Harry Coomar Shumsher Rughooputh Department of Electrical and Electronic Engineering, Faculty of Engineering, University of Mauritius, Re´duit, Mauritius
Soonil Dutt Dharam Vir Rughooputh Department of Physics, Faculty of Science, University of Mauritius, Re´duit, Mauritius
Neural network process vision systems 529 Received March 2001 Revised March 2001 Accepted August 2001
Keywords Cybernetics, Neural networks, Modelling Abstract Froth flotation is a process whereby valuable minerals are separated from waste by exploiting natural differences or by chemically inducing differences in hydrophobicity. Flotation processes are difficult to model because of the stochastic nature of the froth structures and the illdefined chemorheology of these systems. In this paper a hierarchical configuration hybrid neural network has been used to interpret froth images in a copper flotation process. This hierarchical neural network uses two Pulse-Coupled Neural Networks (PCNNs) as preprocessors that ‘convert’ the froth images into corresponding binary barcodes. Our technique demonstrates the effectiveness of the hybrid neural network for process vision, and hence, its potential for use for real time automated interpretation of froth images and for flotation process control in the mining industry. The system is simple, inexpensive and is very reliable.
Introduction Froth flotation is a process whereby valuable minerals are separated from waste by exploiting natural differences or by chemically inducing differences in hydrophobicity. Flotation processes are difficult to model because of the stochastic nature of the froth structures and the ill-defined chemorheology of these systems. These processes are constantly subjected to disturbances due to the random changes in mineral characteristics and changes in the operating conditions of the plants and equipment non-linearities. The structure and appearance of the froth surface reflect considerable information on the flotation process. Consequently, the control of industrial flotation plants is based on the visual appearance of the froth phase. Operators visually monitor the froth surface for an indication of the efficiency of flotation. The systems are usually controlled sub-optimally owing to human error, inadequate training and lack of experience. The interpretation of the froth surface is complicated by the time lag between the occurrence of froth phenomena and the results of the process. Furthermore, different sets of metallurgical parameters may result in a similar visual froth appearance. The authors are grateful to J.S.J. Deventer, Professor of Mineral and Process Engineering at the University of Melbourne, Australia.
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In many flotation concentrators, there is often a lack of adequate process instrumentation (Van Deventer et al., 1995). Only in certain plants will one find particle size monitoring, flotation feed density gauges, flotation pH meters, and on-stream assays of intermediate process streams. The monitoring and control of many plants are consequently often conducted on a largely (informal) heuristic basis, where plant operators attempt to maintain optimal operating conditions based on their experience of the behavior of the plant. As a result, over years of practice, many skilled operators develop an operating strategy based on observations of froth conditions. Changes in process conditions usually occur simultaneously in a number of dimensions – some enhancing and some inhibiting the process efficiency. The plant operator tries to maintain optimum froth conditions and then fine-tune reagents to increase recovery. It can take many years for a novice operator to learn how to “read” the froth and attain proficiency in the art of flotation (Benford and Meech, 1992). So, there is a need for an automated approach, particularly one that is adaptable and is able to learn from past data or recent experience. The present work deals with a neural network based technique to create such an automated system that would perform as well as an expert operator at monitoring and interpreting froth characteristics during the industrial flotation of minerals. Neural network process vision system The use of machine vision for flotation control has been investigated (Woodburn et al., 1994; Symonds and De Jager, 1992). These methods involved the identification of individual elements in the froth structures and they were case-specific, their computational complexity did not allow for an on-line system. Some work (Moolman et al., 1994; Moolman et al., 1995a; Aldrich et al., 1995) has been reported on the use of grey level dependence matrix methods (Sun and Wee, 1983; Siew et al., 1988) to extract features from digitized images of the froth phase (Moolman et al., 1994; 1995b; 1996a; Aldrich et al., 1995). It has been shown that the computed features can be related to practical values such as viscosity, bubble size and the stability and mobility of the froth (Moolman et al., 1995b). Some studies (Hargrave and Hall, 1996; Hargrave et al., 1996; Cilliers et al., 1996) have also been reported on the application of computer vision to flotation froths. Artificial Intelligence techniques have been successfully applied to perform classification tasks in many fields. Artificial neural networks (ANN) have been shown to be of significance for pattern recognition. Some work has been reported on the classification of froth structures in flotation plants on the basis of features extracted by means of digital image processing techniques by using learning vector quantization neural networks for discrete-time classification and self-organised mapping for continuous classification (Aldrich et al., 1995). A back-propagation neural network was used to relate image features to
flotation grade and recovery in order to identify the most important froth characteristic (Moolman et al., 1995b). The computer vision system developed in this work provides a more reliable and sensitive quantification technique than the human eye. Neural-network based computer vision systems for flotation processes rely on feature extraction from digital images of froths by means of digital image processing techniques: grey level dependence methods (spatial grey level dependence (Haralick, 1979), neighbouring grey level dependence (Sun and Wee, 1983). Such feature extraction methods involve considerable computations and are not suitable for real-time on-line applications. Recent experimental and theoretical studies of the visual cortex of the cat highlight the role of temporal processing using synchronous oscillations for object identification (Eckhorn et al., 1990). The original neural network model of Eckhorn when modified according to the proposal of Johnson (Johnson, 1994) is useful for image processing applications. In this paper an artificial neural network has been used to interpret froth images in a copper flotation process. A hierarchical configuration hybrid neural network, developed for this purpose, is shown in Figure 1. This hierarchical neural network uses two Pulse-Coupled Neural Networks (PCNNs) as preprocessors which ‘convert’ the froth images into corresponding binary barcodes. The PCNN used for the image pre-processing is based on Eckhorn’s neuronal model. It is a single layer, two-dimensional, laterally connected network of integrate-and-fire neurons, with a one to one correspondence between the image pixels and network neurons. Each neuron receives input signals from two types of synapses: feeding synapse and linking synapse. The feeding synapse gets its inputs from other neurons whilst the linking synapse obtains its inputs from lateral and feedback connections, both synapses inducing decaying potentials. The integrated signals from the linking synapse plus an offset term are multiplied with the integrating signals from the feeding synapse to produce a membrane voltage that is compared to a local dynamic threshold. When this membrane potential exceeds the value of the dynamic threshold, the neuron fires to produce an output spike; consequently, the threshold is significantly increased. This value then decays until the neuron fires again. The PCNN is fully described elsewhere (Lindblad and Kinser, 1998). When a digital image is applied to the input of a PCNN, the network groups image pixels based on spatial proximity and brightness similarity. P The time signal G[n ] is the number of “on pixels” in each iteration: G½n ¼ ij Y ij ½n where Yij is the output of pixel ij. Thus, the time signal is an encoded onedimensional signature of a two-dimensional image (Icon). The PCNN provides a 1:1 correspondence between images and their time signatures. These time signatures in the form of 8-bit grey level images are then presented to a second PCNN to produce binary barcodes. There is a one-to-one correspondence between these barcoded PCNN outputs and the corresponding input images.
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Figure 1. Block diagram of a hierarchical hybrid neural network for process vision
The froth images obtained from (Moolman et al., 1995b) used for this work represent five types of froth surfaces: (a) Ideal froth surface [Figure 2(a)left]: the bubbles are well-loaded with mineral and display a fenestruliform spot with a watery sheen on top of each bubble, indicating that the froth is neither too brittle nor too stiff. This froth structure indicates optimal process conditions, with the desired reagent flow rates and pulp level. (b) Polyhedral bubbles [Figure 2(b) left]: this type of froth structure signifies a deep, well-drained froth in which adequate separation between the mineral and gangue has been attained. In this way, a froth of high grade can be related to a deep (dry) froth with polyhedral bubbles. (c) Ellipsoidal bubbles [Figure 2(c) left]: the result of a froth that is too tough and hence exhibits resistance against the direction of flow by displaying ellipsoidal bubbles. The presence of ellipsoidal bubbles is a measure of the rigidity of the froth and an indication of the extent to which (in which direction) the froth has to be forced to flow. This froth structure may be caused by too low a pulp level, a specific gravity that is too high, a depressant addition rate that is too low, or by the flotation of a particular type or size of particle. (d) Irregular, ellipsoidal and stiff froth structure [Figure 2(d) left]: sometimes minerals with a peculiar structure (such as the laminar structure of
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Figure 2. Typical froth surfaces with corresponding icons and binary barcodes
valeriite) reveal an irregular structure of the surface froth of high stability. Ellipsoidal bubbles are present, but not with the well-defined clear directional tendency as in (c). This froth is excessively stable, a characteristic detrimental to subsequent mineral processing. This froth structure may also be attributed to a low pulp level or a frother addition rate that is too low. (e) Low mineral content [Figure 2(e) left]: this froth structure can be identified by small, spherical bubbles. The spherical bubbles are representative of a
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shallow, non-drained roth. If the pulp level is too high, the froth phase will be too shallow and this type of froth structure will result. It may also be caused by too much water in the feed (which corresponds to a low specific gravity), which yields a shallow and brittle froth. The time signatures (Icon’s) and the corresponding barcodes are shown in Figures 2(a) to 2(e) (center and right) for these five froth surface images. For an on-line monitoring and control system, froth surface images are captured at regular intervals by a CCD camera. This camera is suspended from a steel frame with anti-vibrational mountings and enclosed by a protective casing. Each image is fed to the hierarchical neural network and the corresponding time signature and barcode are obtained. The time signature is then fed to a previously trained Multi-layer Perceptron Classifier (MLP) to be classified into one of the five froth process classes. Or, alternatively either the one-dimensional binary barcode (BBC-1D) or any part of the two-dimensional barcodes (BBC-2D) can be used for classification using another pre-trained MLP classifier to identify the image to its froth process type. The output from the MLP classifier is used to activate the hardware for controlling the plant. This takes the form of a fuzzy-logic control system that identifies the appropriate control measures to be taken. Conclusions Our technique demonstrates the effectiveness of the hybrid neural network for process vision, and hence, its potential for use for real time automated interpretation of froth images and for flotation process control in the mining industry. The system is simple, inexpensive and is very reliable. The drawbacks of previous monitoring and control systems are overcome. This system can be easily implemented for different types of plants and for different ores. References Aldrich, C., Moolman, D.W., Eksteen, J.J. and Van Deventer, J.S.J. (1995), “Characterization of flotation processes with self-organizing neural nets”, Chem. Eng. Commun., Vol. 139, pp. 25-39. Benford, P.M. and Meech, J.A. (1992), “Advising Flotation operators using a real-time expert system”, Minerals Engineering, Vol. 5, No. 10-12, pp. 1325-31. Cilliers, J.J., Woodburn, E.T., Hall S.T. and Miles, N.J. (1996), “The kinetic modelling of pilot plant flotation using batch froth image data”, Minerals Engineering ‘96, Brisbane, Australia. Eckhorn, R., Reitboeck, E., Arndt, M. and Dicke, P. (1990), “Feature linking via synchronisation among distributed assemblies: Simulations of results from Cat Visual Cortex”, Neural Computation, Vol. 2, pp. 293-307. Haralick, R.M. (1979), “Statistical and structural approaches to texture”, Proceedings of IEEE, Vol. 67, No. 5, pp. 786-803. Hargrave, J.M. and Hall, S.T. (1996), “Diagnosis of concentrate grade and mass flowrate in tin flotation from colour and surface texture”, Minerals Engineering ‘96, Brisbane, Australia
Hargrave, J.M., Miles, N.J. and Hall, S.T. (1996), “The use of grey level measurement in predicting coal flotation performance”, Minerals Engineering, Vol. 9, No. 6, pp. 667-74. Johnson, J.L. (1994), “Pulse-coupled neural nets: translation, rotation, scale, distortion and intensity signal invariance for images”, App. Opt., Vol. 33, pp. 6239-53. Lindblad, T. and Kinser, J.M. (1998), Image Processing using Pulse-Coupled Neural Networks, Springer-Verlag, London. Moolman, D.W., Aldrich, C., Van Deventer, J.S.J. and Stange, W.W. (1994), “Digital image processing as a tool for on-line monitoring of froth in flotation plants”, Minerals Engineering, Vol. 7, No. 9, pp. 1149-64. Moolman, D.W., Aldrich, C., Van Deventer, J.S.J. and Bradshaw, D.J. (1995a), “The interpretation of flotation froth surfaces by using digital image analysis and Neural Networks”, Chemical Engineering Science, Vol. 50, No. 22, pp. 3501-13. Moolman, D.W., Aldrich, C., Van Deventer, J.S.J. and Stange, W.W. (1995b), “The videographic characterization of flotation froths using Neural networks“, in, Neural Networks for Chemical Engineers, Bulsari, A.B., (Eds) Elsevier Science B.V., Amsterdam pp. 525-45. Moolman, D.W., Aldrich, C., Schmitz, G.P.J. and Van Deventer, J.S.J. (1996a), “The interrelationship between surface froth characteristics and industrial flotation performance”, Minerals Engineering, Vol. 9, No. 8, pp. 837-54. Moolman, D.W., Eksteen, J.J., Aldrich, C. and Van Deventer, J.S.J. (1996b), “The significance of flotation froth appearance for machine vision control”, Int. J. Miner. Process., Vol. 48, pp. 135-58. Sun, C. and Wee, W.G. (1983), “Neighbouring grey level dependence matrix for texture classification”, Computer Vision, Graphics and Image Processing, Vol. 23, pp. 341-52. Symonds, P.J. and De Jager, G. (1992), “A technique for automatically segmenting images of the surface froth structures that are prevalent in flotation cells”, Proc. of the 1992 South African Symposium on Communications and Signal Processing, University of Cape Town, Rondebosch, South Africa, pp. 111-5 Siew, L.H., Hodgson, R.M. and Wood, E.J. (1988), “Texture measures for carpet wear assessment”, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 10, No. 1, pp. 92-105. Van Deventer, J.S.J., Aldrich, C. and Moolman, D.W. (1995), “The tracking of changes in chemical processes using computer vision and self-organizing maps”, ICNN’95, pp. 3068-73 Woodburn, E.T., Austin, L.G. and Stockton, J.B. (1994), “A froth based flotation kinetic model”, Trans. I ChemE, 72A, pp. 211-56.
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A new approach to solve a diffusion-convection problem M. Inc Department of Mathematics, Firat University, Elazig, Turkey
Y. Cherruault Received March 2001 Revised March 2001 Accepted August 2001
Universite´ Pierre et Marie Curie, Laboratoire MEDIMAT, Paris, France Keywords Cybernetics, Decomposition method, Noise Abstract We use the Adomian decomposition method to study a non-linear diffusion-convection problem (NDCP). The decomposition method has been applied recently to a wide class of non-linear stochastic and deterministic operator equations involving algebraic, differential, integrodifferential and partial differential equations and systems. The method provides a solution without linearization, perturbation, or unjustified assumptions. An analytic solution of NDCP in the form of a series with easily computable components using the decomposition method will be determined. The non-homogeneous equation is effectively solved by employing the phenomena of the self-cancelling ‘noise terms’ whose sum vanishes in the limit. Comparing the methodology with some known techniques shows that the present approach is highly accurate.
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1. Introduction The method developed by (Adomian, 1983, 1986a, 1986b, 1984; Adomian and Rach, 1985, 1988) since the beginning of the 1980’s gives the solution as an infinite series quickly converging towards an accurate solution. The Adomian decomposition method in applied mathematics is widely considered now by most mathematicians. The method is very reliable and effective and provides the solution in terms of rapid convergent series. The series used is a series of functions rather than terms as is the case in Taylor series. The decomposition method is considered a generalization of Taylor series, but the expansion is about u0 and not about x ¼ 0: The method is applied recently to a wide class of differential and integral equations, stochastic and deterministic problems, linear and non-linear, in many cases. In addition the method addresses all kinds of Mathematical Physics and Mathematical Biological models in a very strong way and has promising results (Laffez and Abbaoui, 1996; Sanchez et al., 2000; Adomian et al., 1996). A critically important problem in frontier science and technology is the physically correct solution of non-linear and stochastic systems modelled by differential or partial differential equations for general initial or boundary conditions. The usual procedures of analysis necessarily change such problems is essential ways in order to make them mathematically tractable by established methods. Unfortunately, these changes necessarily change the solutions,
therefore, they can deviate sometimes seriously, from the actual physical behaviour. These procedures include linearization techniques, perturbation methods, and restrictions on the nature and magnitude of stochastic processes. The avoidance of these limitations so that physically correct solutions can be obtained would add in an important way to our insights into the natural behaviour of complex systems. The advantages of this method to other methods is more simplicity, give better results and with time saving because in this method convergence is especially rapid in the non-linear and nonhomogeneous equations. Using decomposition method we give the solution both x- and t-direction which in the two solutions equals each other (Adomian and Rach, 1990; Wazwaz, 1997a). This kind of solution gives the facility, if we can’t obtain the solution in the x-direction, we can use the t-direction. Thus, when we compute the solution in t-direction, we take the integral of the terms as a constant which depends on x variable, so the calculations will be more easy. In (Van Tonningen, 1995), it was found that, unlike other series solution methods, the decomposition method is easy to program in engineering problems. Relatively few papers deal with the comparison of this method with other existing techniques. In (Bellomo and Manaco, 1985), a useful comparison between the decomposition method and the perturbation method showed the efficiency of the decomposition method compared to the tedious work required by the perturbation techniques. In (Rach, 1987; Bellomo and Sarafyan, 1987; Wazwaz, 1998a), the advantage of decomposition method over the Picard’s method and Taylor series method have been emphasized. In this article we will concentrate on the goal of obtaining an analytic solution of NDCP in the form of a series by utilising the Adomian decomposition method (Adomian, 1983; Adomian and Rach, 1985). The problem posed as a model test problem by Sincovec (Sincovec and Madsen, 1975), the equation, in u(x,t ), is
›u ›2 u ›u ¼ 2 2 c þ CðuÞ; a # x # 1; t $ 0 ð1:1Þ ›t ›x ›x where c is a constant and C(u ) is some reasonable non-linear function of u which is chosen as a potential energy. The boundary conditions and initial condition posed are uð0; tÞ ¼ f 1 ðtÞ; t $ 0
›u ð1; tÞ ¼ f 2 ðtÞ; t $ 0 ›t uðx; 0Þ ¼ gðxÞ; 0kxk1:
ð1:2Þ
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Physically, this problem can represent a simple model of fluid flow, heat flow or other phenomenon, in which an initially discontinuous profile is propagated by diffusion and convection, the latter with a speed of c (Sincovec and Madsen, 1975).
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2. Analysis of the method The decomposition method is now known as an efficient and accurate tool for deriving analytical solutions without linearization, perturbation or computerintensive methods for large classes of deterministic and stochastic linear and non-linear operator equation in different fields of applied mathematics, engineering and physics. For convenience, the Appendix briefly discusses the basic ideas of the decomposition method but is not a substitute for the several books. To apply the decomposition method, we write (taking c ¼ 1) Equation (1.1) in an operator form Lt u ¼ Cðx; tÞ 2 ux þ Lx u
ð2:1Þ
where Lt and Lx are the differential operators, Lt ¼
› ; ›t
Lx ¼
›2 : ›x 2
ð2:2Þ
It is clear that Lt is invertible and L21 t is the one-fold integration from 0 to t. Applying the inverse operator L21 t to (2.1) yields 21 21 21 L21 t Lt uðx; tÞ ¼ Lt CðuÞ 2 Lt ðux Þ þ Lt Lx u
from which it follows that 21 21 uðx; tÞ ¼ gðxÞ þ L21 ð2:3Þ t CðuÞ 2 Lt ðux Þ þ Lt Lx u: P1 P1 u ¼ n¼0 un and the non-linearity is written as CðuÞ ¼ n¼0 An where the An are the specific non-linearities in this Equation (2),
A0 ¼ Cðu0 Þ; A1 ¼ u1 Cð1Þ ðu0 Þ; A2 ¼ u2 Cð1Þ ðu0 Þ þ ð1=2Þu21 Cð2Þ ðu0 Þ; ... ......
ð2:4Þ
from which it follows that the solution is given by (2.2) with the above An, or " # 1 X uðx; tÞ ¼ gðxÞ þ L21 An ðx; tÞ t
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n¼0
" 2 L21 t
1 X
#
"
un þL21 t Lx
n¼0
1 X
# un ðx; tÞ :
ð2:5Þ
n¼0
x
The components of the decomposition series solution uðx; tÞ ¼
1 X
un ðx; tÞ
ð2:6Þ
n¼0
in (2.2) yield the term-by-term components u0 ¼ gðxÞ;
ð2:7Þ
21 21 u1 ¼ L21 t ðA0 Þ 2 Lt ðu0 Þx þ Lt Lx ðu0 Þ;
ð2:8Þ
21 21 u2 ¼ L21 t ðA1 Þ 2 Lt ðu1 Þx þ Lt Lx ðu1 Þ;
ð2:9Þ
.. . 21 21 unþ1 ¼ L21 t ðAn Þ 2 Lt ðun Þx þ Lt Lx ðun Þ for n $ 0:
ð2:10Þ
As a result, the terms of the u0 ; u1 ; u2 ; . . . are identified and the exact solution may be entirely P determined by using the approximation uðx; tÞ ¼ limn!1 fn ; where fn ¼ n21 k¼0 uk ðx; tÞ: We note here that the convergence question of this technique has been formally proved and justified by (Cherruault et al., 1992; Cherruault and Adomian, 1993; Abbaoui and Cherruault, 1995; Abbaoui and Cherruault, 1994). In conjunction with (2.8) and (2.10), all components of u(x,t ), (2.6) can be formally established. The decomposition method provides a reliable technique that requires less work if compared with traditional techniques. Adomian and Rach (1992) and Wazwaz Wazwaz (1997b) have investigated the phenomena of the self-cancelling ‘noise terms’ whose sum vanishes in the limit. An important observation was made that ‘noise terms’ appear for nonhomogeneous cases only. Further, it was formally justified that if terms in u0 are cancelled by terms in u1, even though u1 includes further terms, then the remaining non-noise terms in u0 constitute the exact solution of the equation.
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It is worthwhile to mention that the non-homogeneous equation quickly solved by observing the self-cancelling ‘noise terms’ whose sum vanishes in the limit. The decomposition method provides a reliable technique that requires less work if compared with the traditional techniques. To give a clear overview of the methodology, we have chosen to present two examples.
3. Examples Example 1. We write the general form (1.1) in the following way
›u ›2 u ›u 2 2 þ c 2 CðuÞ ¼ gðx; tÞ; a # x # 1; t $ 0 ›t ›x ›x and taking c ¼ 1; CðuÞ ¼ ð›=›tÞf ðuÞ we obtain get ut 2 uxx þ ux 2 ð›=›tÞf ðuÞ ¼ gðx; tÞ:
ð3:1Þ
where g(x,t ) is a function depends on x and t. Let g ¼ e 2t cosx þ 2e 22t sinxcosx and f ðuÞ ¼ uux : The initial conditions are uðx; 0Þ ¼ sinx; uð0; tÞ ¼ uðp; tÞ ¼ 0:
ð3:2Þ
Let Lt ¼ ›=›t and Lx ¼ ›2 =›x 2 and, write the equation as Lt u ¼ g þ Lx u 2 ux þ ð›=›tÞf ðuÞ:
ð3:3Þ
21 Operating P1 with Lt defined P as the one-fold integration from 0 to t and writing u ¼ n¼0 un and f ðuÞ ¼ 1 n¼0 An are generated for f ðuÞ ¼ uux ; we obtain the decomposition components
u0 ¼ uðx; 0Þ þ L21 t ðgðx; tÞÞ;
ð3:4Þ
21 21 ð3:5Þ unþ1 ¼ L21 t Lx un 2 Lt ðun Þx þ Lt ð›=›tÞAn ; n $ 0: P Then, since 1 n¼0 un is a (rapidly) converging series the partial sum Fn ¼ P n21 i¼0 ui is our approximate to the solution, where An are Adomian polynomials (Adomian, 1984, 1994, 1997; Adomian and Rach, 1991), given by › A0 ¼ u 0 u0 ; ›x
A1 ¼ u 0
› › u1 þ u1 u0 ; ›x ›x
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› › › A2 ¼ u0 u2 þ u1 u1 þ u2 u0 ; ›x ›x ›x ...
541 ......
ð3:6Þ
We can calculate the above terms u0 ; u1 ; . . .; un as given. However, since we calculate approximately, we can simplify the integration by approximating g(x,y ) by a few terms of its double Maclaurin series representation. Then, e 2t ¼ 1 2 t þ
t2 x2 ; sinx ¼ x; cosx ¼ 1 2 2 2
so that t2 x2 x2 2 g < 12tþ 12 þ 2ð1 2 2t þ 2t ÞðxÞ 1 2 : 2 2 2 Hence, u0 ¼ uðx; 0Þ þ L21 t ðgðx; tÞÞ ¼xþt2t
x2 t 2 x2 2 þ t 2 2 2xt þ tx 3 þ 2t 2 x 2 t 2 x 3 ; 2 2 4
ð3:7Þ
21 21 u1 ¼ L21 t Lx u0 2 Lt ðu0 Þx þ Lt ð›=›tÞA0
x x2 t2 ¼ 2t þ t 2 x 2 t 3 2 t 2 þ 2 . . .: 4 2 6
ð3:8Þ
Examining the components u0 and u1 in (3.7) and (3.8), we can easily observe that second term, third term and fourth term in (3.7) and the first term, fourth term and fifth term in (3.8) are the self-cancelling ‘noise’ terms. Consequently, we find that uðx; tÞ ¼
1 X
un ¼ u0 þ u1 þ u2 þ . . . þ un
n¼0
x x t2 t3 uðx; tÞ ¼ x 2 xt þ t 2 2 t 3 þ ··· ¼ x 1 2 t þ 2 þ ··· 2! 3! 2 6
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Hence, the exact solution of (3.1) given by uðx; tÞ ¼ e 2t sinx:
ð3:9Þ
This can be justified through substitution. Example 2. We consider homogeneous NDCP. If we take in (1.1), c ¼ 1 and CðuÞ ¼ 2u 2 þ uux þ u we get ut þ ux 2 uxx þ u 2 2 uuxx 2 u ¼ 0; 0 # x # 1; tl0:
ð3:10Þ
The initial conditions posed are uðx; 0Þ ¼ e x ; 0 # x # 1
ð3:11Þ
uð0; tÞ ¼ e t ; tl0
ð3:12Þ
Following the outline scheme (3.10) rewritten in a standard form Lt u þ ux 2 Lx u þ u 2 2 uuxx 2 u ¼ 0
ð3:13Þ
Thus, 21 21 21 21 uðx; tÞ ¼ uðx; 0Þ 2 L21 t ðux Þ þ Lt ðNuÞ 2 Lt ðMuÞ þ Lt Lx u þ Lt u ð3:14Þ P P1 where Nu ¼ u 2 ¼ 1 n¼0 Bn ; Mu ¼ uuxx ¼ n¼0 C n and the Bn, Cn are the appropriate Adomian polynomials generated for the specific non-linearity’s in this equation B0 ¼ u20 C 0 ¼ u0 ð›2 =›x 2 Þu0
B1 ¼ 2u0 u1 B2 ¼ u21 þ 2u0 u2
and
C 1 ¼ u0 ð›2 =›x 2 Þu1 þ u1 ð›2 =›x 2 Þu0 C 2 ¼ u0 ð›2 =›x 2 Þu2 þ u1 ð›2 =›x 2 Þu1 þ u2 ð›2 =›x 2 Þu0 ... ......
... ð3:15Þ
......
Thus the solution is given by (3.13) with the above Bn, Cn, or ! ! ! 1 1 1 X X X uðx; tÞ ¼ uðx; 0Þ 2 L21 un þL21 Bn 2 L21 Cn t t t n¼0
þ L21 t Lx
1 X n¼0
un
n¼0
x
! þ L21 t
1 X
n¼0
! ð3:16Þ
un
n¼0
The decomposition series solution u(x, t ) into term components
P1
n¼0 un ðx; tÞ
yield the term-by-
u0 ¼ uðx; 0Þ ¼ e x ;
ð3:17Þ
21 21 21 21 x u1 ¼ 2L21 t ðu0 Þx þ Lt ðB0 Þ 2 Lt ðC 0 Þ þ Lt Lx ðu0 Þ þ Lt ðu0 Þ ¼ e t ð3:18Þ
21 21 21 21 x u2 ¼ 2L21 t ðu1 Þx þ Lt ðB1 Þ 2 Lt ðC 1 Þ þ Lt Lx ðu1 Þ þ Lt ðu1 Þ ¼ e
t2 ð3:19Þ 2!
and similarly we find 21 21 21 21 x u3 ¼ 2L21 t ðu2 Þx þ Lt ðB2 Þ 2 Lt ðC 2 Þ þ Lt Lx ðu2 Þ þ Lt ðu2 Þ ¼ e
t3 ð3:20Þ 3!
Consequently, we find that uðx; tÞ ¼
1 X
un ¼ u0 þ u1 þ u2 þ ··· þ un ¼ e x þ e x t þ e x
n¼0
t2 t3 þ e x þ ··· 2! 3!
is the approximate solution. If we denote the approximation to n terms by Fn, we have the improving approximations. F1 ¼ u 0 ¼ e x F2 ¼ u0 þ u1 ¼ e x ð1 þ tÞ t2 x F3 ¼ u0 þ u1 þ u2 ¼ e 1 þ t þ 2! .. . t2 t3 x Fn ¼ u0 þ u1 þ u2 þ u3 ··· þ un21 ¼ e 1 þ t þ þ þ ··· : 2! 3! Hence, limFn ¼ e xþt uðx; tÞ ¼ n!1
ð3:21Þ
is the analytic solution which clearly satisfies the homogeneous NDCP (3.10). The truncated error by replacing the exact solution u(x, t ) by F3 tends to 0, since xþt x t t2 t2 x kuðx; tÞ 2 F3 k ¼ e 2 e 1 þ t þ ¼ e e 2 1 þ t þ 2! 2! which obviously tends to 0.
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Table I shows the relative errors obtained by using the approximation F7, i.e. 7 terms only. The approximation we achieved is to better the 98.5 per cent of the exact value. It is evident that an approximation within 99.9 per cent of the exact value may be achieved if 15 terms of the decomposition are used. The Adomian decomposition method avoids the difficulties and massive computational work by determining the analytic solution. We compare the approximation solution of (3.10) with the exact solution of the equation Table I. Numerical approximations shows a high degree of accuracy and in most cases Fn, the n-term approximation is accurate for quite low values of n. The solution is very rapidly convergent by utulizing the Adomian’s decomposition method (Adomian, 1986a). The numerical results we obtained justify the advantage of this methodology, even in the few terms used the approximation is accurate.
Discussion and conclusion The decomposition method (Adomian, 1983; Adomian and Rach, 1985) does not change the problem into a convenient one for use of linear theory. It therefore provides more realistic solutions. It provides series solutions which generally converge very rapidly in real physical problems. When solutions are computed numerically, the rapid convergence is obvious. The method makes unnecessary the massive computation of discretized methods for solution of partial differential equations. No linearization or perturbation is required. It provides effective procedures for analytical solution of a wide and general class of dynamical systems representing real physical problems. There are some rather significant advantages over methods which must assume linearity, “smallness”, deterministic behaviour, stationarity, restricted kinds of stochastic behaviour, uncoupled boundary conditions, etc. The method has features in common with many other methods, but it is distinctly different on close examination, and one should not be misled by apparent simplicity into superficial conclusions.
x
Table I. Absolute errors obtained for example 2
t
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5
0.316818E-06 0.458460E-06 0.155290E-05 0.697240E-06
0.495750E-06 0.455101E-06 0.276170E-05 0.259612E-05
0.111811E-05 0.548992E-06 0.599983E-05 0.480834E-05
0.204481E-05 0.541429E-06 0.104200E-05 0.658139E-06
The Adomian decomposition methodology is extremely efficient in supplying analytical solutions in linear and non-linear models in applied fields. The rise of non-linear terms is vital to progress in many applied sciences and this nonlinearity has been handled efficiently by the decomposition method. The Adomian decomposition methodology attacks non-linear problems in a like manner as linear problems, thus removing the deficiency of linearization. For more details see (Adomian, 1996, 1997, 1998; Adomian et al., 1997; Wazwaz, 1995a, 1995b, 1995c, 1996, 1998b, 1999; Datta, 1989, 1990; Adomian and Rach, 1993, 1985; Guellal et al., 1997; Deeba and Khuri, 1998; Repaci, 1990; Kaya and Inc, 199a, 199b; Inc and Bildik, 2000). References Adomian, G. (1983), Stochastic systems, Academic Press, New York. Adomian, G. (1984), “A new approach to non-linear partial differential equation”, J. Math. Anal. Appl., 102, pp. 420-34. Adomian, G. (1986a), Non-linear Stochastic Operator Equation, Academic Press, San Diego, CA. Adomian, G. (1986b), “Stochastic water reservoir modelling”, J. Math. Anal. Appl., Vol. 115, pp. 233-4. Adomian, G. (1994), Solving Frontier Problems of Physics: The decomposition method, Kluwer Academic Publishers, Boston. Adomian, G. (1997), “On KdV type equations”, Appl. Math. Comput., Vol. 88, pp. 131-5. Abbaoui, K. and Cherruault, Y. (1994), “Convergence of Adomian’s method applied to differential equations”, Comput. Math. Applic., Vol. 28, No. 5, pp. 103-9. Abbaoui, K. and Cherruault, Y. (1995), “New ideas for proving convergence of decomposition methods”, Comput. Math. Applic., Vol. 29, No. 7, pp. 103-8. Adomian, G. and Rach, R. (1985), “Polynomial non-linearities in differential equations”, J. Math. Anal. Appl., Vol. 109, pp. 90-5. Adomian, G. and Rach, R. (1988), “Coupled differential equation and coupled boundary conditionsM”, J. Math. Anal. Appl., Vol. 112, pp. 112-29. Adomian, G. and Rach, R. (1990), “Equality of partial solution in the decomposition method for linear or non-linear partial differential equations”, Comput. Math. Applic., Vol. 19, No. 12, pp. 9-12. Adomian, G. and Rach, R. (1991), “Transformation of series”, App. Math. Lett., Vol. 4, pp. 69-71. Adomian, G. and Rach, R. (1992), “Noise Terms in Decomposition Solution, Series”, Computers Math. Appl., Vol. 24, No. 11, pp. 61-4. Adomian, G., Cherruault, Y. and Abbaoui, K. (1996), “A nonperturbative analytical solution of Immune response with time-delays and possible generalization”, Math. Comput. Modelling, Vol. 24, No. 10, pp. 89-96. Bellomo, N. and Monaco, R.A. (1985), “Comparison between Adomian’s decompositions methods and perturbation techniques for non-linear random differential equations”, J. Math. Anal. Appl., Vol. 110, pp. 495-502. Bellomo, N. and Sarafyan, D. (1987), “On Adomian’s decomposition method and some comparisons with Picard’s iterative scheme”, J. Math. Anal. Appl., Vol. 123, pp. 389-400. Cherruault, Y. and Adomian, G. (1993), “Decomposition method:A new proof of convergence”, Math. Comput. Modelling, Vol. 18, No. 12, pp. 103-6.
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Cherruault, Y., Saccomandi, G. and Some, B. (1992), “New results for convergence of Adomian’s method applied to integral equations”, Math. Comput. Modelling, Vol. 16, No. 2, pp. 85-93. Laffez, P. and Abbaoui, K. (1996), “Modelling of the thermic exchanges during a drilling. Resulation with Adomian’s decomposition method”, Math. Comput. Modelling, Vol. 23, No. 10, pp. 11-4. Rach, R. (1987), “On the Adomian (decomposition) method and comparisons with Picard’s method”, J. Math. Anal. Appl., Vol. 128, pp. 480-3. Sincovec, R.F. and Madsen, N.K. (1975), “Software for non-linear partial differential equations”, ACM Trans. Math. Software, p. 1. Sanchez, F., Abbaoui, K. and Cherruault, Y. (2000), “Beyond the thin-sheet approximation Adomian’s decomposition”, Optics Com., Vol. 173, pp. 397-401. Van Tonningen, S. (1995), “Adomian’s decomposition method: a powerful technique for solving engineering equations by computer”, Computers Education J., Vol. 5, No. 4, pp. 30-4. Wazwaz, A.M. (1997a), “Equality of partial solution in the decomposition method for partial differential equations”, Intern. J. Computer Math., Vol. 65, pp. 293-308. Wazwaz, A.M. (1997b), “Necessary Conditions for the Appearance of Noise Terms in Decomposition Solution Series”, Appl. Math. Comput., Vol. 5, pp. 265-74. Wazwaz, A.M. (1998a), “A comparison between Adomian decomposition method and Taylor series method in the series solutions”, Appl. Math. Comput., Vol. 97, pp. 37-44.
Further reading Adomian, G. (1988), “A review of the decomposition method in applied Mathematics”, J. Math. Anal. Appl., Vol. 135, pp. 501-44. Adomian, G. (1997), “Optical propagation in random media”, Appl. Math. Comput., Vol. 88, pp. 127-9. Adomian, G., Rach, R.C. and Meyers, R.E. (1997), “Numerical integration analytic continuation and decompositioon”, Appl. Math. Comput., Vol. 88, pp. 95-116. Adomian, G. (1996), “The Kadomtsev-Petviashvili equation”, Appl. Math. Comput., Vol. 76, pp. 76-95. Adomian, G. (1997), “Explicit solutions of non-linear partial differential equations”, Appl. Math. Comput., Vol. 88, pp. 117-26. Wazwaz, A.M. (1995a), “On the solution of the fourth order parabolic eqsby the decomposition method”, Intern. J. Computer Math., Vol. 57, pp. 213-17. Wazwaz, A.M. (1995b), “A new approach to the nonlinear advection problem: An application of the decomposition method”, Appl. Math. Comput., Vol. 72, pp. 175-81. Wazwaz, A.M. (1995c), “The decomposition method for approximate solution of the Goursat problem”, Appl. Math. Comput., Vol. 69, pp. 299-311. Wazwaz, A.M. (1996), “A reliable technique for solving the weakly singular second-kind Volterratype Integral equations”, Appl. Math. Comput., Vol. 80, pp. 287-99. Wazwaz, A.M. (1998b), “A reliable technique for solving the wave equations an infinite onedimensional medium”, Appl. Math. Comput., Vol. 92, pp. 1-7. Wazwaz, A.M. (1999), “A reliable modification of Adomian decomposition method”, Appl. Math. Comput., Vol. 102, pp. 77-86. Datta, B.K. (1989), “A new approach of the wave equation: An approximation of the decomposition method”, J. Math. Anal. Appl., Vol. 142, pp. 6-12.
Datta, B.K. (1990), “A technique for approximate solutions to Schro¨dinger-like equations”, Comput. Math. Appl., Vol. 21, pp. 61-5. Adomian, G. and Rach, R. (1993), “Analytic solution of non-linear boundary-value problems in several dimensions by decomposition”, J. Math. Anal. Appl., Vol. 174, pp. 118-37. Adomian, G. and Rach, R. (1985), “On the solution of algebraic equation by the decomposition method”, J. Math. Anal. Appl., Vol. 105, pp. 141-66. Guellal, S., Grimalt, P. and Cherruault, Y. (1997), “Numerical study of Lorenz’s equation by the Adomian method”, Comput. Math. Applic., Vol. 33 No. 3, pp. 25-9. Deeba, E.Y. and Khuri, S.A. (1998), “The solution of a two-compartment model”, Appl. Math. Lett., Vol. 11 No. 1, pp. 1-6. Repaci, A. (1990), “Non-linear dynamical systems: On the accuracy of Adomian’s decomposition method”, Appl. Math. Lett., Vol. 3 No. 4, pp. 35-9. Kaya, D. and Inc¸, M. (1999), “On the solution of the non-linear wave equation by the decomposition method”, Bull. Malaysian Math. Soc., Vol. 22 No. 2, pp. 151-5. Kaya, D. and Inc¸c, M. (1999), “The decomposition method for approximate solution of the Cauchy problem”, Bull. Inst. Acad. Sinica, Vol. 27 No. 4, pp. 317-21. Inc, M. and Bildik, N. (2000), “Non-perturbative solution of the Ginzburg-Landau equation”, Math. & Comput. Applic., Vol. 5 No. 2, pp. 113-7.
Appendix: The decomposition method Beginning with an equation FuðtÞ ¼ gðtÞ; where F represents a general non-linear ordinary differential operator involving both linear and non-linear terms, the linear term is decomposed into L þ R; where L is easily invertible and R is the remainder of the linear operator. For convenience, L may be taken as the highest order derivative which avoids difficult integration’s which result when complicated Green’s functions are involved. Thus the eqsmay be written (Adomian, 1983, 1984; Adomian and Rach, 1985): Lu þ Ru þ Nu ¼ g where Nu represent the non-linear terms. Solving for Lu, Lu ¼ g 2 Ru 2 Nu: Because L is invertible, an equivalent expression is L 21 Lu ¼ L 21 g 2 L 21 Ru 2 L 21 Nu: If this corresponds to an initial-value problem, the integral operator L 2 1 may be regarded as definite integrals from t0 to t. If L is a second-order operator, L 2 1 is a two-fold integration operator and L 21 Lu ¼ u 2 uðt0 Þ 2 ðt 2 t 0 Þu0 ðt0 Þ: For boundary value problems (and, if desired, for initial-value problems as well), indefinite integrations are used and the constants are evaluated from the given conditions. Solving for u yields u ¼ A þ Bt þ L 21 g 2 L 21 Ru 2 L 21 Nu: P The non-linear term Nu will be equated to P 1 n¼0 An ; where the An are special polynomials to be 21 g: discussed, and u will be decomposed into 1 n¼0 un , with u0 identified as A þ Bt þ L
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1 X
un ¼ u0 þ L 21 R
n¼0
1 X
un 2 L 21
n¼0
1 X
An :
n¼0
Consequently, we can write u1 ¼ 2L 21 Ru0 2 L 21 A0 ;
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u2 ¼ 2L 21 Ru1 2 L 21 A1 ; .. . unþ1 ¼ 2L 21 Run 2 L 21 An ; n $ 0: The polynomials An are generated for each non-linearity so that A0 depends only on u0, A1 depends only on u0 and u1, A2 depends P1 on u0, u1, u2, etc. (Adomian, 1994). All of the un components are calculable, and u ¼ the n-term partial sum n¼0 un : If the series converge P P fn ¼ 1 limfn ¼ 1 i¼0 ui will be the approximate solution since n!1 i¼0 ui ¼ u by definition. It is important to emphasize that the An can be calculated for complicated non-linearities of the form f ðu; u0 ; . . .Þ or f(g(u )). The An are given by: A0 ¼ f ðu0 Þ A1 ¼ u1 f ð1Þ ðu0 Þ A2 ¼ u2 f ð1Þ ðu0 Þ þ ð1=2!Þu21 f ð2Þ ðu0 Þ A3 ¼ u3 f ð1Þ ðu0 Þ þ u1 u2 f ð2Þ ðu0 Þ þ ð1=3!Þu31 f ð3Þ ðu0 Þ A4 ¼ u4 f ð1Þ ðu0 Þ þ ½ð1=2!Þu22 þ u1 u3 f ð2Þ ðu0 Þ þ ð1=2!Þu21 u2 f ð3Þ ðu0 Þ þ ð1=4!Þu41 f ð4Þ ðu0 Þ A5 ¼ u5 f ð1Þ ðu0 Þ þ ½u2 u3 þ u1 u4 f ð2Þ ðu0 Þ þ ½ð1=2!Þu1 u22 þ ð1=2!Þu31 u3 f ð3Þ ðu0 Þ þ ð1=3!Þu31 u2 f ð4Þ ðu0 Þ þ ð1=5!Þu51 f ð5Þ ðu0 Þ and can be found from the stated formula, which has also been generalised to functions of several variables An ¼
n X
cðv; nÞf ðvÞ ðu0 Þ
v¼1
where the second index in the coefficient is the order of the polynomials and the first index progresses from 1 to n along with the order of the derivative. The An depend on components u0 ; u1 ; . . .; un : For f ðuÞ ¼ u 5 ; for example A0 ¼ u50 A1 ¼ 5u40 u1 A2 ¼ 5u40 u2 þ 10u30 u21
A3 ¼ 5u40 u3 þ 20u30 u1 u2 þ 10u20 u31 ... ...... It is to be noted that in this scheme, the sum of the subscripts in each term of the An are equal to n. The c(v,n ) are products (or sums of products) of v components of u whose subscripts sum to n, divided by factorial of the numbers of repeated subscripts. Thus, c(1,3) can only be u3, c(2,3) is 3 u1u2, and cð3; 3Þ ¼ ð1=3!ÞuP 1 : For a non-linear equations in u, one may express any given function f(u ) in the An by f ðuÞ ¼ 1 n¼0 An : Other convenient algorithms have also been developed for much more complicated and multidimensional functions, trigonometric functions, fractional P powers and for some particular functions. It is now established that the series 1 n¼0 An for Nu is equal to a generalised Taylor series about the function u0 and that the series terms approach zero as 1=ðmnÞ! if m is the order of the highest linear differential P operator. Since the series converges and does so very rapidly, the n-term partial sum fn ¼ 1 i¼0 ui , can serve as a practical solution. The limn!1 fn ¼ u: See the references included in Further Reading. The numerical results we obtained confirm the superiority of the method, related to the accuracy level, over the existing techniques of (Adomian, 1983, 1984, 1986a; Adomian and Rach, 1985). Table I clearly indicate how the decomposition methodology yields reliable results much closer to the exact solution.
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An expert system for the selection of strategic planning technique Suresh Subramoniam
Received March 2001 Revised March 2001 Accepted August 2001
College of Engineering, Trivandrum, India
K.V. Krishnankutty Member, Board of Studies, University of Kerala, Trivandrum, India Keywords Cybernetics, Strategic planning, Expert systems Abstract The expert system for the selection of strategic planning technique was developed with a view of assisting an executive in selecting a strategic planning tool based on constraints such as: time, skill, finance, computer, expert advice, form and content of input, and form and content of output required to be generated. The expert system runs on IBM PC/AT or compatible, which is the minimum system requirement, and contains approximately 115 rules in the rule base. The expert system was developed in Turbo PrologTM. The premises which help in concluding a particular strategic planning tool are stored in separate file called ‘STRATEGY.DAT’ and can be consulted at the end of the session by the user. This file saves current session’s premises which concluded the particular strategic planning tool. The system, after recommending a tool, outputs a list of planning phases where the implementation of such a strategic planning tool will succeed. It runs on a backward chaining inference mechanism which is a feature of Turbo PrologTM.
Kybernetes, Vol. 31 No. 3/4, 2002, pp. 550-560. q MCB UP Limited, 0368-492X DOI 10.1108/03684920210422629
Introduction Strategic planning had its birth in the l950’s and reached maturity stage in the l980’s. The size, scope and complexity of the modern business world stresses the need to extend the planning and budgeting horizon beyond the twelve month operating period (Webster et al., 1989). Strategic planning techniques can be broadly classified into five main schools of thought based on style or focus (Taylor, 1984). These are planning as a central control system, a framework for innovation, an organizational change process, a political activity and way of exploring future. Each technique is useful only for a particular style and not the other. The selection of particular strategic planning technique can be simplified by the use of an expert system for a given set of parameters at hand. The step by step process for strategic planning right from formulation to implementation is centered around the selection of appropriate strategic planning technique for the context in many real life situations (Wootton and Horne (1997)). This paper discusses the development of an expert system which will aid the process of selection of an appropriate strategic planning technique which in turn will integrate strategic planning into the core management process.
In any organization, strategic planning ensures proper balance among various competing sub-units and the team for strategic planning headed by the Chief Executive Officer (CEO) (Hopkins, 1991). There are internal considerations and external considerations which need thorough analysis in formulating corporate strategies for achieving a desired level of organizational performance (Hensey, 1991). The current management practice is to treat strategic planning as an ongoing management process which requires clear strategic thinking by a team, headed by CEO, and it cannot be considered as a one time operation for better organizational performance (Hensey, 1991). Even though strategic planning and a variety of tools for its implementation are already available, none of these techniques alone could meet the expected performance. Some corporations failed as a result of some or all the reasons depicted below (Webster et al., 1989): .
Lack of management commitment;
.
Absence or too little emphasis on building the database;
.
Lack of follow-up after defining strategies;
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Planning not in tune with budgeting, in strategy implementation. The formulation of strategy requires large amount of information about the system elements as shown in Figure 1 (Hamermesh, 1983). The same strategic planning technique may not be successful at various stages of the planning process such as (Webster et al., 1989): .
.
Mission statement;
.
Competitive/Environmental analysis;
Figure 1. Factors to be considered for strategy development (Hamermesh, 1983)
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.
Organizational analysis;
.
Planning assumptions;
.
Objectives and priorities;
.
Action plans;
.
Control systems.
A set of 27 strategic planning techniques and the various stages where planning technique implementation can be successful are given in Table I (Webster et al., 1989). There are nine different parameters which should be considered in the selection of the strategic planning tool for a particular
Table I. Strategic Planning Techniques and its relevance in different planning phases (Webster et al., 1989)
*
Strategic planning tool
MS
EA
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Dialectic enquiry Nominal group technique Driving force Stakeholder analysis Pims analysis Market opportunity analysis Value chain analysis Benchmarking Situational analysis (swot) Critical success factors Product life cycle analysis Product market analysis Future studies Multiple scenarios Spire Environmental forecasting and trend analysis Experience curves Competitive analysis (porter) Portfolio analysis Financial modela analysis Metagame analysis Strategic gap analysis Mckinsey 7-s framework Operating budget Management by objectives Sustainable growth model Strategic funds programming
£ £ £ £
£ £ £ £ £ £
Planning phase OA PA OP £ £ £ £ £ £
£ £ £
£ £ £ £ £ £ £ £ £
£ £ £ £ £
AP
CS
£ £ £ £ £ £ £ £ £ £ £ £ £ £ £
£
£
£ £ £ £ £ £ £ £
£ £ £ £ £ £ £
£ £ £
£ £
£ £ £
£ £ £
£
£
£
£ £ £ £
Note: ‘ £ ’ Denotes that the tool can be used for the corresponding planning phase. Legend: MS – Mission statement; EA – Environmental analysis; OA – Organizational analysis; PA – Planning assumptions; OP – Objectives; Priorities; AP – Action plans; CS – Control systems.
planning phase. These parameters can be broadly classified as expected output, required input, implementation time, and resource requirements (expert advice, finance, skills, and computer). Table II (Webster et al., 1989) gives various strategic planning tools and the corresponding parameter values based on which it is defined in the expert system. The domain characteristics, mainly the structure of the problem, clearly indicate that an expert system can be made use of for the selection of an appropriate strategic planning tool for the planning stage under consideration using the available resources in the corporation (Subramoniam, 1992).
Strategic planning technique 553
The expert system The expert system developed is user friendly and does not require programming knowledge for a query session. The constraints of the problem
#
Strategic planning tool
Parameter P1 P2 P3 P4 P5 P6 P7 P8 P9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Dialectic enquiry Nominal group technique Driving force Stakeholder analysis Pims analysis Market opportunity analysis Value chain analysis Benchmarking Situational analysis (SWOT) Critical success factors Product life cycle analysis Product market analysis Future studies Multiple scenarios Spire Environmental forecasting and Trend analysis Experience curves Competitive analysis (porter) Portfolio analysis Financial model analysis Metagame analysis Strategic gap analysis Mckinsey 7-s framework Operating budget Management by objectives Sustainable growth model Strategic funds programming
1 2 3 3 3 1 3 3 1 3 3 3 1 1 3 1 3 3 3 3 3 3 2 3 1 1 1
1 2 2 2 3 2 1 1 2 2 2 2 3 1 1 1 1 1 1 2 1 2 2 2 1 1 1
1 2 3 3 2 2 2 1 2 3 2 2 2 1 2 2 3 1 1 3 1 3 2 2 1 3 2
1 1 1 1 1 3 2 1 1 1 3 3 3 1 3 3 2 1 3 3 1 2 1 2 1 2 2
1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 2 1 3 1 4 1 1 1 1 1 1 3 3 1 1 5 4 5 6 6 1 1 1
1 1 1 2 3 4 5 6 1 1 7 10 8 4 4 4 9 10 4 11 4 12 13 14 14 15 14
1 1 1 1 2 1 1 3 3 3 3 1 4 3 5 1 6 1 7 8 1 1 3 8 9 8 1
1 1 2 3 4 5 6 7 8 1 9 10 11 11 12 13 14 2 10 15 9 16 17 18 19 20 21
Legend: P1 – Time; P4 – Skill; P7 – Content of input; P2 – Expert; P5 – Computer; P8 – Form of output; P3 – Finance; P6 – Form of input; P9 – Content of output.
Table II. Strategic Planning Techniques and the corresponding parameter values
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Figure 2. Expert System Architecture
at hand can be input to the system, as and when the system prompts for it through window menus, while running the program. The expert system was developed in Turbo PrologTM (Townsend, 1988) and contains approximately 115 rules in the rule base. The expert system runs on IBM PC/AT or compatible, which is the minimum computer configuration required. The backward chaining which is a feature of Turbo PrologTM is the inferencing mechanism for this expert system. Figure 2 shows the expert system architecture. The static database or rule base contains a complete set of rules which defines various strategic planning tools. The dynamic database is unique for each session and this stores the user input for all nine parameter values (discussed later) for the particular query session. The user option for each parameter or bound values are stored in the dynamic database as well as in a file and can be referred to at the end of the session for finding the probable reason for selecting a particular strategy by the expert system. In the case of absence of a rule which succeed in finding an appropriate strategic planning tool for the current set of parameter bindings which depicts the constraints at hand, the user will get a negative response. For any query session, the nine parameter values which completely defines a strategy in the rule base is accepted by the expert system in an interactive mode. The nine parameters are: time, expert advice, finance, skill, computer, content of the input data, form of input data, content of the output to be generated, and the form of output to be generated. The different parameters listed above can bind with one of the choices input for that parameter by the user during a query session. Figures 3 – 11 present selection menu for different parameters. The program edits the user input and accepts only allowed values for a particular parameter. All menus loop until a valid option is chosen and is implemented through a ‘repeat’ predicate. Out of the above nine parameter inputs, time and financial constraints have subjective choices in the corresponding menus. In the case of time constraint,
the choice extensive, minimum and moderate depict different scenarios. The actual time required for the application of a particular technique may vary from one organization to other. If additional benefits are anticipated by choosing a technique which requires more survey results and analysed data, then extensive time requirement can be justified but not otherwise (Webster et al., 1989). As an example, if Market Opportunity Analysis is taken, time requirement may vary from a few weeks to five years. In order to yield good results from Market Opportunity Analysis, a detailed product to product analysis which takes at least five years is the minimum time requirement. But, if the same technique is implemented after doing some empirical research which takes only a few weeks, the results may not be highly encouraging for the same corporation. So wherever time is considered as a major constraint, it is better to use a technique which requires minimum or moderate time requirements. For the financial constraint, the three choices are expensive, moderately expensive, and not expensive. The cost for implementing a technique can be studied by summing up the corresponding salaries which is a function of participant’s time and the opportunity cost assigned to competing demands (Webster et al., 1989). The major cost factors can be attributed to data collection, computer analysis, and use of strategic management consultants. It is possible find a rough cost estimate of the above factors for the business situation under study. If it is found to be expensive, one or more of the above major cost factors can be eliminated and a new strategic planning technique which will suit the budget can be employed. This analysis requires frequent consultation with the expert system in order to find a technique which suits financial constraint at hand. If the budget tolerates all the above cost factors, then a strategic planning technique which is more expensive and effective can be employed for the planning phase under study. The financial constraints can be labeled as expensive, moderately expensive and not expensive based on organization’s financial capability in meeting the above cost factors, which will result in higher proportionate profits by the implementation of the planning technique, as suggested by the expert system. Sample session The main program[1] ‘STRATEGY.PRO’ was run on an IBM PC compatible and Figure 3 to Figure 13 clearly depicts a query session in full. Since many of the parameter values are typical of the problem being studied, back tracking is allowed only in the case of two parameters, namely, input content and output content. The system follows a menu driven style and accepts the user inputs with adequate editing. Whenever a strategy is concluded by the system for a set of input parameter values, backtracking is prevented by using a cut which is a powerful Prolog feature for preventing backtracking. If a particular session fails to find a proper strategic planning tool, backtracking is done and the
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Figure 3. Expert System menu for time constraint
Figure 4. Expert System menu for expert requirement
Figure 5. Expert System menu for finance constraint
Figure 6. Expert System menu for skill constraint
Figure 7. Expert System menu for computer requirement
system accepts new inputs for parameter values for input content and output content alone. This is done before trying to fire a new rule which will suggest an alternative solution since many of the other parameters depend directly on financial constraint as discussed before. The meta rule which confirms the premises of the strategy for the sample session is given below:
Criteria (“Situational Analysis (SWOT)”):xpositive(time,1), xpositive(expert,2),
Strategic planning technique
xpositive(finance,2),
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Figure 8. Expert System menu for input data format
Figure 9. Expert System menu for input content
Figure 10. Expert System menu for output data format
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xpositive(skill,1), xpositive(computer,1), xpositive(input_form,4), xpositive(input_content,1),
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xpositive(output_form,3), xpositive(output_content,8). Once a particular strategy is confirmed, the system outputs the name of the corresponding strategic planning tool with a list of planning phases where
Figure 11. Expert System menu for output content
Figure 12. Expert System solution for the current session
success rate of such a planning tool can be very high (Figure 12). The system is also provided with the capability to store the bound values of the various parameters for the current session separately in a file called ‘STRATEGY.DAT’ (Figure 14).
Conclusions The development of an expert system for strategic planning selection is done mainly with a view of assisting an executive in choosing an appropriate strategic planning tool for the planning phase. It covers a variety of tools which are used commonly in industry. The system may not provide the best
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Figure 13. Expert System prompt for another query session/quitting
Figure 14. Various parameter bindings for the current session as saved in STRATEGY.DAT
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alternative but can induce strategic thinking, databased decisions and a structured disciplined approach in the problem-solving process. Note 1. Program listing can be obtained from the authors upon request.
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References Hamermesh, Richard G. (1983), Strategic management, Harvard Business Review Book series, John Wiley and Sons, Inc., USA. Hensey, Melleville (1991), “Essential success factors for strategic planning”, Journal of Management in Engineering, Vol. 7, No. 2, pp. 167-77. Hopkins, William E. (1991), “Engineers as strategic managers: performance assessment”, Journal of Management in Engineering, Vol. 7, No. 2, pp. 213-23. Taylor, B. (1984), “Strategic Planning-Which style do you need?”, Long Range Planning, Vol. 17, No. 3, pp. 51-62. Webster, James L., Reif, William E. and Bracker, Jeffrey S. (1989), “The manager’s guide to strategic planning tools and techniques”, Planning Review, Vol. 17, No. 6, pp. 4-15. Wootton, S. and Horne, T. (1997), Strategic Planning: The nine step programme, Kogan Page Ltd., London.
Further reading Carl Townsend. Introduction to Turbo Prolog, Sybex Inc., USA, Indian Edition (l988). Suresh Subramoniam. Expert Systems: Guidelines for managers, Industrial Management & Data Systems, Vol. 92, No. 4, 23 – 25 (l992). Bernard Taylor and John Harrison, The Manager’s case book of business strategy, Heinemann Professional Publishing Ltd., UK (l990). Richard Koch, The Financial Times guide to strategy, Pitman Publishing, UK, (l995). Raeth, P.G. (1990). Expert Systems: A software methodology for modern applications, IEEE Computer Society Press, Los Alamitas, CA.
The current issue and full text archive of this journal is available at http://www.emeraldinsight.com/0368-492X.htm
Semantics of L(MT): a language for ecological modelling J. L. Uso´-Dome`nech and P. Sastre-Vazquez Departament de Matema`tiques. Universitat Jaume I. Castello´n. Spain Keywords Cybernetics, Systems theory, Language Abstract The authors continue developing a Linguistic Theory of the Ecological Systems, but in terms of the Semantics. Each symbol (transformed function) is syntactically a lexeme, carrying an associate sememe or atomic semantic unit. Each sememe can be decomposed in semes or quantic semantic unities. They are studied and semantic systems, associated with the syntactic, that serve them as suprastructure, in terms of two levels: the quantic and the atomic. Also, it is demonstrated that for all models of the complex reality, there will always exist a most complex model from the syntactic and semantic point of view.
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561 Received March 2001 Revised March 2001 Accepted August 2001
Introduction The concept of a model has been frequently used on an intuitive and reduced scale; however, no model has all the characteristics of the reality that it represents. It is only a model concerning certain properties and relationships. A model should be called “authentic” when it represents a defined, determined system. Said defined model, is the isomorphic and homomorphic image of a system that exists in reality. On the other hand, the idea of model itself comes from common language, re-thought by the philosophical tradition. The Platonic School is somehow evoked though the double function through which reasoning is exercised: “concept” and “image” are intern and dialectically models of one another. Until now modelling has been considered based on the technical perspectives (use) of the branches of reasoning in which they are applied, but for Semantics, which is a general theory of human language and representation, presents itself as a problem of unknown reach. Language, in this way, acquires a new dimension: it makes a scheme from a graph of our action, establishes rules for our operations over beings and our exchanges with the others. According to the project, diverse references are picked up from the world and the direct categories of senses are chosen in an appropriate way. The Ecological Systems as a whole, possesses certain characteristics that clear up the original and proper form of the function of these, even though its methodology is not separated from the rest of the models. On the other hand, they must be separated from the models based on physical, mechanical causes. The existence of feed-backs expresses the conditions of adaptation, regulation The authors want to thank to the Professor Daniel Berend (Department of Mathematics and Computer Sciences of the University Ben-Gurion of the Negev, Israel) for the help developed in the development of this theory.
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or structured response from equally structured signals that have been constructed on the scientifically literature basis belonging to the theories of Biology, Ecology and other related areas. The possibility of establishing models in Ecological Systems is scarce if the elements and relationships cannot be characterised unequivocally. The structure of a set of ecological systems can never be limited totally to a model. Said models are only approximations. No matter what the model, a representation of a natural being can at best only be a homomorphous model. Modelling is homorphic mapping, that is, constructing on image (a model) of reality by abstracting a particular aspect (Higashi and Burns, 1991). The model itself (semiotic system) will always be homorphic in relation to reality or Ontological System (Villacampa and Uso´-Dome`nech, 1999). However, at the same time there can be cases of homomorphism and isomorphism even in models which describe the same part of reality. In mathematics it can be easily determined if isomorphism or homomorphism exists. On the other hand, it is not always easy to establish if two given physical systems are isomorphic or homomorphic. Up to what point is, an ecological system of the physical world a model of the other? And the same question can be asked about the relationship which exists between said physical system and a formal mathematical system. As Frey (1972) says, to answer this question, first all of the elements between which the correspondence must be established have to be characterised unequivocally in both systems as they are being presented and the existing relationships must be definable unequivocally in all its properties. A specific effort to overcome the ambiguity and facility of intuition has to be made. With the use of a mathematical expression a plane of objectivity can be reached or at least get close to it as far as possible. However, a mathematical expression is a formality. All formality is an object language (Bach, 1964). The functions of the object language are objectivity, systemisation and communication of our knowledge. It is not to deal with propositions, signs and various calculations characteristic of the object language (Mathematics), and that means that the metalanguage (natural language) dominated over them. The linguistic structuring of reality is not a perceptual structuring, but a semantic restructuring which reorganises the elements schematically on another level of meaning. The first level of meaning being perceptive. Language, or the system of concepts does not reflect reality, but creates a reality over which we can communicate. Looking at this process of reasoning of knowledge, and paraphrasing Morin (1977), we use a second class cybernetics in which we use language to know language, that is to say where resourcefulness is the norm where there is no possible linearity where there are only feedback processes. The authors have undertaken a theory of a language of Text-Model of the Ecological Systems, denominated L(MT), developing a methodology (Uso´Dome`nech et al., 1995; 1997a; 2000b; Villacampa et al., 1997), a syntax
(Villacampa and Uso´-Dome`nech, 1999; Villacampa et al., 1999a; 1999d), a textual theory and its statistical laws of text (Villacampa et al., 1999b; SastreVazquez et al., 1999; 2000) and the studies of linguistic entropy (Uso´-Dome`nech et al., 2000c). The authors in this paper deal with the principle of a semantic Theory of Models of Ecological Systems using concepts that are characteristic of semantics such as “meaning”, “seme”, “sememe”, etc. The hypothesis is as follows: (1) The dynamical models of Ecological Systems are built in a object formal language. (2) This language has a syntactic and semantic component. (3) In every dynamic model of an ecological system its behaviour is given by flow equations that describe a specific matter and energetic process of the system. The flow equations can be built from external and internal variables of the system. (4) It is possible to define for each variable, an associated field of transformed functions (lexemes) and a semantic for them for each one. The semantic field of the variable forms its semantic vocabulary. (5) The unity of meaning is the sememe. The quantum is the seme. The semantic vocabulary is formed by the set of semes. (6) The ensemble of all semantic vocabularies forms the semantic lexicon (dictionary). (7) We can approach the semantic system from two points of view: the quantum level and level of the semes and the atomic level or level of sememes. Linguistic theory of ecological systems We suppose a Suprasystem (Villacampa and Uso´-Dome`nech, 1999) formed by one Ontological System, the Observer, and the Semiotic System and we assume the next hypothesis: (1) A concept is a type of internal structure characterised by the control that the semantic content has over outputs (behaviours) of the system. A system has concepts, as far as its internal states constitute a match, some degree of co-ordination between input and output although it does not determine what concepts the system has. The concepts of a system depend on what type of information the system has been set up with. A semantic structure does not constitute a concept until there has been a match between the representative and functional notes, that is, until it has become a cognitive structure. However, no matter what the concept involved it is something determined by its semantic properties (Dretske, 1981). Therefore a concept is a holon (Koestler, 1967; Patten, 1980), that is
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a structure with two sides; one that looks backwards towards its informative origins and the other which looks forward to the effects and consequences. No structure constitutes a cognitive concept unless it has both aspects, but what gives a structure its conceptual identity is its ethiological specificity. The holon has made a model of effective input zt from actual physical input zt0 based on its state at time t : mt : XðtÞxZ t0 ! Z t : This function represents the knowledge or “noospheric” ability to abstract reality and respond to the abstraction rather than to the absolute reality (Patten, 1981). (2) The model as such is a complex cognitive structure (cognitive system), internally flexible, made up of subsystems (sentences, words) that can be replicable. Particular relations among these components fall into two groups: the more generalised relations between classes of components (syntax), leading to concepts like style and organisation (grammar), and secondly particularised relations among individual components giving a unique picture (semantics). A number of constraints are related to space, forced one dimensional chain. (Margalef, 1991). From this point of view, the model constitutes a language that follows the platonic idea of “Organum” to communicate with each other about things. According to Chomsky (1965; 1969) a language “without phonetic components” must have a syntactic component and a semantic component. Patte (1972) said: “The concepts of constraint and language are very general, and closely related at the deep level”. However, he does not explain how this idea should be developed in order to obtain useful information. To make any progress, it is necessary to show that a proper abstraction of the concept of natural language produces a suitable theoretical framework for its description. In second place, this formal language somehow needs to encode the mathematical constraints obeyed by the /under study. It is remarkable fact that the properties of language which Harris (1972) considers universal and essential, and which are relevant to a mathematical formulation are, apparently, also satisfied by mathematical (equations) sequences. (3) Mathematical modelling Ecological Systems is the process of producing semiotic systems of mathematical relations with the rules defined by the syntax of the formal language with a homomorphisme respect to conceptual semiotic system and/or Ontological Reality. The semantic systems We understand by syntax of a formal system the study of properties of system considered in if same, in its deductive possibilities. For semantics of a system is understood the study relationships between system and certain dominance of objects that system is capable of representing. The semantics is,
since, necessarily united to possibilities of interpretation of system. However, it can be supported that distinction between syntax and semantics is not rigorous; its limits are not perfectly labeled, since up until now, it has not been able to find a formal distinction criterion (Ladrie`re, 1957). For the purposes that we attempt to explain we can say, that syntax is referred to relationships between the sign and words of formal system, while semantic is referred to relationships between the said formal system expressions and objects expressed by them. Definition 1. We define a semantic field (Trier, 1931) as a part of the vocabulary joined closely, where each particular sphere is divided, classified and organised so that the elements contribute to define their surroundings. Each one of the symbols of Vocabulary Vx, can be considered as a sememe from meaning point of view. Definition 2. The seme is the meaning’s characteristic feature (Pottier, 1967). It is the primary unit or quantum of significance, not susceptible to independent fulfilment, and made into a semantic configuration or sememe. Definition 3. The sememe is defined (Pottier, 1967) as the set of semes. The lexeme’s significance contents would be its sememe. The symbol (Uso´Dome`nech et al., 1997a, b; Villacampa et al., 1999a; Villacampa and Uso´Dome`nech, 1999) will be a sememe from Semantic point of view. For example, the transformed function of third order exp(sin(x )) will be sememe exp(sin(x )), and semes x, sin, exp. Definition 4. All symbols of a syntactic vocabulary Vx, can be considered the Semantic Field of measurable symbol x. Let S m ¼ ðT m ; Rm Þ a system and x [ T m one primitive symbol (Villacampa and Uso´-Dome`nech, 1999). The symbol of zero order (f 0x) will have one semantic mean or seme only, that we denote s1x.’s A symbol of first one (f 1x) will have two semes s1x and. s2x. For example sin x has two semes s1x ¼ x; s2x ¼ sin x; etc. Let m be the cardinal set of first order symbols (f 1x). The said number m is arbitrary, that is to say, it depends on Modeller. In Table I the number of sememes and semes are specified. Set of semes
Cardinal of set of semes
z 0x ¼ {s1x } z 1x ¼ {s1x ; s2x } z 2x ¼ {s1x ; s2x ; s3x } z 3x ¼ {s1x ; s2x ; s3x ; s4x } ...... z nx ¼ {s1x ; s2x ; s3x ; . . .snx } ......
1 mþ1 m2 þ m þ 1 m3 þ m2 þ m þ 1 ...... m n þ m n21 þ . . . þ 1 ......
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Table I. Set of semes of a symbol x
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We can approach semantic system from two points of view: the quantum system or level of semes and atomic system or level of sememes.
The Q-system The Q-system has semes as elements. The associative field of semes for a primitive symbol x is (Table I): We denote as zix the set {sjx} being sjx [ z ix ði ¼ 0; . . .; n; j ¼ 1; . . .; m n þ n21 m þ . . . þ 1Þ the semes or quantic unities of semantic and n the order of symbol. Definition 5. We define the Semantic Q-Vocabulary of order one of symbol x, and we denoted as Sx the set formed by all semes of semantic field of said symbol x. cardSx ¼ :0 : We consider one subset J1x , Sx : Such as J1x ¼ {z0x ; z1x ; . . .; znx }: Whose cardinal will be a n integer number. The cardinal of J1x will be 1
cardJx ¼ ðn þ 1Þ þ n·m þ ðn 2 1Þm 2 þ ðn 2 2Þm 3 þ ··· þ m n Consequence 1. Each syntactic vocabulary of first order V1x of a primitive symbol x, has associated one semantic Q-vocabulary of first order J1x. In Linguistic Theory, an operator is a linguistic element that is used to constitute a phrasic physical structure. Said operator while universal is an “immanent datum” and empty of sense, that acquires in what is particular. It is tried in short, to a conceptual analysis. For us, ^S will be a semantic operator, being the particular semantic sense of one elementary mathematical operation or seme of addition, product, division and logic connections. It is from point of view of Modeller “. . .it is added to. . .”, “. . .it is multiplied by. . .”, etc.. We can consider ^S as an operation that it is not commutative, is not associative and it has not neuter element necessarily Definition 6. We define Semantic Product and we denote as ^, the semantic relationships between all elements of first semantic vocabulary J1x of primitive symbol x and all elements of first semantic vocabulary J1y of other primitive symbol y through an operator. ^ works as a cartesian product but it contains all semantic operators ^S. The Semantic product between two semantic sets z1x and z1y established a binary semantic relationship between all elements of z1x and z1y : j;v i;u i j;v j zix ^zjy ¼ {ðsi;u x ^S sy Þjsx [ zx ^ sy [ zy ; i; j ¼ 0; 1; . . .; n}
being ^S the semantic operator. Of the same way are defined the following semantic Q-vocabularies of order superior to 1. Definition 7. We define Semantic Q-Vocabulary of order two J2xy as formed by:
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J2xy ¼ {zux ^zvy ; zux [ J1x ; zvy [ J1y } Definition 8. by:
We define Semantic Q-Vocabulary of order three J3xyz as formed
567 J3xyz
¼
{zux ^zvy ^zwz ; zux
[
J1x ; zvy
[
J1y ; zwz
[
J1z }
Definition 9. We define Semantic Q-Vocabulary of order n Jnx1. . .xn as formed by: Jnx1 ...xn ¼ {zux1 ^. . .^zwxn ; zux1 [ J1x1 ; . . .; zwxn [ J1xn } Definition 10. We shall call Semantic Q-Lexicon the set of all semantic Q-vocabularies of any order and we shall denote as L. L ¼ {J1x1 ; . . .; J1xn ; J2x1 x2 ; . . .; J2xn21 xn ; J3x1 x2 x3 ; . . .; J3xn22 xn21 xn ; . . .; Jnx1 ...xn } Definition 11. We call Semantic primary Q-Lexicon and we denote as P the 1 1 set of all semantic Q-vocabularies of first order P ¼ {Jx1 ; . . .; Jxn } The complementary of P shall be denoted as R such as L¼P < R The complementary set R defines the semantic Q-relations between elements of Semantic primary Q-Lexicon P. Let L the semantic Q-lexicon and we shall make a partition such as L ¼ {J1x1 ; . . .; J1xn ; J2x1 x2 ; . . .; J2xn21 xn } < {J3x1 x2 x3 ; . . .; J3xn22 xn21 xn ; . . .; Jnx1 ...xn } ¼ LE < LC We shall make in LE other partition as 1
1
2
2
LE ¼ {ðJx1 ; . . .; Jxn Þ; ðJx1 x2 ; . . .; Jxn21 xn Þ} Theorem 1. The set LE is a simple system-linkage (Lloret-Climent et al., 1998) denoted by (P, r). Proof. A simple system-linkage S ¼ ðM; RÞ is the pair formed of a set object M and a set of binary relations, in such a way that R # PðMXMÞ: The set object is P ¼ {J1x1 ; . . .; J1xn }
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And the relational object is r defined by: r ¼ {Jx1 x2 ; . . .; Jxn21 xn }
r ¼ {J2x1 x2 ; . . .; J2xn21 xn }
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1
1
1
1
¼ {zux1 ^zvx2 ; . . .; zwxn21 ^zzxn ; zux1 [ Jx1 ; zvx2 [ Jx2 ; . . .; zuxn21 [ Jxn21 ; zzxn [ Jxn } ) r # PðP^PÞ
Therefore LE ¼ ðP; rÞ is a simple system-linkage being r ¼ R 2 ‘ and ‘ the relations formed by semantic Q-vocabularies of order higher one. A Definition 12. The Semantic Q-system is an ordered pair {P, ‘} of P and ‘ sets, being P or set object the Semantic primary Q-Lexicon and ‘ a set of semantic relationships such as ‘ # P½Xn21 i¼1 P^i P Being X an operation product of semantic relationships ^i between Q-vocabularies of first order and that permit existence of Q-vocabularies of order higher to one. This operation is equivalent to Generalised Cartesian Product. The relational set is not formed by binary relations as classic concept of system but n-tuplets. The concept of system is enlarged when semantic concepts has been introduced in theory. An elementary semantic Q-vocabulary L0 is formed by one quantic semantic unity, it is to say J0x ¼ {s0x } ¼ z0x Let L0 a set L0 , LE such as L0 ¼ {J0x1 ; J0x2 ; . . .; J0xn } ¼ {z0x1 ; z0x2 ; . . .; z0xn } Definition 13. We define Q-Semantic Base Model and we shall call SS ¼ ðL0 ; ‘0 Þ the system determined by elementary semantic Q-lexicon L0 of all primitive symbols and the relational set ‘0 formed by ‘0 ¼ L0 ^L0 : Consequence 2. The Base Model of Zeigler (1984), (Villacampa and Uso´Dome`nech, 1999) has associated a Q-Semantic Base Model that serve it as suprastructure. Definition 14. We define the Q-Supreme Semantic Model and we shall call SS* ¼ ðP* ; R* Þ; being P* the Semantic primary Q-Lexicon of all primitive symbols and R* the set formed for semantic Q-vocabularies of order higher one. Consequence 3. The Supreme Semiotic Model (Villacampa and Uso´Dome`nech, 1999) has associated a Q-Supreme Semantic Model that serve it as suprastructure. Theorem 2. The Q-Semantic Base Model is a subsystem of Q-Supreme Semantic Model SS * ¼ ðP* ; R* Þ: Proof. Let SS ¼ ðL0 ; ‘0 Þ be the Q-Semantic Base Model and SS * ¼ ðP* ; R* Þ the Q-Supreme Semantic Model. Like
P* ¼ {J1x1 ; J1x2 . . .; J1xn }
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¼ {{z0x1 ; z1x1 ; . . .; znx1 }; {z0x2 ; z1x2 ; . . .; znxn }; . . .; {z0xn ; z1xn ; . . .; znxn }} ¼ {z0x1 ; z0x2 ; . . .; z0xn } < {z1x1 ; . . .; znx1 ; z1x2 ; . . .; znxn ; . . .; z1xn ; . . .; znxn } ¼ L0 < {z1x1 ; . . .; znx1 ; z1x2 ; . . .; znxn ; . . .; z1xn ; . . .; znxn } ! L0 , P* being R* ¼ {J1x1 ; J1x2 ; . . .; J1xn }^{J1x1 ; J1x2 ; . . .; J1xn } and J0xi , J1xi ) r0 , R* Theorem 3. All Supreme Semiotic Model SL0 * ¼ ðL0 * ; R0 * Þ will be submodel of other Supreme Semiotic Model SL1 * ¼ ðL1 * ; R1 * Þ: Proof. The cardinal of vocabulary of n-order of a nþ1 primitive symbol x is (Villacampa and Uso´-Dome`nech, 1999) is card V x ¼ mm2121 : It is possible to nþ2 find an order n þ 1 such as the cardinal shall be card V x ¼ mm2121 and the cardinal of L0* shall be cardL0 * ¼ vð1 þ m þ m 2 þ . . . þ m n Þ being v the cardinal of primitive symbols or variables in the Base Model. Like cardL1 * ¼ vð1 þ m þ m 2 þ . . . þ m n þ m nþ1 Þ therefore cardLo * , cardL1 * : A Theorem 4. All Q-Supreme Semantic System S S0 * ¼ ðP0 * ; R0 * Þ will be subsystem of other Q-Supreme Semantic System S S1 * ¼ ðP1 * ; R1 * Þ: Proof. It is demonstrated of the same way that Theorem 3. A The A-system From the moment that is conferred an accurate meaning to lexemes of syntactic system, putting them on correspondence with the entities of functional mathematical universe, we will obtain representation of said system. The lexeme is converted therefore into sememe. That is to say f ) S: This representation operation is not something designed to add to the lexemic symbols of those which had been abstracted in its presentation. The formal semiotic system can be considered as an abstraction of its representations and of its presentations. But in such abstraction exists a dialectical unit in how much the syntactic system lacks sense without the existence of semantic associated and per se, this is an absurdity. The semantic system constitutes the superstructural unit of syntactic system, constituting an object called Semiotic System. In our particular methodology each transformed function (Uso´-Dome`nech et al., 1997a) was defined by as a symbol (Sastre-Vazquez et al., 1999; 2000; Villacampa et al., 1999b; Villacampa and Uso´-Dome`nech, 1999). From the point of view of syntax this constitutes a lexeme. Each lexeme has associated one
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sememe according to what is understood. Previously we have defined the sememe as the set of semes. We are obligated to introduce a new definition the one which would consider to sememe as a set of semes related by the logic operation of the conjunction. A sememe S i ði ¼ 0; . . .; mÞ of order i will be S i ¼ s 0 ^ s 1 ^ . . .s i ; i ¼ 0; . . .; m being m the arbitrational number that depends on the modeller. The A-system has sememes as elements. The associative semantic field of sememes for a primitive symbol x is (Table II): Definition 15. We define Semantic A-Vocabulary of order one of primitive symbol x, and we denoted as Qx the set formed by all sememes of semantic field of said symbol x. cardQx ¼ :0 : We consider one subset V 1Sx , Qx : The cardinal of A-Vocabulary of first order VSx will be the same of syntactic vocabulary of lexemes (Villacampa and nþ1 Uso´-Dome`nech, 1999), is to say cardV Sx ¼ mm2121 : For the same reason, upon equaling lexeme with sememe at symbolic level or presentation, we will be able to establish the second, third,. . ., n order A-vocabularies of the same way that was established syntactically. The operator ^ has the semantic sense defined in paragraph 3.1, being denoted also as ^S. Definition 16. We define A–vocabulary of order two V2Sxy as formed by: V 2S xy ¼ {S i ^S S j ; S i [ V 1x ; S j [ V 1y } Definition 17. We define A-vocabulary of order three V3Sxyz as formed by: V 3Sxyz ¼ {S i ^S S j ^S S k ; S i [ V 1x ; S j [ V 1y ; S k [ V 1z } Definition 18. We define A-vocabulary of order n V nx1 x2 ...xn the one formed as: V nx1 x2 ...xn ¼ {S i ^S S j ^S . . .^S S w ; S i [ V 1x1 ; S j [ V 1x2 ; . . .; S w [ V 1x2 } Definition 19. We call Semantic primary A-Lexicon and we denote as L1S the set of all semantic A-vocabularies of first order
Table II. Set of sememes of a symbol x
Set of sememes
cardinal of set of sememes
{S 0x} {S 1x} {S 2x} {S 3x} ...... {S nx} ......
1 M m2 m3 ...... mn ......
L1S ¼ {V 1x1 ; V 1x2 ; . . .; V 1xn }
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Definition 20. We shall call Semantic A-Lexicon the set of all semantic A-vocabularies of any order and we shall denote as LS. LS ¼ {V 1x1 ; . . .; V 1xn ; V 2x1 x2 ; . . .; V 2xn21 xn ; . . .; V nx1 ...xn } An elementary semantic A-vocabulary LS0 is formed by one quantic semantic unity, it is to say Definition 21. We define A-Semantic Base Model and we shall call S AS ¼ ðLS0 ; RS0 Þ the system determined by the elementary semantic A-lexicon LS0 of all primitive symbols and the relational set RS0 formed by RS0 , LS0 ^LS0 Consequence 4. The Base Model of Zeigler has associated a A-Semantic Base Model that serve it as suprastructure. Consequence 5. The Base Model of Zeigler has associated a A-Semantic Base Model that serve it as suprastructure and the suprasuprastructure is Q-Semantic Base Model. Definition 22. We define the A-Supreme Semantic Model and we shall call S AS * ¼ ðLS * ; RS * Þ; being LS* the Semantic primary A-Lexicon of all * ¼ L * ^L * primitive symbols and RS* the set formed for RS0 S S Consequence 6. The Supreme Semiotic Model has associated a A-Supreme Semantic Model that serve it as suprastructure and the suprasuprastructure is A-Supreme Semantic Model. Theorem 5. The A-Semantic Base Model is a subsystem of A-Supreme Semantic Model S AS * ¼ ðLS * ; RS * Þ: Proof. Let S AS ¼ ðLS0 ; RS0 Þ be the A-Semantic Base Model and S AS * ¼ ðLS * ; RS * Þ the A - Supreme Semantic model. LS0 , LS * ; RS0 , L0 ^L0 and RS * ¼ LS * ^LS * ) L0 ^L0 , LS * ^LS * ) RS0 , RS * A Observations (1) Give an Ontological System and a Semiotic System there will exist always an other Semiotic System more complex from syntactic point of view. (2) Given a Semiotic System and its associated Semantic System there will always exist other Semantic System more complex from semantic point of view.
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Considerations 1.- Let S1 be a Semiotic System corresponding to a given Ontological System V. The Semiotic Base System (Base Model) and Supreme Semiotic System (in terms of syntax) of V will be SByS1* respectively. Of all what is preceding will be deduced a string formed by a growing ordered chain of semiotic systems such that: T ¼ {S1 ; S2 ; . . .; SB ; S*1 ; S*2 ; . . .kS1 , S2 , . . . , SB , S*1 , S*2 , . . .} cardT ¼ :0 With this is outlined the whole. But to be outlined the whole as essential category leads us to a practice of a very different order. It removes us at the same time from the strict rationality of knowledge to interrogate us on relationship of Observer with Ontological Reality considered as a whole. In effect, “to think the whole” remains definitely as it would be of scientific analysis, and these by two essential reasons: of a side, because all “unit”, it will be of nature that it will be, only it can be understood in a widest system of which it is not more than an element, and this system requests at the same time to be studied in the same way (Go¨del, 1931). This is not more than a system that includes all subsystems and it can not be thought of as the same until the end. In addition, because scientific object and scientific activity are homogenious and this activity is always, until when it combines and reconstructs, analytical, and concerns variables that it has been isolated previously. At the same time it exists an apparent contradiction between string T whose cardinal is the Cantor’s first transfinite and such string whose superior bound is V in such a way that would be presented thus: T ¼ {S1 ; S2 ; . . .; SB ; S1* ; S*2 ; . . .V} cardT ¼ r; r [ N This is a set composed of a string and of a superior bound V. Adopting Leibniz’s nomenclature will be called a homogone set. Is this a first transfinite ordinal v?. There are other considerations, of course, which can be discussed at length. The theory operates with a certain strings (lexemes, sememes, semes, systems) that admits a transfinite when its component x travels an infinity merely potential. Theory determines a natural unit u for measuring of quality x. In experimental revelation of physical process (calculation limitation, etc.), the potential infinity of x is abridged, refused, being reduced to a quantum of form X ¼ k:u being k an abstract natural number. But k is not exactly given, an ownership condition k [ ½k1 ; k2 such that if x , k1 ·u has to be fulfilled, the string possesses quality expressed by it being not - finite and if x . k2 ·u quality expressed by its transfinite. These qualities of strings, which are rigorous in theory, appear diffuse in experimentation. There are other considerations, of course, which can be discussed at length.
(1) The immense multiplicity of lexeme;sememe, semes to indicate behaviours multiplicity of their own Reality. Logical and mathematical symbols are conventions to express abstract formulations of a universal abstraction, inherent to each content, that is to say, have a force for each content. But one must attempt to conceive them in the wake of abstraction of live and real act of the Thought, that it does not operate with an extracted content from itself, but exclusively of Reality, and that in the Thought appears in a way ideal. Further discussion may be required on this concept. (2) Observer is not more than a singular being, object between other objects of Ontological System, carrying the world in its cognitive inside and world is in me in the same operation by means of the one which I perceive me in him. (3) The production notion yields fully counts of the crux of Conscious Thought, while the worked object takes a sense for Man, as a human product. The comprehension of sense, of meaning is, precisely, the symbolic transposition of production operations in an intentional operations system, in those which subject is appropriated ideally of object reproducing in his conscience. Patten (2000) says: “All models have in common that they encode experience and always involve signs, signals, syntax, semantics and an ability to decode and derive meaning from what is encoded”.
The production idea (experience, ability) comes implicit. The model is a tool, an equipment, in which ideal production of object in the position of judgement finds his authentic basis in real production in which are begotten mathematics abstractions of formal language. Discussion (1) In a previous paper (Villacampa and Uso´-Dome`nech, 1999), the authors have defined the syntactic bases from a language to which has been denominated LðM T Þ: Generally, it is easy to confuse meaning with the interpretation or decoding of the received message. Each semantic units (semes and sememes), have associated a meaning and a decoding possibility. But if the order of sememe increases, and therefore, the number of semes, the interpretation or decoding leaves making more and more difficult. We pause here with limits imposed by the knowledge and human psychology. The language binary character (informativeexpression of the transmitter) forces us to reopen the meaning problem as a dual structure construed by the significance and the significant. (Villacampa et al., 1999b) The significance is something which a process even has before of that the concept, which defines, it
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exists. It is the semantic component of the information emitted by the process as the indication or the source. Then it is independent of the observer existence. The significant is what appears when the process concept is showed up and it is united to a certain context. It is, when the process appears as a syntagmatic set element, this being considered as cognitive structure set of the processes. It depends on the observer. It is equivalent to the interpretation of Pierce (Hoffmeyer, 1996) that it may be defined as its transformation into a new sign being itself a sign. We can distinguished between having significance in a process, as an inherent propriety to itself, and to be significant when it is related with the rest of the reality processes considered as system. So, the significance is an ontological system propriety, whereas the significant will be of the semiotic systems or meaning system. (2) The adjustment of the lexical units to experimental data (explanation) versus its interpretation, is a semantic problem of decoding that accomplish a Semantic Principle of Uncertainty (Uso´-Dome`nech et al., 1997b; 2000a). (3) Another inherent problem to any language, including the language LðM T Þ; is the duality synchrony-dyachrony. A language can be studied according to its double perspective: the synchrony and the dyachrony. We have outlined as much of the syntactic as the semantics vision from a synchronous or restricted dyachronic point of view. Our first supposition is the stability of the Ontological System, that is to say, the conservation of variables and relationships among these, during a certain period of time. This is a rough approach of reality. But in this form, any metasystem (Klir, 1985) of the text system considered, it will be always a subsystem of the other Supreme Text, as it has been observed in the Theorem 5, and this way ad infinitum. In such a way that the Modeller opposes a specific replacement function, and he will be able to move from a metasystem to another that includes it, that is to say, from a Metatext to another of which the first one is not more than a simple subtext.
References Bach, E. (1964), An Introduction to Transformational Grammars, Holt, Rinchart and Wilson, New York. Chomsky, N. (1965), Aspects of the Theory of Syntax, MIT Press, Cambridge, Massachusets. Chomsky, N. (1969), Syntactic Structures, Mouton, La Haye. Dretske, F.I. (1981), Knowledge & the flow of information, Bradford Books, Publishers. Frey, G. (1972), Die Mathematisierung Unserer Welt, W. Kohlammer Verlag, Stuttgart. Go¨del, K. (1931), “U¨ber formal unentscheidbare Sa¨tze der Principia Mathematica und verwandter Systeme”, I. Monatshefte f. Mathem. u. Physik, 38, pp. 173-98.
Harris, Z. (1972), “Mathematical structures of language”, in, Interscience tracts in pure and applied mathematics, number 21 John Wiley & Sons, London, Sydney, Toronto pp. 6-19. Hoffmeyer, J. (1996), “The global semiosphere”, in, Semiotics Around the World: Synthesis in Diversity, June Proc. Fifth Congress Int. Assoc. Semiotic studies, Berkeley, California pp. 13-8. Higashi, M. and Burns, T.P. (1991), “Enrichment of ecosystem theory”, in, Theoretical Studies of Ecosystems, Higashi, M., Burns, T.P. (Eds), Cambridge University Press, Cambridge, New York pp. 1-38. Klir, G.J. (1985), Architecture of Systems Problem Solving, Plenum Press, New York. Koestler, A. (1967), The Ghost in the Machine, Macmillan, New York. Ladrie`re, J. (1957), Les limitations internes des formalismes, E. Neuwolaerts, Louvain. Lloret-Climent, M., Villacampa-Esteve, Y. and Uso´-Dome`nech, J.L. (1998), “System Linkage: Structural Functions and Hierarchies”, Cybernetics and Systems: an International Journal, Vol. 29, pp. 35-45. Margalef, R. (1991), “Networks in Ecology”, in, Theoretical Studies of Ecosystems, Higashi, M., Burns, T.P. (Eds), Cambridge University Press, Cambridge, New York pp. 41-57. Morin, E. (1977), La Me´thode, Seuil, Paris. Patte, H.H. (1972), “Laws and constraints, symbols and languages”, in, Towards a Theoretical Biology 4. Essays, Waddington, C.H. (Eds), Edinburgh University Press, Edinburg. Patten, B. (1980), “Systems Approach to the Concept of Niche”, Synthese, Vol. 43, pp. 155-81. Patten, B. (1981), “Systems theory of the ecological niche”, Amer. Nat., Vol. 117, pp. 893-922. Patten, B. (2000), Holoecology. (Manuscript). Pottier, B. (1967), Pre´sentation de la linguistique, fondements d’une the´orie, Klincksieck, Paris. Sastre-Vazquez, P., Uso´-Dome`nech, J.L., Villacampa, Y., Mateu, J. and Salvador, P. (1999), “Statistical Linguistic Laws in Ecological Models”, Cybernetics and Systems: an International Journal, Vol. 30, No. 8, pp. 697-724. Sastre-Vazquez, P., Uso´-Dome`nech, J.L. and Mateu, J. (2000), “Adaptation of linguistics laws to ecological models”, Kybernetes (In press). Trier, J. (1931), Der Deustche Wortschatz im Sinnbezirk des Verstandes, Carl Winter, Heidelberg. Uso´-Dome`nech, J.L., Villacampa, Y., Stu¨bing, G., Karjalainen, T. and Ramo, M.P. (1995), “MARIOLA: a model for calculating the response of mediterranean bush ecosystem to climatic variations”, Ecological Modelling, Vol. 80, pp. 113-29. ´ Uso-Dome`nech, J.L., Mateu, J. and Lopez, J.A. (1997a), “Mathematical and Statistical formulation of an ecological model with applications”, Ecological Modelling., Vol. 101, pp. 27-40. Uso´-Dome`nech, J.L., Mateu, J., Villacampa, Y. and Salvador, P. (1997b), “Adjustment versus meaning? Towards a quantic theory of ecological models”, in Ecoystems and Sustainable Development I, Advances in Ecological Sciences, Uso´, J.L., Brebbia, C., Power, H. (Eds), WIT Press, Southampton-Boston vol Vol I. pp. 375-93. Uso´-Dome`nech, J.L., Villacampa, Y., Mateu, J. and Sastre-Vazquez, P. (2000a), “Uncertainty and Complementary Principles in Flow Equations of Ecological Models”, Cybernetics and Systems: an International Journal, Vol. 31, No. 2, pp. 137-60. Uso´-Dome`nech, J.L., Mateu, J. and Lopez, J.A. (2000b), “MEDEA: software development for prediction of Mediterranean forest degraded areas”, Advances in Engineering Software., Vol. 31, pp. 185-96.
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Uso´-Dome`nech, J.l., Sastre-Vazquez, P. and Mateu, J. (2000c), “Syntax and First Entropic Approximation of L(MT): A Language for Ecological Modelling”, Kybernetes (In press). Villacampa, Y. and Uso´-Dome`nech, J.L. (1999), “Mathematical Models of Ecological Systems. A Linguistic Vision”, International journal of General Systems, Vol. 28, No. 1, pp. 37-52. Villacampa, Y., Uso´, J.L., Vives, F. and Lloret, M. (1997), “A populational model of the reproductive behaviour of Mediterranean bushes: a case of Cistus albidus L”, in, Ecosystems and Sustainable Development I, Advances in Ecological Sciences, Uso´, J.L., Brebbia, C., Power, H. (Eds), WIT Press, Southampton-Boston Vol. I, pp. 395-403. Villacampa, Y., Uso´-Dome`nech, J.L., Mateu, J. and Sastre, P. (1999a), “Generative and Recognoscitive Grammars in Ecological Models”, Ecological Modelling, Vol. 117, pp. 315-32. Villacampa, Y., Corte´s, M., Vives, F., Uso´, J.L. and Castro, M.A. (1999c), “A new computational algorithm to construct mathematical models”, in, Ecosystems and Sustainable Development II, Advances in Ecological Sciences, Brebbia, C., Uso´, J.L. (Eds), WIT PRESS, Southampton-Boston Vol. II. pp. 323-30. Villacampa-Esteve, Y., Uso´-Dome`nech, J.L., Castro-Lopez-M, A. and Sastre-Vazquez, P. (1999b), “A Text Theory of Ecological Models”, Cybernetics and Systems: an International Journal, Vol. 30, No. 7, pp. 587-607. Zeigler, B.P. (1984), Multifacetted Modelling and Discrete Event Simulation, Academic Press, London.
The current issue and full text archive of this journal is available at http://www.emeraldinsight.com/0368-492X.htm
Approximation of the solution for a class of first order p.d.e. by Adomian method
Solution for a class of first order p.d.e. 577
S. Khelifa Universite´ des Sciences et de la Technologie Houari Boumedie`ne, Institut de Mathe´matiques, El Alia, Algerie Universite´ Pierre et Marie Curie, Laboratoire MEDIMAT, Paris, France
Received March 2001 Revised March 2001 Accepted August 2001
Y. Cherruault Universite´ Pierre et Marie Curie, Laboratoire MEDIMAT, Paris, France Keywords Adomian polynomials, Cybernetics, Decomposition method Abstract Aim’s to show how to approach the solution for a class of first order p.d.e. using Adomian decomposition method. Discusses the generalities of the method and a-dense curves. Outlines the new approach and provides applications of its use.
1. Introduction Let us consider the problem: 8 > < ›u 2 ›u ¼ Fðt; x; uðt; xÞÞ; ›t ›x > : uð0; xÞ ¼ u0 ðxÞ;
t.0 x [ R;
ð1Þ
In fact, we want to approximate the solution of this problem using both: Adomian decomposition method and a-dense curves. More precisely, our idea is to “densify” the set ½0; þ1½£R by an a-dense curve, and to take the “restriction” of the first order p.d.e. (occurring in (1)) on the previous a-dense curve. We hope that the new problem we will get is a differential equation with initial condition. Hence, using Adomian decomposition method we can get a very good approximation for the solution of the second problem, and consequently a good solution of (1). 2. Generalities on Adomian decomposition method, and on a-dense curves 2.1 Adomian decomposition method In this section paragraph, we shall try to present succinctly but precisely, the principles of the Adomian decomposition method. This method had been initiated by the G. Adomian in the beginning of the 80’s, in an empiric way and
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without theoretical foundations (cf Adomian, 1986; Adomian and Rach, 1985). The joint author of this paper Y. Cherruault was the first scientist to establish rigorous bases for this method, and to justify convergence results and generalizations of the technique (cf Cherruault, 1988, 1994b). Under his direction and with his collaborators in the MEDIMAT laboratory, numerous works and publications have been realized (cf Abbaoui and Cherruault, 1994, 1995; Abbaoui et al., 1995; Guellal and Cherruault, 1995; and others). In 1995 one of his numerous disciples, K. Abbaoui presented his Ph.D. thesis where he developed the mathematical foundations of this method (cf Abbaoui, 1995). The Adomian decomposition method can be used for the resolution of nonlinear (or linear) equations of different kinds: algebraic, differential, partial differential, integral, etc. . . , and with particular efficiency in the nonlinear case (Cherruault, 1998). The basic principles of the method are simple and in the same time natural (intuitively at least). The originality of this technique in comparison with other resolution methods, based generally on the linearization of the nonlinear terms, lies in the decomposition of the nonlinear operator in a polynomial series, preserving the nonlinear character. The construction of the polynomial series is made in a way permitting us to calculate the terms of the series in a recurrent way. Let H be a Hilbert space, N a non linear operator from H into H, and f an element of H. If we have to solve the following general functional equation, written in its canonical form: u 2 N ðuÞ ¼ f
ð2Þ
then we search for the unknown u, and we develop the operator N under the form of infinite series: u¼
þ1 X
un
and
n¼0
N ðuÞ ¼
þ1 X
An
n¼0
the un are functions belonging to H, and the An are polynomials depending on u0 ; u1 ; . . .; un ; . . . called: “Adomian polynomials” associated to N and obtained by the relationships: ! þ1 þ1 X X N l i ui ¼ l i Ai ðu0 ; u1 ; . . .; ui ; . . .Þ i¼0
i¼0
or " !# þ1 X dn i n!An ¼ n N l ui dl i¼0 where l is a parameter introduced by convenience.
ð3Þ l¼0
The last identity, involves that An only depends on u0 ; u1 ; . . .; un : We return now to the equation (2), and we replace u and N by there respective expressions, we obtain:
Solution for a class of first order p.d.e.
u0 þ u1 þ . . . þ unþ1 þ . . . ¼ f þ A0 ðu0 Þ þ . . . þ An ðu0 ; u1 ; . . .; un Þ þ . . . The identity is satisfied for: 8 > > > > > > > > >
> > > unþ1 ¼ An ðu0 ; u1 ; . . .; un Þ > > > > > .. : .
ð4Þ
then it is clear that we can determine the uk in a recurrent way. The exact solution of equation (2) is given by: u¼
þ1 X
un
n¼0
where the ui’s are given by (4). Remark 2.1.1. For more details on the convergence of the method and on the the truncation error or accuracy, see: Abbaoui, 1995; Cherruault, 1998; Khelifa and Cherruault, 2000.
2.2 a-dense curves The essential idea governing a-dense curves defined in Rn, consists in expressing the n variables by means of one variable. In other words, we want to “fill” Rn with a parametric curve (see Cherruault, 1998; Mora and Cherruault, 1997; Mora et al., 2000; Sagan, 1994; Ziadi and Cherruault, 1998). At the beginning, these curves have been introduced to solve local and global optimization problems. Many research workers of the MEDIMAT laboratory were interested in them and initially it was Y. Cherruault, who formulated them with A. Guillez using the Alienor global optimization method (see Cherruault, 1991; 1994a; 1999). Definition 2.2.1. Let S be a subset of Rn. We say that S is a-dense in Rn if and only if: ;P [ Rn ; ’P * [ S
such that
where k·k designs the euclidean norm in Rn.
kP 2 P * k # a
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Example 2.2.1 (1) The Archimedes spiral defined by: r ¼ a:u;
580
u$0
is pa-dense in R2. (2) The curve defined by: 8 1 > > > < a ðx 2 2kaÞ; 2ka # x # ð2k þ 1Þa; k ¼ 0; . . .; m y¼ 1 > > > : 2 a ðx 2 2kaÞ; ð2k 2 1Þa # x # 2ka; k ¼ 1; . . .; m where m [ N* and a ¼ ð2m þ 1Þ21 ; is a-dense in the square ½0; 1 £ ½0; 1: 3. A new approach for solving partial differential equations Let us return to the problem (1). It admits a unique local solution (see Courant and Hilbert, 1962) as soon as (u0, F ) belongs to C1 ðRÞ £ C1 ðR3 Þ: We are looking for u(t, x ) under the form u (t(s ), x(s )), where the subset: C ¼ {Y [ R2 ;
Y ¼ ðtðsÞ; xðsÞÞ;
s [ ½0; s0 }
has to be an a-dense curve in ½0; A £ R: Hence with the notation: zðsÞ ¼ uðtðsÞ; xðsÞÞ; we can write: 8 dz ›u dt ›u dx > > > < ds ¼ ›t ds þ ›x ds ›u ›u > > > : ›t 2 ›x ¼ Fðuðt; xÞÞ Then, by identification, it is natural to take: dt ¼ 1; ds
dx ¼ 21 ds
and
dz ¼ FðzÞ ds
xð0Þ ¼ a
and zð0Þ ¼ u0 ðaÞ
with the initial conditions: tð0Þ ¼ 0;
which corresponds to the classical resolution technique used for first order p.d.e. and called the characteristic method. So the using of Adomian
decomposition method for the calculus of the solution z(s ) of the equation: 8 > < dz ¼ FðzÞ; s . 0 ds ð5Þ > : zð0Þ ¼ u0 ðaÞ
Solution for a class of first order p.d.e. 581
will give an approximation of the solution of problem (1) by means of the truncated Adomian series. Remark 3.1. The functions t(s ), x(s ) describe straight lines because their equations are: {tðsÞ ¼ s;
xðsÞ ¼ 2s þ a}
, x ¼ 2t þ a
We can prove easily that the set: S * ¼ {ðw; yÞ [ R2 such that y ¼ 2w þ ma; m [ Z; w . 0} is a-dense in ½0; þ1½£R; with a . 0: Proposition 3.1. We assume that (u0, F ) belongs to C1 ðRÞ £ C1 ðR3 Þ: Then the unique solution of problem (1) can be obtained by solving the following differential system (6), and conversely: 88 dt > > > > ¼ 1; tð0Þ ¼ 0 > > < > ds > > > > dx > > : ds ¼ 21; xð0Þ ¼ a ð6Þ > > > > > > > dza ¼ Fðz Þ; z ð0Þ ¼ u ðaÞ; a [ R > a a 0 : ds Demonstration (i) Let u(t,x ) be the unique solution of (1), then we set: tðsÞ ¼ s;
xðsÞ ¼ 2s þ a;
za ðsÞ ¼ uðs; 2s þ aÞ:
We can easily verify the relationships: dt ¼ 1; ds
tð0Þ ¼ 0 and
dx ¼ 21; ds
xð0Þ ¼ a
dza ›u dt ›u dx ›u ›u ¼ 2 ¼ Fðuðs; 2s þ aÞÞ; þ ¼ ds ›t ds ›x ds ›t ›x then:
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dza ¼ Fðza Þ ds and za ð0Þ ¼ uð0; aÞ ¼ u0 ðaÞ: (ii) Let (t(s ),x(s ),za(s )) be the unique solution of (6); it is clear that: tðs; aÞ ¼ s;
xðs; aÞ ¼ 2s þ a
or equivalently, we can write: sðt; xÞ ¼ t;
aðt; xÞ ¼ t þ x
Denote by vðs; aÞ ¼ za ðsÞ and set uðt; xÞ ¼ vðsðt; xÞ; aðt; xÞÞ: Therefore: *Þ uð0; xÞ ¼ vð0; xÞ ¼ zx ð0Þ ¼ u0 ðxÞ **Þ
›u ›u ›v ›a ›v ›s ›v ›a ›v ›s 2 2 FðuÞ ¼ þ 2 2 2 FðvÞ ›t ›x ›a ›t ›s ›t ›a ›x ›s ›x
›u ›u ›a ›a ›v ›s ›s ›v 2 2 FðuÞ ¼ 2 þ 2 2 FðvÞ ›t ›x ›t ›x ›a ›t ›x ›s
›u ›u ›v ›v ›v 2 2 FðuÞ ¼ ð1 2 1Þ þ ð1 2 0Þ 2 FðvÞ ¼ 2 FðvÞ ›t ›x ›a ›s ›s then:
›u ›u dza 2 2 FðuÞ ¼ 2 Fðza Þ ¼ 0 ›t ›x ds Consequence 3.1. The resolution of problem (1) has been brought back to the resolution of problem (5) on straight lines t ¼ 2x þ a; a [ R: The Adomian canonical form of (5) is: Z s zðsÞ ¼ u0 ðaÞ þ FðzðwÞÞdw 0
The Adomian decomposition method can now be used for computing the solution z(s ): Z sX þ1 þ1 X vk ðsÞ ¼ u0 ðaÞ þ Ak ðv0 ; v1 ; . . .; vk Þdw k¼0
and therefore:
0 k¼0
8 v ðsÞ ¼ u0 ðaÞ > < 0 Z
Solution for a class of first order p.d.e.
s
> : vnþ1 ðsÞ ¼
An ðv0 ; v1 ; . . .; vn Þdw
0
where An are Adomian polynomials associated to F, and defined by (see Abbaoui, 1995): 8 A0 ðv0 Þ ¼ Fðv0 Þ > > < X ðv1 Þp1 ðvn Þpn jpj A ðv ; v ; . . .; v Þ ¼ . . . F ðv0 Þ n 0 1 n > > p! pn ! : jnpj¼n 1 where: F jpj ðv0 Þ ¼
dp F ðv0 Þ du p
jnpj ¼ p1 þ 2:p2 þ . . . þ n:pn
and
jpj ¼
n X
pi
i¼1
The nine first Adomian polynomials are given in appendix. Proposition 3.2. Suppose that u0 is an element of C1 (R). If F p(v0) exists and jFjpj ðv0 Þj # M , expð21Þ for every p in N, then the Adomian decomposition series of general term vk(s) converges to za(s ) unique solution of (5). Moreover if we denote wk ðsÞ ¼ v0 þ v1 ðsÞ þ . . . þ vk ðsÞ; we have: kza 2 wk k1 #
ðk þ 1Þk M kþ1 ; ðk þ 1Þ! ð1 2 M : expð1ÞÞ
;k [ N*
Demonstration. This is a consequence of theorem 6.1 and theorem 6.2 of Khelifa and Cherruault (2000). Remark 3.2. Consequently, if we want to know the approximative value of the solution u(t, x ) at a point M(xM, yM), we first determine the section of the adense curve S* containing M (it is clear that aM ¼ yM þ xM Þ; then we apply Adomian decomposition method or the repasting technique (see Benabidallah, 2000), and furnish a very good approximation of u(M ). Remark 3.3. This process offers a second possibility: we can give an approximate solution of u(t,x ) on a section of the a-dense curve S*. Remark 3.4. We can also give an approximation of the solution u(t, x ). In order to do that, it suffices to proceed as we have done in the second part of the proof of proposition 3.1: we consider a as the second variable (s is the first one),
583
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and denote vðs; aÞ ¼ za ðsÞ; knowing that: ( t¼s , x ¼ 2s þ a
(
s¼t a¼tþx
therefore we obtain: uðt; xÞ ¼ vðt; t þ xÞ 4. Applications 4.1 First application Consider the test function uðt; xÞ ¼ expðt 2 þ xÞ solution of: 8 > < ›u 2 ›u ¼ ð2t 2 1Þu; t . 0 ›t ›x > : uð0; xÞ ¼ expðxÞ; x [ R then by Adomian decomposition method we determine the solution of: 8 > < dz ¼ ð2s 2 1Þz; s . 0 ds > : zð0Þ ¼ expðaÞ; a [ R a. Consider the point M(3.5, 23.5). The equation of the straight line containing M is: x ¼ 2t (that is to say a ¼ 0). Then we determine z11 0 ðsÞ the truncated Adomian series involving 12 terms corresponding to a ¼ 0: Numerical essays have proved that the previous function converges only locally. Hence if we want to get a good approximation of u(M ) we must necessarily use the repasting technique (see Benabidallah, 2000). Indeed, z~11 0 ðsÞ furnish a fine approximation: jexp 354 2 z~11 juðM Þ 2 z~11 0 ð3:5Þj 0 ðsM Þj ¼ # 1:85 £ 1026 11 11 z~0 ðsM Þ z~0 ð3:5Þ where z~11 0 ðsÞ is the obtained function by some pasting process. b. For example, with a ¼ 2; the error function is given by: 12 ðsÞ ¼ expðs 2 2 s þ 2Þ 2 z11 2 ðsÞ where z11 2 ðsÞ is the truncated Adomian series involving 12 terms. Numerical essays proved that 12(s ) is equal to zero on [0, 2]. (See Figure 1).
Solution for a class of first order p.d.e. 585
Figure 1. Graph of the function 12(s)
In this case the variable on x-axis is s, s [ ½0; 4 and Figure 1 represents the graph of the function 12(s ). c. In accordance to the remark 3.4, we can compute z11 a ðsÞ also noted v11(a, s ), and we obtain an approximation of u(t, x ) in a neighbourhood of t ¼ 0 : u11 ðt; xÞ ¼ v11 ðx þ t; tÞ Numerical trials proved that the function:
dðt; xÞ ¼ uðt; xÞ 2 u11 ðt; xÞ is equal to zero on ½0; 2 £ ½A; B for any A and B. (see Figure 2) On x-axis the variable is t, t [ ½0; 2; on y-axis the variable is x, x [ ½ – 10; 10 and Figure 4.1c represents the graph of the function d(t, x ). 4.2 Second application Let us consider the problem: 8 ›u ›u u > < 2 ¼ þ gðt; xÞ; ›t ›x 1 þ u > : uð0; xÞ ¼ u0 ðxÞ; x [ R
t.0
In this case, the sections of the a-dense curve S* are also: tðsÞ ¼ s;
xðsÞ ¼ 2s þ a
ð7Þ
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and the problem (7) can be brought back to the problem: 8 > < dz ¼ z þ gðs; 2s þ aÞ; ds 1 þ z > : zð0Þ ¼ u0 ðaÞ:
s.0
Numerical essays had been realized with: gðt; xÞ ¼ expðt 2 xÞ and
u0 ðxÞ ¼ 1
On x-axis the variable is t, t [ ½0; 1 and on y-axis the variable is x, x [ ½21; 5: In fact Figure 3 represents the graph of the function u6(t, x ), on ½0; 1 £ ½21; 5: We recall that u6(t, x ) is the truncated Adomian series involving 7 terms which approximates the solution of (7), with the data considered above.
4.3 Third application In this part, we want to solve the previous problem (7) with different boundary conditions:
Figure 2. The graph of the function d(t, x)
8 ›u ›u u > > 2 ¼ þ gðt; xÞ; > > < ›t ›x 1 þ u uð0; xÞ ¼ u0 ðxÞ; x , 5 > > > > : uðt; 5Þ ¼ v ðtÞ; t . 0
t.0
0
587
In this case, the sections of the a-dense curve are given by: tðsÞ ¼ s;
xðsÞ ¼ 2s þ a;
tðsÞ ¼ s þ a 2 5;
s . 0 for a # 5
xðsÞ ¼ 2s þ 5;
s . 0 for a . 5
Then we solve the problem by Adomian decomposition method. This leads for a # 5; to the differential equation: 8 dw w > < ¼ þ gðs; 2s þ aÞ; ds 1 þ w > : wð0Þ ¼ u0 ðaÞ
s.0
and for a . 5; we have to solve the following differential equation by Adomian technique: 8 dc c > < þ gðs þ a 2 5; 2s þ 5Þ; ¼ ds 1 þ c > : cð5Þ ¼ v0 ða 2 5Þ
s.0
A numerical simulation had been performed with: gðt; xÞ ¼ 2t 2 x;
u0 ðxÞ ¼ 2:
x2 5
Solution for a class of first order p.d.e.
and
v0 ðtÞ ¼ 2
Figure 4.3 illustrates the results. Indeed, we can find the graph’s sketch of u9(t, x ) on the domain D , R2 defined by: D ¼ ½0; 2:5 £ ½21; 2:5 < ½0; 2:5 £ ½2:5; 5 < ½2:5; 4 £ ½2:5; 5 It is an approximation of the solution u of the equation:
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Figure 3. Graph of the function u6(t, x) on [0, 1] £[ 2 1, 5]
Figure 4. Graph of the function u9(t, x) on the domain D
8 ›u ›u u > > 2 ¼ þ 2:t 2 x; > > < ›t ›x 1 þ u uð0; xÞ ¼ ð0:08Þx 2 ; x , 5 > > > > : uðt; 5Þ ¼ 2; t . 0
Solution for a class of first order p.d.e.
t.0
589
u9(t, x ) is the truncated Adomian series involving 10 terms. As previously, on x-axis the variable is t and on y-axis the variable is x. We recall that Figure 4 represents the graph of the function u9(t, x ), on the domain D. Remark 4.1. All the numerical tests have been realized with the scientific software: Mathematica 3.0.
5. Generalization to a class of first order p.d.e Let V be an open set of the plane, and C0 a regular curve in V. Let be given the following first order p.d.e.: 8 ›u ›u > < aðx; yÞ þ bðx; yÞ ¼ Gðx; y; uðx; yÞÞ ›x ›y ð8Þ > : uðsÞ ¼ u0 ðsÞ on C 0 We get the parametric expression of C0: C 0 ¼ {ðx; yÞ [ R2 such that x ¼ x0 ðsÞ and y ¼ y0 ðsÞ; s [ ½0; 1} As in §3, we can obtain the sections of an a-dense curve by solving: 8 dxs > > > < dt ¼ aðxs ðtÞ; ys ðtÞÞ; xs ð0Þ ¼ x0 ðsÞ > dys > > : dt ¼ bðxs ðtÞ; ys ðtÞÞ; ys ð0Þ ¼ y0 ðsÞ This leads to the differential equation: 8 > < dzs ¼ Gðxs ðtÞ; ys ðtÞ; zs ðtÞÞ dt > : zs ð0Þ ¼ u0 ðx0 ðsÞ; y0 ðsÞÞ easily solved by Adomian decomposition method.
ð9Þ
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Remark 5.1. Under the hypotheses: – a, b and G are continuously differentiable – jaj þ jbj – 0 – aðx0 ðsÞ; y0 ðsÞÞy00 ðsÞ 2 bðx0 ðsÞ; y0 ðsÞÞ:x00 ðsÞ – 0; ;s [ ½0; 1 the problem (8), admits a unique local solution (see Courant and Hilbert, 1962). Proposition 5.1. Under hypotheses of remark 5.1, the unique solution of problem (8) can be obtained from the unique solution of: 88 dxs > > > > ¼ aðxs ðtÞ; ys ðtÞÞ; xs ð0Þ ¼ x0 ðsÞ > > < > dt > > > > dys > > : dt ¼ bðxs ðtÞ; ys ðtÞÞ; ys ð0Þ ¼ y0 ðsÞ ð10Þ > > > > > > > dzs ¼ Gðx ðtÞ; y ðtÞ; z ðtÞÞ; z ð0Þ ¼ u ðx ðsÞ; y ðsÞÞ > s s s s 0 0 0 : dt and conversely. Proof (i) Let u(x, y ) be the solution of (8), then we set: zs ðtÞ ¼ uðxs ðtÞ; ys ðtÞÞ where the couple (xs(t ), ys(t )) is solution of: 8 dxs > > > < dt ¼ aðxs ðtÞ; ys ðtÞÞ; xs ð0Þ ¼ x0 ðsÞ > dys > > : dt ¼ bðxs ðtÞ; ys ðtÞÞ; ys ð0Þ ¼ y0 ðsÞ therefore: dzs ›u dxs ›u dys ›u ›u ¼ þ ¼ aðxs ; ys Þ þ bðxs ; ys Þ dt ›x dt ›y dt ›t ›x ¼ Gðxs ; ys ; uðxs ; ys ÞÞ Then: dzs ¼ Gðxs ðtÞ; ys ðtÞ; zs ðtÞÞÞ dt Moreover:
zs ð0Þ ¼ uðxs ð0Þ; ys ð0ÞÞ ¼ uðx0 ðsÞ; y0 ðsÞÞ ¼ u0 ðx0 ðsÞ; y0 ðsÞÞ: (ii) Let (xs(t ), ys(t ), zs(t )) be the solution of (10), and denote vðt; sÞ ¼ zs ðtÞ knowing that: aðx0 ðsÞ; y0 ðsÞÞy00 ðsÞ 2 bðx0 ðsÞ; y0 ðsÞÞsx00 ðsÞ – 0; We can write (locally): ( x ¼ xðt; sÞ
( ,
y ¼ yðt; sÞ
;s [ ½0; 1
t ¼ tðx; yÞ s ¼ sðx; yÞ
then we set: uðx; yÞ ¼ vðtðx; yÞ; sðx; yÞÞ: Thus: (*) – For ðx; yÞ [ C 0 ; there exists s [ ½0; 1 such that: x ¼ x0 ðsÞ
and
y ¼ y0 ðsÞ
so: uðx; yÞ ¼ uðx0 ðsÞ; y0 ðsÞÞ ¼ uðxs ð0Þ; ys ð0ÞÞ ¼ vð0; sÞ ¼ zs ð0Þ or: uðx0 ðsÞ; y0 ðsÞÞ ¼ u0 ðx0 ðsÞ; y0 ðsÞÞ (*) – In addition we can evaluate the expression: Aðx; yÞ ¼ aðx; yÞ
›u ›u þ bðx; yÞ 2 Gðx; y; uðx; yÞÞ ›x ›y
›v ›s ›v ›t ›v ›s ›v ›t Aðx; yÞ ¼ aðx; yÞ þ þ bðx; yÞ þ ›s ›x ›t ›x ›s ›y ›t ›y 2 Gðx; y; vÞ
›s ›s ›v Aðx; yÞ ¼ aðx; yÞ þ bðx; yÞ ›x ›y ›s
›t ›t ›v þ aðx; yÞ þ bðx; yÞ 2 Gðx; y; vÞ ›x ›y ›t but by definition: aðx; yÞ ¼ aðxðt; sÞ; yðt; sÞÞ ¼
›x ðt; sÞ ›t
Solution for a class of first order p.d.e. 591
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bðx; yÞ ¼ bðxðt; sÞ; yðt; sÞÞ ¼
›y ðt; sÞ ›t
hence:
592
Aðx; yÞ ¼
›x ›s ›y ›s ›v ›x ›t ›y ›t ›v þ þ þ ›t ›x ›t ›y ›s ›t ›x ›t ›y ›t
2 Gðx; y; uÞ
Aðx; yÞ ¼
›s ›x ›s ›y ›v ›t ›x ›t ›y ›v þ þ þ ›x ›t ›y ›t ›s ›x ›t ›y ›t ›t
2 Gðx; y; vÞ We know that:
›s ›x ›s ›y ›s þ ¼ ¼ 0; ›x ›t ›y ›t ›t
›t ›x ›t ›y ›t þ ¼ ¼1 ›x ›t ›y ›t ›t
because s and t are independent variables. At last:
›v dzs 2 Gðx; y; vÞ ¼ 2 Gðxs ðtÞ; ys ðtÞ; zs ðtÞÞÞ ¼ 0 ›t dt
Aðx; yÞ ¼
aðx; yÞ
›u ›u þ bðx; yÞ ¼ Gðx; y; uðx; yÞÞ ›x ›y
A Definition 5.1. The integral curves defined by (xs(t ), ys(t ), zs(t )), solutions of (10), are called characteristic cures of the p.d.e. occurring in (8). Proposition 5.2. Under hypotheses of remark 5.1, and: if G jpj(u0(s )) exists and jG jpj ðu0 ðsÞÞj # M , expð21Þ for all p in N, then the Adomian decomposition series associated to the problem (9) converges to zs(t ) unique solution of (9). Moreover: kzs 2 wk k1 #
ðk þ 1Þk M kþ1 ; ;k [ N* ðk þ 1Þ! ð1 2 M : expð1ÞÞ
where wk is the truncated Adomian series of ðk þ 1Þ terms obtained by using Adomian decomposition method exactly like in consequence 3.1. Demonstration. It is a consequence of theorem 6.1 and theorem 6.2 of Khelifa and Cherruault (2000). Remark 5.2. We can generalize without any difficulty, the previous results to the case of a linear first order p.d.e. with variable coefficients in Rn:
8 n X ›u > > ai ðx1 ; . . .; xn Þ ¼ Gðx1 ; . . .; xn ; uðx1 ; . . .; xn ÞÞ < › x i i¼1 > > : uðsÞ ¼ u0 ðsÞ on G * following exactly the same processes used above. 6. Conclusion In conclusion, we can say that the Adomian decomposition method can be perfectly adapted to the approximation of the solution of a class of first order partial differential equations (specially in the nonlinear case). We saw also in this paper that we can choose the desired approximation: we can find the approximated value of the solution in a given point of the domain; we can compute the approximated solution on a a-dense curve; and we can also determine an approximation of the solution u(t, x ) on all the domain (under the condition that (a, s ) can be expressed explicitly in function of (t, x )). Much remains to be done: However, for example how to solve the other first order partial differential equations using Adomian decomposition method, and more generally the nonlinear partial differential equations with initial and boundary conditions which are not easily expressed in the canonical form. References Abbaoui, K. (1995), Les fondements mathe´matiques de la me´thode de´compositionnelle d’Adomian et application a` la re´solution de proble`mes issus de la biologie et de la me´decine, The`se de doctorat de l’Universite´ de Paris VI. Adomian, G. (1986), Nonlinear stochastic operator equations, Academic Press, New York, NY. Abbaoui, K. and Cherruault, Y. (1994), “Convergence of Adomian method applied to differential equations”, Mathematical and Computer Modelling, Vol. 28, No. 5, pp. 103-9. Abbaoui, K. and Cherruault, Y. (1995), “New ideas for proving convergence of decomposition method”, Computers Math. Applic., Vol. 29, No. 7, pp. 103-8. Adomian, G. and Rach, R. (1985), “On the solution of algebraic equation by decomposition method”, Mathematical Analysis and Applications, Vol. 105, No. 1, pp. 141-66. Abbaoui, K., Cherruault, Y. and Seng, V. (1995), “Practical formulae for the calculus of multivariable Adomian polynomials”, Mathematical and Computer Modelling, Vol. 22, No. 1, pp. 89-93. Benabidallah, M. (2000), Application de la me´thode d’Adomian pour l’approximation de la solution globale pour une classe d’e´quations diffe´rentielles, The`se de Magister de l’U.S.H.B. Cherruault, Y. (1988), “Convergence of decomposition method”, Kybernetes, Vol. 18, No. 2, pp. 31-8. Cherruault, Y. (1991), “New deterministic methods for global optimization and application to biomedicine”, International Journal of Biomedical Computing, Vol. 27, pp. 215-29. Cherruault, Y. (1994a), “Global optimization in biologie and medicine”, Mathematical Computing Modelling, Vol. 20, No. 6, pp. 119-32.
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Cherruault, Y. (1994b), “Convergence of decomposition method and application to biological systems”, International Journal of Bio-medical Computing, Vol. 36, pp. 193-7. Cherruault, Y. (1998), Mode`les et me´thodes mathe´matiques pour les sciences du vivant, Presse Universitaire de France, Paris. Cherruault, Y. (1999), Optimisation: me´thodes locales et globales, Presse Universitaire de France, Paris. Courant, R. and Hilbert, D. (1962), Methods of mathematical physics, Interscience John Wiley & Sons Vol. II. Guellal, S. and Cherruault, Y. (1995), “Application of decomposition method to identify the distributed parameters of an elliptical equation”, Mathematical and Computer Modelling, Vol. 21, No. 4, pp. 51-5. Khelifa, S. and Cherruault, Y. (2000), “New results for the Adomian method”, Kybernetes, Vol. 29, No. 3, pp. 332-54. Mora, G. and Cherruault, Y. (1997), “Characterization and generation of a-dense curves”, Computers Math. Applic., Vol. 33, pp. 83-91. Mora, G., Cherruault, Y. and Ziadi, A. (2000), “Functional equations generating space-densifying curves”, Computers Math. Applic., Vol. 39, pp. 45-55. Sagan, H. (1994), Space filling curves, Springer-Verlag. Ziadi, A. and Cherruault, Y. (1998), “Generation of a-dense curves in a cube of Rn”, Kybernetes, Vol. 27, No. 4, pp. 1-10.
Appendix The nine first Adomian polynomials are obtained as follows: A0 ðv0 Þ ¼ Fðv0 Þ A1 ðv0 ; v1 Þ ¼ v1 F ð1Þ ðv0 Þ A2 ðv0 ; v1 ; v2 Þ ¼ v2 F ð1Þ ðv0 Þ þ 12 ðv1 Þ2 F ð2Þ ðv0 Þ A3 ðv0 ; . . .; v3 Þ ¼ v3 F ð1Þ ðv0 Þ þ v1 v2 F ð2Þ ðv0 Þ þ 16 ðv1 Þ3 F ð3Þ ðv0 Þ A4 ðv0 ; . . .; v4 Þ ¼ v4 F ð1Þ ðv0 Þ þ v1 v3 þ 12 ðv2 Þ2 F ð2Þ ðv0 Þ þ 12 ðv1 Þ2 v2 F ð3Þ ðv0 Þ þ
4 ð4Þ 1 24 ðv1 Þ F ðv0 Þ
A5 ðv0 ; . . .; v5 Þ ¼ v5 F ð1Þ ðv0 Þ þ ðv1 v4 þ v2 v3 ÞF ð2Þ ðv0 Þ þ 12 ððv1 Þ2 v3 þ v1 ðv2 Þ2 ÞF ð3Þ ðv0 Þ 1 þ 16 ðv1 Þ3 v2 F ð4Þ ðv0 Þ þ 120 ðv1 Þ5 F ð5Þ ðv0 Þ
A6 ðv0 ; . . .; v6 Þ ¼ v6 F ð1Þ ðv0 Þ þ v1 v5 þ v2 v4 þ 12 ðv3 Þ2 F ð2Þ ðv0 Þ þ 12 ðv1 Þ2 v4 F ð3Þ ðv0 Þ þ v1 v2 v3 þ 16 ðv2 Þ3 F ð3Þ ðv0 Þ þ 16 ðv1 Þ3 v3 þ 14 ðv1 Þ2 ðv2 Þ2 F ð4Þ ðv0 Þ þ
4 1 ð5Þ 24 ðv1 Þ v2 F ðv0 Þ
þ
6 ð6Þ 1 720 ðv1 Þ F ðv0 Þ
A7 ðv0 ; . . .; v7 Þ ¼ v7 F ð1Þ ðv0 Þ þ ðv1 v6 þ v2 v5 þ v3 v4 ÞF ð2Þ ðv0 Þ þ 12 ðv1 Þ2 v5 F ð3Þ ðv0 Þ þ v1 v2 v4 þ 12 v1 ðv3 Þ2 þ 12 ðv2 Þ2 v3 F ð3Þ ðv0 Þ þ 16 ðv1 Þ3 v4 F ð4Þ ðv0 Þ þ ð12 ðv1 Þ2 v2 v3 þ 16 v1 ðv2 Þ3 F ð4Þ ðv0 Þ þ 241 ðv1 Þ4 v3 F ð5Þ ðv0 Þ 1 1 þ 121 ðv1 Þ3 ðv2 Þ2 F ð5Þ ðv0 Þ þ 120 ðv1 Þ5 v2 F ð6Þ ðv0 Þ þ 5040 ðv1 Þ7 F ð7Þ ðv0 Þ
A8 ðv0 ; . . .; v8 Þ ¼ v8 F ð1Þ ðv0 Þ þ v1 v7 þ v2 v6 þ v3 v5 þ 12 ðv4 Þ2 F ð2Þ ðv0 Þ þ 12 ðv1 Þ2 v6 þ v1 v2 v5 þ v1 v3 v4 þ 12 ðv2 Þ2 v4 þ 12 v2 ðv3 Þ2 F ð3Þ ðv0 Þ þ ð 16 ðv1 Þ3 v5 þ 12 ðv1 Þ2 v2 v4 þ 14 ðv1 Þ2 ðv3 Þ2 þ 12 v1 ðv2 Þ2 v3 ÞF ð4Þ ðv0 Þ þ 241 ðv2 Þ4 F ð4Þ ðv0 Þ þ 241 ðv1 Þ4 v4 þ 16 ðv1 Þ3 v2 v3 F ð5Þ ðv0 Þ þ 121 ðv1 Þ2 ðv2 Þ3 F ð5Þ ðv0 Þ 1 1 1 þ 120 ðv1 Þ5 v3 þ 481 ðv1 Þ4 ðv2 Þ2 F ð6Þ ðv0 Þ þ 720 ðv1 Þ6 v2 F ð7Þ ðv0 Þ þ 40320 ðv1 Þ8 F ð8Þ ðv0 Þ
Solution for a class of first order p.d.e. 595
The current issue and full text archive of this journal is available at http://www.emeraldinsight.com/0368-492X.htm
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A cybernetic approach to the multiscale minimization of energy function: Grey level image segmentation Pilar Arques, Patricia Compan˜, Rafael Molina, Mar Pujol and Ramo´n Rizo Grupo de Visio´n, Gra´ficos e Inteligencia Artificial (VGIA), Departamento de Ciencia de la Computacio´n e Inteligencia Artificial, Universidad de Alicante, Apdo. Alicante, Spain Keywords Cybernetics, Computers, Image processing Abstract Segmentation is an important topic in computer vision and image processing. In this paper, we sketch a scheme for a multiscale segmentation algorithm and prove its validity on some real images. We propose an approach to the model based on MRF (Markov Random Field) as a systematic way for integrating constraints for robust image segmentation. To do that, robust features and their integration in the energy function, which directs the process, have been defined. In this approach, the image is first transformed to different scales to determine which one fits better to our purposes. Then, it is segmented into a set of disjoint regions, the adjacent graph (AG) is determined and a MRF model is defined on the corresponding AG. Robust features are incorporated to the energy function by means of clique functions and optimal segmentation is then achieved by finding a labeling configuration that minimizes the energy function using Simulated Annealing.
1. Introduction The idea of analysing signals at several resolutions has been paid much attention in the field of computer vision. Multiscale is a way of efficiently and effectively representing data with the aim of reducing computational complexity. Multiscale can be thought of as a data structure that produces a successive condensed representation of the information in a given image. The most obvious advantage of this type of representation is that it provides a way of reducing the computational cost of several image operations (Li, 1995). Image segmentation is a low-level vision task, which is a fundamental research area. Segmentation has been extensively treated in the image processing and computer vision literature as a pre-processing step in many high-level vision tasks like image interpretation or object recognition. Although there is a wide variety of image segmentation techniques that are well treated in (Haralick and Shapiro, 1993), it is difficult to satisfy all the properties for the Kybernetes, Vol. 31 No. 3/4, 2002, pp. 596-608. q MCB UP Limited, 0368-492X DOI 10.1108/03684920210422656
This work has been supported by the Spanish Comisio´n Interministerial de Ciencia y Tecnologı´a (CICYT), project number TAP98-600.
optimal set of segmented regions. The resulting segmented images generally depend on the predetermined threshold values. Since the MRF model has a powerful capacity to integrate some visual information, it has been applied to solve the problems that have been exposed previously. This model has been used to some vision problems such as image restoration and segmentation (Arques et al., 1999a, 1999b; Geman and Geman, 1984; Geman, 1998; Li, 1995; Modestino and Zhang, 1994). But due to the great number of pixels on which the MRF is to be defined, it usually takes a lot of computational time to obtain optimal labels, which makes difficult to apply the MRF model to real scene domains. In such cases, multiscale can be applied to reduce the computational complexity. Segmentation schemes based on the concept of multiscale have been proposed in literature, and more recently a segmentation approach under multiscale using Markov random field has been proposed (Li, 1995). In this paper, we are interested in developing multiscale Markov Random Field models. Before applying our MRF image segmentation scheme, the optimal scale must be determined (Section 2). Then the original image is scaled to the optimal size, a preliminary segmentation using a MPMT algorithm is made, and finally an MRF model on the segmented regions is defined (Section 3), so that the number of sites to be considered is significantly reduced, thus reducing the computational time required. To find the optimal segmentation result the Simulated Annealing algorithm is used (Section 4). Finally, in (Section 5) the results obtained by experimentation are shown. 2. Optimal scale determination Multiscale techniques are extremely useful for reducing computational time in high cost algorithms. That is the case of MRF image segmentation schemes. Very often, the aim of a segmentation system is obtaining the regions that make up the image as quickly as possible rather than a perfect definition of these regions. Therefore, using a scale scheme may be a good way of reducing the number of pixels in the image. This is the case, for instance, of a visionguided robot that must answer in real time to external stimuli. Nevertheless, scaling can only be applied up to a critical point. At this point, the segmented image obtained becomes too corrupted to be useful. The first step in our system is determining the critical point, that is, the optimal scale. In order to obtain the critical point, an empirical method has been used. Several images have been scaled from 2 up to 8 times. Each one has been segmented using our segmentation scheme. In the first steps of the investigation, the researcher determined the optimal segmentation among all the resulting images. In order to mechanise the process, a measure of the difference between the result and the optimal segmentation (drawn by the expert) has been implemented. This distance can be defined as follows: Let r(x,y ) be the distance between two pixels x and y of an image X, then the
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shortest distance between a pixel x [ X and a region A # X is dðx; AÞ ¼ inf{rðx; aÞ : a [ A}: The Hausdorff distance between two region A and B # X can be defined as the maximal distance from a pixel of region A to the closest pixel of B, that is: H ðA; BÞ ¼ max{sup dðx; BÞ; sup dðx; AÞ} x[A
ð1Þ
x[B
Nevertheless, Hausdorff distance is very sensible to noise, so a new distance is defined (Baddeley, 1992), using the p-order mean instead of sup: 2 3p1 X 1 ð2Þ Dp ðA; BÞ ¼ 4 jdðx; AÞ 2 dðx; BÞjp 5 N x[X where N ¼ total number of pixels in X. Transforms can be used, so the expression obtained is: 2 31p X 1 p ð3Þ jwðdðx; AÞÞ 2 wðdðx; BÞÞj 5 Dp ðA; BÞ ¼ 4 N x[X where w(t ) can be, for instance, min{t,c} for a fixed c . 0: The distance Dp(A,B ) measures how similar images A and B are. In Section 5, result is shown and the critical point is determined. It is also shown some relations that can be considered between the features of the image and the optimal scale. 3. MRF definition From the first segmentation made with the MPMT algorithm (Arques et al., 1999a, 1999b), in which it is assumed that there exists a over-segmentation of the image, the Adjacent Graph is constructed and then MRF is defined (Geman and Geman, 1984; Modestino and Zhang, 1994). 3.1 Markov random fields The concept of an MRF is a direct extension of the concept of a Markov process, which gives a tool to analyze spatial or contextual dependencies between physical phenomena. Wong (1968) introduced the definition for a twodimensional continuos MRF. The definition of a discrete MRF is more useful for our purposes, a generalization of the concept of a Markov chain. In formal terms we have the following: Let S be a finite set of N sites, and G ¼ {Gs ; s [ S} be a neighborhood system for S, i.e. a collection of subsets of S for which s Gs ; ;s [ S and s [ Gr ; iff r [ Gs ; ;r; s [ S:
Let F ¼ {F s ; s [ S} be any family of random variables indexed by s [ S; and suppose, for simplicity, that these variables take values on some finite sets {Qs}. We will call any possible sample realization f : ðf S1 ; . . .; f SN Þ; f Si [ QSi ; a configuration of the field. Let V be the set of all possible configurations and let P be a probability measure in V. Definition 1. F is a MRF with respect to G if: (1) PðF ¼ f Þ . 0;f [ VððF ¼ f Þ denotes the event ðF s ¼ f s ; ;s [ SÞ). (2) PðF s ¼ f s =F r ¼ f r r – sÞ ¼ PðF s ¼ f s =F r ¼ f r r [ Gs Þ for every s [ S: Definition 2. Given a system of neighborhoods on a lattice, we define a clique C as either a single site, or a set of sites of the lattice, such that all the sites that belong to C are neighbours of each other. There is no obvious intuitive relation between the form of the conditional probability distributions and the qualitative behaviour of the sample fields. To overcome these difficulties, we need an alternative way of defining an MRF. Theorem 1. (Hammersley-Clifford). If F is an MRF on a lattice S with respect to the neighborhood system G, the probability distribution of the configurations generated by it will always have a definite form, which is a Gibbs distribution: Pf ð f Þ ¼
1 1 exp 2 U ð f Þ Z T
ð4Þ
where U( f )¼energy function, Z ¼ normalizing constant, T is the temperature. X ð5Þ V cð f Þ Uð f Þ ¼ c
where C ranges over the cliques associated with the given neighborhood system.
3.2 Energy function and robust estimators In our segmentation model (Arques et al., 1999a, 1999b), we have defined the energy function like: X U ðCE=R; FÞ ¼ ð6Þ V c ðCE=R; FÞ c
where CE is the actual label configuration, R is the region process (measure of the region’s homogeneity) and F is the boundary process (measure of the discontinuity in the limits of adjacent regions).
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V c ðCE=R; FÞ ¼ Vecij ·maxðjmi 2 mj j; js2i 2 s2j jÞ ð7Þ þ a·ð1 2 Vecij Þ·maxðjmi 2
600
mj j; js2i
2
s2j jÞ·ð1=F ij Þ
where a is a control parameter and ( 1 if i; j have the same region label Vecij 0 otherwise F ij ¼
1 X ap nij p[Aij
ð8Þ ð9Þ
nij is the number of pixels in the common boundary between i and j regions, Aij is the set of edges in the common boundary between i and j regions, ap is the number of pixels in the edge p, mi is the average of the intensity of the grey label of the region i and s2i is the variance of the region i. The fact of using the same features for small regions and large ones forces a treatment that is not adequate for large regions. So, we propose to dynamically differentiate the treatment of large and small regions. For large regions (+7 per cent of the image) robust estimators are defined. Robust regression tries to revise estimators that are not so strongly affected by outliers (Roufseeuw and Leroy, 1987). A robust analysis first wants to fit a regression to the majority of the data and then to discover the outliers as those points which possess large residuals from the robust solution. The breakdown process deletes the outliers, removing a percentage of data points in the sample. This percentage is called breakdown point and it must be as big as needed to delete all the outliers. TR ¼ n* breakdown point
ð10Þ
TMR ¼ n 2 2* TR
ð11Þ
where n is the size of the sample, TR is half the size of deleted data and TMR the size of the breakdown sample. Robust average is obtained deleting outliers data points. Robust variance is calculated using in the robust average and the winsorized variance ðS 2w Þ: Winsorizing is a process consisting of substituting the outliers data with the robust mean of the sample, in order to maintain the amount of elements. Robust variance is defined as: S 2T ¼
ðn 2 1ÞS 2w nT 2 1
where nT is the trimmed sample size.
ð12Þ
3.3 Subsampling on a Markov random field The problem we focus on is whether a random field can be subsampled from a Markov random field. We only consider a periodic subsampling scheme. We also do not consider the case where a filtering operation is applied to a Markov random field before subsampling. The filtered case can be handled by applying the results to the filtered random field instead of the original Markov random field. The theorems on the subsampling problem on a Markov random field presented below are the basis of our development (Jeng, 1992). The first one states that if a subsampling scheme satisfies certain conditions, then the subsampling operation does not destroy the Markovian property. The second one states that there exists at least one subsampling scheme that preserves the Markovian property with an increased neighborhood support. Theorem 2. Let S be a MRF. Denote by Sd, Sp the sites that will be dropped and kept by a subsampling scheme H. If one can partition Sd into nonoverlapped regions Di with a finite support in such a way that any two sites in different regions Di are not neighbors of each other, the subsampled random field S0 is a MRF. Otherwise it is not. Lemma. Let S be a MRF with a neighborhood support G. There is a partition L1 ; L2 ; . . .; L r of the lattice L such that L i > L j ¼ B and .) In view of the difficulty of dealing with these topics, a Resource Center for Cyberculture Studies (RCCS) is very welcome, at: . A particularly welcome feature is its undertaking to review relevant books and to make the reviews accessible on its website. The Director of the Center is David Silver and its physical location is the University of Washington in Seattle. An early announcement was made on the cyberspaceand-society list in June 2001 (at which time the Center was located in the University of Maryland) by Jeremy Hunsinger (see below) and at that time the number of books that had been reviewed was over 85 and were said to cover ‘‘a range of topics, from online culture, communities, and identities to hypertext, digital literacy, and online pedagogy to Internet policy, the digital divide, and online privacy.’’ A visit to the RCCS site in October 2001 revealed a wealth of sources of information, including, under the heading: ‘‘Book of the Month’’ a list of one, two or three books for every month since July 1997, with the possibility of clicking on any title to see a comprehensive review. In addition there is an annotated bibliography, under the four headings of Cyberculture in Context (8 entries), Virtual Communities (9 entries), Community Networks (3 entries) and Virtual Identities (5 entries). The introductory note under the last heading refers to a cartoon in the New Yorker with the caption: ‘‘On the Internet, no one knows you are a dog’’ and claims that a challenging problem is to examine the effect of online interaction on the notion of selfhood. Of the entries under the heading of Cyberculture in Context, two are by authors from the MIT Media Lab, one of them its director. As this headings introductory note says, the participation of the Media Lab is probably the best possible guarantee that the predictions will become reality. As might be
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expected the entries are generally enthusiastic, though one is said to display a trace of neo-Luddism. Some of the discussion is about the best way to foster a communal atmosphere in Internet use. The authors tend to deprecate the unsociable situation where most users are focused on a particular short-term objective, or communication only with established contacts. There is of course mention of MOOs and MUDs, types of virtual-reality multi-user domain in which the participants appear as avatars, so that true physical characteristics and even gender are hidden. There seems to be a view that Virtual Reality will play an increasing part in Internet exchanges, though just how it will operate is not clear. The site has a wealth of links to other sources of information, including a Directory of Electronic Theses and Dissertations in the Humanities, and the opportunity to subscribe to a listserv facility called cyberculture-announce, which will distribute notices of relevant events and publications. Another site dealing with similar interests is the Center for Digital Discourse and Culture (CDDC) of Virginia Polytechnic Institute in Blacksburg. This is the affiliation of Jeremy Hunsinger, mentioned earlier. The address is: . There is a listserv facility, to which subscribers are invited. The site also provides a hub for a Cyber Studies resources WebRing. This was begun in January 1998 and when visited in October 2001 it had links to a total of 48 sites spread over various countries, many of them referring to subdivisions of the general area as cybersociology and cyberanthropology. One site is devoted to the topics of Gender, Sex and the Web, and a number of others make some reference to gender. A concern with gender is also apparent in many messages to the cyberspace-and-society listserv, and is somewhat surprising since Internet communications are on the whole irrespective of gender. The site with the heading of Gender, Sex and the Web has the address: and has links to a very large number of other sites dealing with variations on the theme. London-AI This very useful facility, managed by Sunny Bains and operating as a list within the UK Mailbase, is undergoing changes. It obviously demands time and effort, and financial help to pay for assistance has been provided by The Society for the Study of Artificial Intelligence and Simulation of Behaviour (AISB). This was an unfair burden on members of AISB, since many subscribers to the list were not AISB members, and the arrangement has been discontinued. Attempts were made to find a way of continuing, but list subscribers did not rush to join the AISB, or even to send messages of support, when it was put to them that such action might save the London-AI list. The demise of the London-AI list in its familiar form was announced by Sunny Bains in a message distributed to the list on 23rd October 2001. Fortunately there is an offer from Tom Smith of the University of Sussex to continue the facility using a WEB-based scheme that will operate with less
manual effort. Details will be announced shortly. As Sunny Bains says: ‘‘It may take a while beforewe get a calendar as rich or diverse as it has been using the current method, but this seems the only possible compromise to shutting it down completely.’’ She is to be thanked and congratulated for her efforts in establishing this valuable facility.
Internet commentary
Cyrillic keyboard Several of the main fonts provided with Windows have variants allowing for languages other than English, and alphabets other than the familiar Latin one. Windows can be set up to operate in any specified combination of languages, and the one in effect at any time is indicated in a corner of the screen and can be changed using the mouse pointer. The most-used printing fonts, including Arial, Helvetica and Times New Roman, are provided in alternative versions, so that, besides for example the basic Arial there are variants Arial Cyrillic, Arial Turkish, Arial CE (for Central European) and so on. Every font, including these variants, includes the English (Latin) alphabet as well as the necessary additional characters. In the case of a font with the appendage of Cyrillic, these include the complete Cyrillic alphabet. When the computer is set to work in Russian, one such font comes into use and the keyboard is made to refer to the Cyrillic characters. Both alphabets can be accessed from the keyboard, because the language setting can be freely switched between Russian and English. Characters can also be selected onscreen using the mouse pointer and the character map, but this is an extremely tedious way to input more than a few words. Messages to be sent by e-mail can be made to use a Cyrillic font by entering the ‘‘Format’’ pull-down menu of Outlook Express, and then ‘‘Encoding’’ within it, and then ‘‘Cyrillic (Windows)’’, when the message is set up. (If the ‘‘reply’’ facility is used, and the incoming message was in a Cyrillic font, the new message automatically uses it also.) A difficulty, however, is that the standard distribution of the Cyrillic letters over the keys has no relation to the familiar QWERTY keyboard. This is inconvenient for a user familiar only with the QWERTY layout, even if a keyboard is obtained that is labelled with both alphabets. An alternative is to alter the assignment of the Cyrillic letters to keys so that they are mapped, as far as possible, onto the letter of the English alphabet that sound most similar. A complete mapping is not possible since the Cyrillic alphabet has more letters than the English one, and some have to spill over onto non-alphabetic characters. Nevertheless, it is enormously easier to transfer from QWERTY to a homophonic or anglophonic keyboard than to the standard Cyrillic one. Keyboard drivers that allow homophonic Cyrillic keyboards can be downloaded free of charge from: . A variety of options is offered and one that is certainly effective and convenient is a package called WinKey due to Gavin Helf. With this installed, double-clicking on its WinKey icon activates it (apparently remaining so until the computer is restarted). While WinKey is active and the
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system is set to operate in Russian the user has the choice of two keyboard layouts and can toggle between them by pressing the ‘‘scroll lock’’ key. One layout is homophonic Cyrillic and the other is the familiar QWERTY for the English alphabet, so that documents or messages that mix the alphabets are readily produced, slightly more conveniently than by switching the computer between languages, and with the advantage in some contexts that everything is done without leaving the ‘‘Cyrillic’’ font variant.
Alex M. Andrew
Book reviews Coordination of Internet Agents: Models, Technologies, and Applications edited by Andrea Omicini, Franco Zambonelli, Matthias Klusch and Robert Tolksdorf Springer Heidelberg 2001 ISBN 3-540-41613-7 xxv + 523 pp. hardback, £34.00 or $49.95 The central topic of this book is important and topical. Although much of the fascination and value of the Internet can be attributed to its largely spontaneous development, there is also a need for structure. This is perhaps most readily apparent in connection with security and commercial transactions, but the possibilities go well beyond this, and agent-based operation is now talked about in much the same way as object-oriented programming. Computation can be a shared activity of a community of agents. Thinking along these lines is clearly behind such recent developments as the extensible markup language XML and the C# (C sharp) language introduced by Microsoft. XML allows for inter-agent communications initiated by the software rather than the user. The book has eighteen chapters by different groups of authors. These are divided into six Parts, and there is an introductory Preface that includes an overview of the contents. There is no subject index, but there is a common reference list of no fewer than 660 items. It is interesting to note that a sprinkling of these refer to societal organisation in general and in particular that the book by Axelrod (1984) is included. The topic is treated in the book in a very general, academic way, with comparison of alternative overall models for coordinated action. The Preface does not give clear orientation for a reader not already involved in the topic, depending as it does on such terms as ‘‘perspective’’ and ‘‘paradigm’’ and ‘‘infrastructure’’. This is not to say that the treatment is not sound and pertinent; on the contrary it is in the context of diffuse emerging technology such as this that it can be most valuable to stand back and take a general view. In the Preface, the book’s contents are outlined as: In this book, the authors of the chapters write about models, technologies, and applications for the coordination of Internet agents. So, this book starts with a birds-eye view on coordination models (Part I), maps models onto technologies for Internet-based (Part II) and multi-agent (Part III) systems,
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and explores the emerging related issues from both a conceptual and a technological viewpoint (Part IV). Then, it deals with the application issues (Part V) and ends by supplying some visions on the future (Part VI). The two papers in Part I make references to Linda-like systems, with even the comment that references to coordination languages are sometimes assumed to refer to the particular (and early and effective) example so denoted. Early papers describing Linda are Carriero and Gelernter (1989, 1990). The earlier of these describes it purely as a means of achieving parallel operation in a single machine, though the later one refers to its possible use to control parallel operation of distinct computers, possibly programmed in different internal languages, in a local area network. Of the two papers in the final Part, on ‘‘Visions’’, the first describes a marketbased model for resource allocation, requiring a form of currency with which to purchase the services of a resource. Some such regulation will presumably become necessary. The second paper refers to the intriguing idea of computational ecosystems. There is clearly a great deal of valuable and thought-provoking material here. It is in a rather specialised and esoteric form and the book will not be of much help to someone who just wants to learn to use the relevant offerings of the software houses implementing for example XML and C#. Alex M. Andrew References Axelrod, Robert M. (1984) The Evolution of Cooperation, Basic Books, New York. Carriero, N. and Gelernter, D. (1989) ‘‘Linda in context’’, Comm. ACM, Vol. 32, part 4, pp. 444–58. Carriero, N. and Gelernter, D. (1990) ‘‘Tuple analysis and partial evaluation strategies in the Linda precompiler’’, in Gelernter, D., Nicolau, A. and Padua, D. Languages and Compilers for Parallel Computing, Pitman, London and MIT Press, Cambridge, Mass., pp. 114–25.
Soft Computing: New Trends and Applications by L. Fortuna, G. Rizzotto, M. Lavorgna, G. Nunnari, M.G. Xibilia and R. Caponetto Springer London 2001 xii + 257 pp. ISBN 1-85233-308-1, ISSN 1439-2232 paperback, with CD ROM, £29.50 or $49.95 Advanced Textbooks in Control and Signal Processing Series The term ‘‘soft computing’’ is attributed to Lotfi Zadeh and indicates a range of techniques that produce useful results without the rigorous analysis that is a feature of traditional methods. Specific mention is made of fuzzy logic and artificial neural nets, as well as the optimisation techniques of evolutionary
algorithms and simulated annealing. It is acknowledged that there are numerous conference proceedings and other publications that treat these topics separately, but here a selection of the most useful is covered in one volume. This is valuable because important applications have been made that combine methods so as to produce systems that can be described as neuro-fuzzy, fuzzygenetic or as using fuzzy cellular neural networks. The emphasis is firmly on applications and the treatment is driven by the needs of industry rather than by other considerations such as speculation about the working of real nervous systems. All of the topics are developed from, essentially, first principles, and a number of quite advanced applications are described. To cover all this in a fairly small book it is necessary that the treatment is in places somewhat terse though always chatty and readable. Each chapter is followed by a set of references that will give useful support. The references are mainly to modern works and do not necessarily honour the pioneers, so that there is, for example, no mention of McCulloch and Pitts. Attention is necessarily focused on those developments that have proved to have practical utility, and the treatment of neural nets is largely concerned with multi-layer perceptrons, with an acknowledgement of Kohonen nets but no mention of the Hopfield variety. The treatment of fuzzy logic, and then of fuzzy control, in the first two chapters after the Introduction, is particularly welcome since it spells out the application of such control in an application of a suitable degree of complexity, namely a device containing a switching transistor, as well as an inductor, capacitor and rectifier, that can be used for DC voltage adjustment. Cellular neural nets also receive extensive treatment. These are regular arrays of neuron-like elements, each with connections only to its immediate neighbours. An example is the game of ‘‘Life’’. Such arrays have important applications in image processing, and their analysis requires the invoking of Chaos Theory. This is developed in the book in considerable detail, introducing the ideas of chaotic dynamics and strange attractors. The development is not mathematically painless, but is probably as near to it as is possible, and the book is worth consulting for this feature alone. The applications mentioned range widely. One chapter is devoted to description of a robot for picking citrus fruit, where the complex dynamics of the robotic arm were allowed for using neural nets in the controller. The system is still experimental and some alternative arrangements are discussed, with no data about actual performance in an orchard, but even so it is a development that will arouse widespread interest. The book comes with a CD ROM, which seems to have been an afterthought as there is no mention of it in the book. The author of the CD ROM introduces himself as Francesco Zivillica. On it there is an assortment of additional pieces of theory along with animated demonstrations, a number of them interactive. The demonstrations include the formation of swirling spiral patterns that are important attractor states of cellular neural nets, as well as some of the remarkable effects achievable in the game of ‘‘Life’’. There is also a presentation
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of a means of modelling by use of a fuzzy-genetic algorithm that appears to be additional to the material presented in the book itself, and animated views of some mobile robots. The book, with its CD ROM, is a valuable addition to the literature. It is suitable as a teaching text, and will certainly facilitate practical exploitation of combinations of the different kinds of soft computing. The treatments of such topics as fuzzy control and chaos theory will be welcomed as instructive texts by those who find it easiest to get to grips with abstruse theories in specific practical contexts. Alex M. Andrew
Guided Evolution of Society: A Systems View by Bela H. Banathy Kluwer Dordrecht 2001 xviii + 415 pp. ISBN 0-306-46382-2 hardback, £65.00 or $95.00 Contemporary Systems Thinking Series The aim of this book is no less than its title suggests, namely the exploration of the means whereby the human race might consciously guide its own future evolution as a society. It is acknowledged that evolution of physical attributes has virtually ceased, and has been replaced by evolution of mental and cultural characteristics. These characteristics have evolved to a stage where we are able to view ourselves and to speculate about our own origins and to form theories of evolution and so on. An important part of this has been the evolution of consciousness, and now we also have consciousness of evolution. A reasonable next step is to utilise this understanding and to plan to guide future evolution in favourable directions. The change is to be brought about by peaceful and democratic means and this is no less than a blueprint for the future of humanity. The suggestion raises a number of obvious questions, one of which is about the degree to which evolution can usefully be guided. Banathy refers to an Evolutionary System (ES) and a distinct Evolutionary Guidance System (EGS). It is certainly true, as demonstrated by achievements of plant and animal breeding (though these are not examples quoted by Banathy) that evolution of physical features can usefully be guided, in these cases by an EGS that is distinct from the ES. In the social context the EGS has to be a part of the ES and it is not clear that the two can usefully be separated, especially since the ultimate goals of the EGS are presumably set within the ES. The notional separation is reminiscent of that of an ‘‘operational automaton’’ and a ‘‘learning automaton’’ in discussions of learning or self-organising systems.
Another point that is readily illustrated by plant and animal breeding is that there are dangers in trying to advance more rapidly than the level of understanding warrants, as illustrated by recent concerns about genetic manipulation. Similarly, in the socio-economic context, overregulated planned economies tend to be short-lived, presumably because their planning was based on an inadequate model. The term ‘‘evolution’’ is applied to processes that operate in untidy and uncharted environments where haste can be disastrous. Nevertheless, Banathy argues persuasively for the conscious guidance of social evolution, as a means of unifying and strengthening scattered initiatives aimed at improving the human condition. This is mainly to be achieved by education and by the establishment of a thoroughly democratic system of government based on the example of ancient Greece, where citizens were required to engage in discussions of topical issues in the agoras of their cities. (Modern guidebooks usually refer to agoras as market places but they clearly had other functions.) In the first part of the book the development of ideas on evolution is reviewed, with the contributions of many authors reviewed in some detail, starting with Lamarck and Darwin, and contemporaries of the latter including Thomas Henry Huxley and Alfred Russell Wallace. The suggestion that evolution can be seen continuing as intellectual and social development is attributed initially to Herbert Spencer and the contributions of many others are acknowledged. In recent years the views of Richard Dawkins have been influential, particularly his discussion of memes, corresponding, in cultural evolution, to genes in biology. This is an important idea since memes can account for the emergence of characteristics such as altruism, although the gene is, according to Dawkins’ own memorable description, selfish. Although not noted by Banathy, possible explanations of the emergence of altruism were considered earlier by a number of workers including J.B.S. Haldane. Some references are attached to a note by Andrew (2001). It is interesting that one of the writers reviewed in Banathy’s book, namely S. Gould, expresses reservations about the correspondence between biological and cultural evolution. He points out important differences, such as the fact that species that separate in biological evolution do not recombine, whereas separate threads of cultural development may merge. The development of humans through various stages is also reviewed, up to the agricultural revolution followed by the industrial and now informationa1 revisions. The curious (but long-lived) deviation represented by the Neanderthals is discussed in a fair amount of detail, essentially as also treated by Mithen (1998). The reservations expressed here about the central argument of the book should not obscure the fact that it is erudite, persuasive and written with great sincerity and concern for the future of humanity. The central message is one of hope that people can get together and agree on concerted action, and it is hardly necessary to observe that the current world situation, for example with regard to greenhouse emissions, deforestation and nuclear
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weapons, does not give much grounds for such hope. Banathy is clearly aware of the difficulties and argues that it is good to have an idealistic goal even of it is not reached. He also draws comfort from a quotation from Margaret Mead (on his page 355) where she says that: ‘‘a small group of thoughtful, committed citizens can change the world: indeed, it’s the only thing that ever does’’. Whatever its credibility, there is a great deal here that has a profound bearing on current issues Alex M. Andrew References Andrew, A.M. (2001) ‘‘Faith, physics and biology’’, Forum contribution, Kybernetes, Vol 31, Nos 1/2, in the press. Mithen, S.J. (1998) The Prehistory of the Mind: a search for the origins of art, religion and science, Phoenix, London (earlier edition, Thames and Hudson, London, 1996).
The Emerging Consensus in Social Systems Theory Kenneth C. Bausch Kluwer Academic/Plenum Publishers New York – London – Moscow 2001 i-xxvi – 419 pp. ISBN 0-306-46539-6 USD – 69.50 /EUR – 79.95 /GBP – 49.50 Keywords Publication, Cybernetics, Social Systems, Sytems Theory It is said that this book ‘‘will become the bible’’ in social systems theory (Professor Filion, University of Mon Real Business School), ‘‘a lucid account of modern systems’’ (Hector Sabelli, Director of the Chicago Centre for Creative Development) and ‘‘this book does not produce a metanarrative for our post modern age -it does, however, reveal the terrain upon which such a narrative will run’’ (Bela H. Banathy, President of the International Systems Institute and Alexander Christakis President of CWA Ltd in the book’s Forward.) These are worthy recommendations from distinguished people, but then buying a book is very much a personal matter. Do we want the text to learn more about Social Systems? Are we looking for a reference text or are we seeking a ‘‘course text’’? This book we are told provides in its Parts 1-4 all of these things. It is both comprehensive in its coverage and still, in under 500 pages, provides excellent detail. Not much in the field is left out and the author, we are informed, in the course of some 40 years of inquiry into the human condition has delved intensively into philosophy, orthodox theology, Eastern religions social and political ideology, psychology, and systems theory. It is clear from the book that the author, by adapting the methodology of interactive management, has synthesized the thought of such major systemic thinkers as Habermas, Lumann and many others. There are five discernible areas into
which he has concentrated these thoughts; the practice and ethics of design; the structure of the social world, communication, cognition; epistemology. We are also informed that in deciding to purchase this text ‘‘the choices are straight forward: invest a few years trolling the philosophy, sociology, and systemic theory sections in the library; or read the coherent perspective offered’’. That is the opinion of David Ing of the TBM AdvanceBusiness Institute who also adds that it is a ‘‘masterful synthesis of social theory in a systemic framework’’ and that ‘‘the breadth of his (Bausch) understanding of social theory and systemic theory is impressive’’. On a less profound note the book is undoubtedly well structured with a useful and revealing introduction that leads into Part One – Background; Part two – Incorporating Human Participation into Systems Theory and Design; Part Three – Advances in the area of Social, Cognitive and Evolutionary Theory (includes four chapters on Luhmann; one each on Kampis and Goertze I) and finally Part Four – Five emerging Synthesis. This completes the five contexts that the author says ‘‘grew out of my fascination with applying systems theory to social processes’’ (See also Book Reports – Kybernetes Vol. 31 No. 1, 2002) D.M. Hutton Index to Book reviews The index to volume 30, 2001 was published Kybernetes Volume 30 Number 9/10 2001, pp 1356–64. For completeness the reviewers of the books listed are included below: Book reviews (C.J.H. Mann) REVIEWER AYESH, A., Essential Dynamic HTML Fast: Developing an Interactive Web Site, No. 2, pp. 220–1. A.M. Andrew BROOKS, R.A., Cambrian Intelligence: The Early History of the New AI, No. 1, pp. 105–7. A.M. Andrew CHRISTIANINI, N. and SHAW-TAYLOR, J., An Introduction to Support Vector Machines and Other Kernel-based Learning Methods, No. 1, pp. 109–12. A.M. Andrew CUNNINGHAM, P. and FROSCHI, F., Electronic Business Revolution: Opportunities and Challenges in the 21st Century, No. 1, pp. 104. C.J. Harwood DAUM, B. and SCHELLER, M., Success with Electronic Business Systems, No. 1, pp. 103–4. C.J. Harwood GABBAY, D.M. and de RIJKE, M. (Eds), Frontiers of Combining Systems 2, No. 1, pp. 108–9. A.M. Andrew GOLDBERG, K. (Ed), The Robot in the Garden: Telerobotics and Telepistemology in the Age of the Internet No. 2, pp. 218–20. A.M. Andrew
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HARTLEY, R. and ZISSERMAN, A., Multiple View Geometry in Computer Vision, No. 9/10, pp. 1335–6. A.M. Andrew KARPENKO, A., Logical Investigations-7, No. 9/10, pp. 1339–41. A. Adamatzky KUSIAK, A., Computational Intelligence in Design and Manufacturing, No. 9/10, pp. 1333–4. A.M. Andrew LIN, Y., General Systems Theory: A Mathematical Approach, No. 2, pp. 216–17. Ibrahim Garrol MENZEL, P. and D’ALUISIO, F., Robo Sapiens: Evolution of a New Species, No. 9/10, pp. 1337–9. A.M. Andrew PLATTINI, M. and DIAZ, O. (Eds), Advanced Database Technology and Design, No. 2, pp. 223–4. D.M. Hutton ROSEN, K.H. (Ed.), Handbook of Discrete and Combinational Mathematics, No. 1, pp. 114–15. D.M. Hutton SEUL, M., O’GORMAN, L. and SAMMON, M.J., Practical Algorithms for Image Analysis: Description, Examples, and Code, No. 1, pp. 112–13. A.M. Andrew SINHA, N.K. and GUPTA, M.M., Soft Computing and Intelligence Systems: Theory and Applications, No. 1, pp. 104–5. C.J. Harwood TURUNEN, E., Mathematics Behind Fuzzy Logic, No. 2, pp. 223–3. A.M. Andrew
Book reports Highly Parallel Computations: Algorithms and Applications M.P. Bekakos WIT Press -Southampton UK Nov. 2001 456 pp ISBN 1-85312-748-5 £l59.00/US$245.0000/Euro 254.00 Keywords Publication, Cybernetics, Computing Described as a ‘‘state-of-the -art survey from experts in the field’’ this book highlights the remarkable progress made in parallel computing in the last few decades. When its originators developed both computer systems and the appropriate algorithms to take advantage of parallelism they promised a revolutionary change in both the way computers were built and also the way in which we tackled our problems. Cyberneticians and systemists still have problems that could be solved using the strategies outlined in this text. Many, however are reluctant to use what has now been so laboriously developed in this field. Perhaps this text is the answer because it does give an insight, to we are told, to various highly parallel computational methods as well as a survey of the applications of processor network architectures. The chapters presented are self-contained and many are quite obviously written by the leaders in this field. For example, the contributions by Professor D.J. Evans and colleagues are based on the wealth of their experience in this area. The book is in three Sections: I. General -purpose parallel computing and metacomputing II. Special parallel architectures and systolic application studies III. Neural algorithms and applications The text contributes to the WIT’s Press Advances in High Performances Computing Series, and has been designed for graduate/post graduate students and researchers working in the area of supercomputing. Cyberneticians and systemists will find it particularly useful as a means of getting up-to-date in parallel computing methodologies. Even if they cannot afford to buy personal copies it will make a useful addition to the department library. In educational and some business establishments it is clearly a text that should join the reference books and manuals in the computing section not only as a source for practitioners working in the area but also for those who need to demonstrate the state-of-the art of parallel computing to their students or course participants. Further details from: www.witpress.com
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Kybernetes, Vol. 31 No. 3/4, 2002, pp. 627-631. # MCB UP Limited, 0368-492X
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Information Warfare: How to Survive Cyber Attacks Michael Erbschloe Osborne/McGraw-Hill New York 2001 316 pp ISBN 0-07-213260-4 $29.99 Keywords Cybernetics, Computing, Publication Having the word ‘‘Cyber’’ in the title of a book will not attract either cyberneticians or systemists to the text. For many it is a much mis-used word. The author, however, is a keen user of the word and has coined it to pre-fix many others viz;. cyberterrorist, cyberspace, etc. The topic he has chosen for his book is now a particularly important one. He describes the new type of ‘‘war’’ that has broken out between those who want to protect their information and those who for so many different reasons want to gain access to it, and as he puts it mount ‘‘a deadly cyber attack’’ on a system. The type of attack is described and the defence is discussed in great depth. The author is well versed in the cyberterrorists subculture and provides some insight into the scenario that one must expect. As a result he does offer a variety of strategies for combating such attacks and on how a systems information should be protected. These attacks are not going to diminish, quite the contrary, unless governments worldwide can jointly address the problem chaos will undoubedly reign. In the meantime it is a question of defence in depth as in the conventional war. We need to know the attack weapons and develop our defences. Erbschloe offers good advice for the protection of our information infrastructures and tells us what can be done, particularly by national governments to fight the information war. More details of the book from: http://www.osborne.com
JAVA Programming: From the Beginning Kim King W.W. Norton 2001 ISBN 0–393–97437–5 £5.00 At this price the book has to be a bargin for anyone who wants to know about Java, whether they have or have not got programming experience. It starts with an introduction to computer terminology which in itself is encouraging.
Enough syntax is given so that programs can be introduced. The reader progresses chapter by chapter to learn more about objects, classes and syntax. There are chapters that cover graphics, interfaces, some basic algorithms and the built-in I/O Java classes. In addition, a website is available to provide source code for students and aids for course lecturers. It is regarded by many reviewers as an excellent text which provides an introduction to both Java and to programming. The one disadvantage is that it covers Java 1.1 but many of the features transfer to all versions. Well thoughtout and informal in style makes it a very suitable training or course text. C.J.H. Mann Book Reviews and Reports Editor
Cabell’s Directory of Publishing Opportunities in Management Volume I A-G: Volume II H-I Volume III J: Volume IV K-W Cabell Publishing Company Beaumount, Texas 2001 1973 pp i-xxi ISBNO-911753-15-X $149.95 US addresses (add $50 for surface mail $100 for airmail for all addresses outside US) Keywords Publication, Cybernetics, Management. The Cabell Directory, we are told by the publisher, has the simple objective of helping you to publish your ideas. It aims to provide the information needed to select those journals that are most likely to publish your work and to do this it provides an index that classifies the journals into twenty-two topic areas. It also gives details of the journal’s type of review process which, of course, is essential for any aspiring contributor. Although it focuses on journals that specialise in the area of Management there are other directories available that cover other areas of specialisation. The Directory is easy to use and its four volumes certainly cover the field of management. It provides useful advice on how to get your article published. See page 1374 for the Kybernetes entry. One would expect to see these volumes on the shelves of a good library rather than in ones own collection of reference texts. Whether the basic information can be gleaned by searching the Net is debatable. No doubt the publishers must be considering an online version or providing the material on disk. Other Directories are being published on Business (new 2001-2); Computers (2002): Education (2001); Psychology (2001): Biology (2002); Engineering (2002). For further information visit: www.cabells.com
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System Development: A Strategic Framework Michael Bronzite Springer-Verlag 2001 ISBN 0–85233–176–3 £19.95 Any book that can help those involved with systems development and with projects will be welcome. The media is forever telling us about the disasterous IT projects that have cost the world’s taxpayers an enormous amount of money. In the UK alone millions of pounds sterling have been lost on systems that did not achieve their goals. The book deals with some well-known projects and draws its information from them. It gives us the benefit, with hindsight, of the ‘‘pros and cons’’ of the delivered systems. Useful information is provided about the delivery of such IT systems. The analysis is well presented and structured. Some reviewers would suggest that the text does not live up to its title and that a greater emphasis should have been placed on the development strategy that should be employed in such developments. Even so, it contains much valuable material that will assist system designers and managers in their attempts to deliver viable systems. Index to Book reports The index to volume 30, 2001 was published Kybernetes Volume 30 Number 9/10 2001, pp 1356–64. For compleeness the reviewers of the books listed are included below: Book reports (C.J.H. Mann) ACKOFF, R., Exploring Personality: An Intellectual Oddyssey, No. 1, p. 118. ADAMATZKY, A., Computing in Nonlinear Media and Automata Collectives, No. 9/10, pp. 1342–4. BRITISH COMPUTER SOCIETY (BCS) TASK FORCE, Intellectual Property Rights in Software, No. 2, p.225. GALBRAITH, P., BLUM, W., BOOKER, G. and HUNTLEY, I.D., Mathematical Modelling Teaching and Assessment in a Technology Rich World, No. 2, pp. 225–6. GROSSING, G., Quantum Cybernetics – Towards a Unification of Relativity and Quantum Theory via Causal Modelling, No. 1, p. 116. JACOBSON, I., The Road to the Unified Software Development Process, No. 2, pp. 226–7.
MARTONEN, T.B. (Ed.), Medical Applications of Computer Modelling: Cardiovascular and Ocular Systems, No. 2, p. 227. METZGER, R. and WEN, Z., Automatic Algorithm Recognition and Replacement: A New Approach to Program Optimisation, No. 1, pp. 118–19. PHAM, D.T. and KARABOGA, D., Intelligent Optimisation Techniques, No. 1, pp. 117–8. RASKIN, J., The Humane Interface, No. 1, pp. 116–17.
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Software review ISAAC: Irreducible Semi-Autonomous Adaptive Combat Developer: Andrew Ilachinski Center for Naval Analyses, Alexandria VA22302, U.S.A. Keywords Combat simulation, Automata collectives, Social dynamic, Pattern formation, Complex systems, Artificial life, Distributed intelligence Abstract How does behaviour of individual combatants determine outcomes of the battle? Why lattice gas can be applied to simulation of a battlefield? What are most common patterns of a battle? One can find answers on these, and many other, questions if familiarise himself with the ISAAC simulation software. The software is briefly tackled in the review.
Kybernetes, Vol. 31 No. 3/4, 2002, pp. 632-638. # MCB UP Limited, 0368-492X
A discrete simulation of crowds, collectives and societies is one of the most advanced disciplines in the field of artificial life and complex systems (Bonabeau et al, 1999; Helbing, 1992; Helbing and Molnar, 1995; Epstein and Axtell, 1996; Axelrod, 1997). Typically, mass pools of digital entities obey simple rules of local interaction and inhabit a two-dimensional discrete space. They interact with one another and form complex patterns during their evolution. A correspondence between local interaction rules and patterns they implicitly produce is usually a subject of investigation. An evolution of discrete collective is called a population dynamic when entities consume each other. This is a battlefield when they destroy each other in order to fulfil tasks different from simply supporting their existence. Majority of the software related to a battlefield simulation is based mainly on a virtual reality and employs ‘‘see-the-battlefield’’ paradigm. No attention is usually paid to development of tactical thinking on a large-scale. No resources at all are allocated to simulation of minimalist agents engaged in a combat operation. Noticeably, more spatially oriented war gaming software is well spread in the computer games world. However, again it is rather a superficial imitation than true simulation of complex combatant systems. Recent development of simulation tools for battlefields perfectly mirrors the evolution of simulation in natural sciences. It gradually shifts from differential equations, which do not account for spatial dynamics of simulated systems, toward cellular automata, where behaviour of any particular entity influences system’s progression. A first step from Lanchester equations to discrete models has been already done by Woodcock et al. (1988), who designed quite simple cellular automata rules to represent development of automata configurations in a sense of a battlefield. Less than a decade later Ilachinski made the second step (Ilachinski, 1996a, 1996b; 1997a; 1997b). He designed the ISAAC, a simulator of
multiple-agent collectives, parameters of which are interpreted in terms of Software review combat, and space-time dynamic of which simulates development of a battlefield. When designing ISAAC, Ilachinski aimed to ‘‘use complex systems theoretic inspired basic research to develop fundamentally new conceptualisations of combat’’ (Ilachinski, 1997a; 1997b). ISAAC operates models based on a paradigm of discrete entities moving on 633 a discrete lattice and interacting locally to one another. Lattice gases (Frisch et al., 1987) and lattice chemistries (Dab et al., 1990) are best examples. Sometimes, these systems are ignorantly named ‘‘mobile cellular automata’’ (see e.g Miramontes et al, 1993). Why combat? Here is the ironic answer. Because ‘‘. . . individual behaviour under the combat rule is different from individual behaviour in the non-combat case. . .’’ (Epstein and Axtell, 1996). In one of his talks Andrew Ilachinski quotes Carl von Clausewitz: ‘‘War is . . . always the collision of two living forces’’. From mass flow models of forces’ collisions to emerging patterns of colliding combatants – is a main idea of the simulator.
Basics of simulation The ISAAC is a basic core simulation engine, working in DOS. EINSTein forms a Windows interface to the simulation engine. This is an ‘‘. . . interactive toolbox to . . . for the general exploration of combat as a complex adaptive system’’ (Ilachinski, 1997). When discussing particulars of the simulation we keep in mind both versions of the software. An action takes place on a two-dimensional lattice, where two forces are engaged in a battle. Each force has its own flag and battles to capture enemy’s flag or to destroy enemies. Discrete mobile automata – combatants – are elementary entities of the model. They could, for example, represent infantrymen or tanks. The combatants obey simple rules: they act depending on values of their sensors and local states of their environment. Each combatant is supplied with a personality vector. The vector consists of six components, corresponding to alive friendly agent, alive enemy agent, injured friendly agent, injured enemy agent, own flag, and enemy flag. Values of the vector’s components may be set up in a range of 25-50 units from defensive (lowest) to offensive (highest) values (Figure 1). Combatant’s neighbourhood consists of five nested sub-neighbourhoods. They are, from internal to external, as follows: movement range, constraint range, fire range, sensor range and communication range. When the combatant decides which neighbouring lattice node to move in he virtually calculates an image of his neighbourhood inhabitants weighted by offensive/defensive values of its personality vector, e.g. moves toward or away an enemy or a friend.
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Figure 1. The EINSTein simulation screen with combatants’ data displayed
Both rules and personality vectors of combatants may be either constant or dynamically changing during battlefield development. Additional features of the model include local and global commanders, reconstitution fratricide (which reflects a probability to be inured by friendly fire) and attrition. There are several run modes in the simulator, including the following: . .
. .
Interactive (one can change combatants’ parameters on flight); Multiple series (multiple trials of the same combat but with different initial configurations of combatants); Fitness landscape; Genetic algorithm (searches in specified space for the combatants’ parameters optimal for a defined mission).
Space-time configurations of combatants are probably most important data, collectable in result of simulation. They are invaluable. Additionally, one can gather integral data of combatants dynamic. These include .
Force strengths (measured in fractions of alive, injured or killed combatants);
.
Statistics of combatants with specified weapons;
.
Distributions of combatants’ populations;
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Distances to enemy’s flag;
. .
Spatial entropy and territorial possession; and Mission fitness landscape.
It does not take to much time to become acquainted with the simulation engine. Almost immediately one can start to explore a rich space-time dynamic of combatants. Their behaviour varies from simple forward advance and frontal attack, to more complicate penetration or retreat, to even more sophisticated flanking manoeuvers and encirclement. An example of battlefield development is shown in Figure 2. Applications It is difficult to find for sure whether military actively employs the discussed simulator or not. Few results are published anyway. Thus, e.g., Horne and Lauren from the United States Marine Corps and New Zealand Defence Force, respectively, utilised ISAAC in their Operational Synthesis project (Horne and Lauren, 1999; Lauren, 1999) to explore non-linearity, intangibles, and coevolving landscapes of modern battlefields, and to create a model useful for field commanders. These problems are particularly well tackled by ISAAC. Outcomes of non-linearity are intrinsic for locally communicating large-scale automata collectives. It is quite typical for any lattice swarm to drastically change appearance of their global patterns when ‘‘naively small’’ changes in automaton local transition function are made, see e.g. (Adamatzky, 2000). Such person-oriented parameters as a degree of training, state of a morale, ethos, leadership (Horne and Lauren, 1999; Lauren, 1999), despite their high importance, not often been incorporated in simulation systems. One can fully exploit these features in ISAAC via proper definition of agents’ personality vectors. Co-evolving landscapes and adaptability of combatants is also very important issues. In ISAAC the agents’ rules may be either fixed or evolving. When the rules evolve during development of the battlefield the combatants adapt not only to specific of their environment but to tactics adopted by enemy agents as well. Spatial characterisation of simulated combats allows one also to answer the question ‘‘how manoeuvre affects warfare’’ (Lauren, 1999). Thus, as reported in (Horne, 1999), the ISAAC is put to use in the Marine Corps’ Project Albert, dealing with space-time relations between manoeuvres, attrition and engagement the enemy. Computer simulation shows that a degree of manoeuvrability clearly determines a success of goal achievements as well as amount of potential casualties. Further development The simulator is clearly a beta-version. There are a lot of bugs and the program may behave maliciously. Also it is not unusual for the simulator to simply crash. However, this does not spoil a general charm of the software. The simulator is perfect as a ‘‘proof-of-concept’’.
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Figure 2. Snapshots of a simple battlefield simulated in ISAAC. The combatants start at opposite sides of the battlefield. They move toward each other and eventually engage. The simulation stops when a combatant reaches the flag of the opposite side
An integration of the package with other tools for army training and intelligent simulation of battlefields (see e.g. Stone et al, 1996) could be a great idea. Civil applications, e.g. social dynamic, of the simulator can be also quite productive. The ISAAC bears a huge potential for modelling pedestrian flows
(see e.g. Helbing and Molnar, 1995) and collective outbursts like panic, craze Software review and hostility (Smelser, 1962). What is about parallelism? Behaviour of real and simulated combatants is intrinsically parallel. Thus, implementation of ISAAC on a massively parallel hardware could be quite logical. There are examples of successful parallel simulation of battlefields. The MasPaWS, a massively parallel war simulator 637 (Sekharan et al, 1996), is one of them. The MasPaWS is written on Maspar Programming Language and is implemented on a massively parallel SIMB machine with several thousands elementary processors. The elementary processors communicate either locally, as in typical mesh processors, or at long distances via specific routing procedures. Tanks are basic elements of the model. The tanks move on twodimensional lattice. Each tank has a certain radius of perception, obeys simple rules of engagement its enemy, and follows intuitive migration protocols, including reduced path planning utilities. Each processor of the parallel hardware is responsible for path of the simulated battlefield. Such a set-up proves to have reasonable computational costs because the natural parallelism of the battlefield dynamic is exploited. Merging of ISAAC and MasPaWS in a single product could be advantageous. As far as we know (Brandestein et al, 1998) Maui High Performance Computing Center has already ported the ISAAC to the massively parallel processor IBM SP to simulate and investigate models of multiple combats. Distribution The ISCAAC and EINSTein, together with documentation and sample combat scenarios, are downloadable free of charge on its home page at the Center of Naval Research site http://www.cna.org/isaac/. Andrew Adamatzky Intelligent Autonomous Systems Lab University of the West of England
[email protected] References Adamatzky, A. (2000), ‘‘Choosey hot sand: reflection of grain sensitivity on pattern morphology’’, International Journal of Modern Physics C, Vol. 11, pp. 47-68. Axelrod, R. (1997), The Complexity of Cooperation, Princeton University Press. Bonabeau, E., Dorigo, M. and Theraulaz, G. (1999), Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press. Brandestein, A., Horne, G. et al. (1998), High performance computer support for analysis of entity-based simulations of land combat, Maui High Performance Computing Center. http://pipeline.mhpcc.edu/research/ab98/ 98ab12.html Dab, D., Lawnizcak, A., Boon, J.-P. and Kapral, R. (1990), ‘‘Cellular-automata models for reactive systems’’, Physical Review Letters, Vol. 64, pp. 2462–5. Epstein, J.M. and Axtell, R. (1996), Growing Artificial Societies, The MIT Press.
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Frisch, U., d’Humieres, D., Hasslacher, B., Lallemand, P., Pomeau, Y. and Rivet, J.P. (1987), ‘‘Lattice gas hydrodynamics in two and three dimensions’’, Complex Systems, Vol. 1, pp. 649-707. Helbing, D. and Molnar, P. (1995), ‘‘Social force model for pedestrian dynamics’’, Physical Review E., Vol. 51, pp. 4282-6. Helbing, D. (1992), ‘‘A fluid–dynamic model for behaviour of pedestrians’’, Complex Systems, Vol. 6, pp. 391-415. Horne, G. E. (1999), Maneuver warfare distillations: essence not verisimilitude, Proc. 1999 Winter Simulation Conference. http://www.informs-cs.org/wsc99papers/166.pdf Horne, G.E. and Lauren, M.K. (1999), Operational synthesis applied to mutual NZ/US questions In: 34th Ann. Conf. of the Operational Research Society of New Zealand, Hamilton. http:// www.mngt.waikato.ac.nz/orsnz99 Ilachinski, A. (1996a), Land warfare and complexity: Part I. Mathematical background and technical sourcebook, CNA CIM-461. http://www.cna.org/ isaac/lw1.pdf Ilachninski, A. (1996b), Land warfare and complexity. Part II. An assessment of the applicability of nonlinear dynamics and complex systems theory to the representation of land warfare, CNA CRM-68. http://www.cna.org/ isaac/lwpart2.pdf Ilachinski, A. (1997a), Irreducible semi-autonomous adaptive combat (ISAAC): An artificial-life approach to land warfare, CNA Research Memorandum 91-61.10. http://www.cna.org/ isaac/crm9761.htm Ilachinski, A. (1997b), A mobile cellular automata approach to land combat: A user’s guide to an early version of ISAAC, CAN CIM-482. Lauren, K. (1999), Characterising complexity on the battlefield using fractal statistics In: 34th Ann. Conf. of the Operational Research Society of New Zealand, Hamilton. http:// www.mngt.waikato.ac.nz/orsnz99 Miramontes, O., Sole´, R.V. and Goodwin, B.C. (1993), ‘‘Collective behaviour of random-activated mobile cellular automata’’, Physica D, Vol. 63, pp. 145-60. Sekharan, C.N., Ma, Z.-H., Vepulapati, U. B., Porthouse, W.J. and Irwin, A. (1996), ‘‘MasPaWS – A massively parallel war simulator’’, Simulation Practice and Theory, Vol. 4, pp. 265-82. Smelser, N. J. (1962), Theory of Collective Behaviour, The Free Press, New York. Stone, G., Ressler, E. and Lavelle, E. (1996), ‘‘Intelligent simulation of the battlefield (ISB)’’, Expert Systems with Applications, Vol. 11, pp. 227-36. Woodcock, A.E.R., Cobb, L. and Dockery, J.T. (1988), ‘‘Cellular automata: A new method for battlefield simulation’’, Signal, Vol. 42, pp. 39-50.
Announcements
Announcements
June 2002 International Conference on Management of Data and Symposium on Principles Database and Systems – SIGMOD/PODS ’02, Madison, Wisconsin USA, June 2-6 Contact: David DeWitt. Tel: 608-262-1204. E-mail:
[email protected] SACMAT’02: 7th ACM Symposium on Access Control Models and Technologies, Monterey, California, USA, June 3-4 Contact: Ravi S. Sandu. Tel: +l-703-993-1659. E-mail:
[email protected] SoCG’02: 18th Annual ACM Symposium on Computational Geometry, Universitat Politecnica, Barcelona, Spain, June 5-7 Contact: Vera Sacristan. Tel: 34-93-401-6931. E-mail:
[email protected] ECIS’02: The 10th European Conference on Information Systems, Gdanski, Poland, June 6-8 Contact: Stanislaw Wrycza. Tel: +(48 58) 5509400. E-mail:
[email protected] ICDL 2002 – International Conference on Development and Learning, Cambridge Mass., USA, June 13-15 Contact: Juyang Weng. E-mail:
[email protected] SIGMETRICS 02: The International Conference on Measurement and Modelling of Computer Systems, Los Angeles, California, USA, June 15-19 Contact: Richard Muntz. Tel: 310-825-3546. E-mail:
[email protected] PLDI 02: Conference on Programming Language Design and Implementation (ACM SIGPLAN 2002), Berlin, Germany, June 17-19 Contact: Jens Knoop. Tel: 49-231-755-5810. E-mail:
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[email protected] IEA/AIE’02: Fifteenth International Conference on Industrial and Engineering Applications of Artificial Intelligence & Expert Systems, Cairns Convention Centre, Cairns, Australia, June 17-20 Contact: Moonis Ali. Tel: 512-245-3409. E-mail:
[email protected]. URL: http://iea2002aie.bsee. swin.edu ISMM’02: 2002 International Symposium on Memory Management, Berlin, Germany, June 20-21 Contact: Hans J. Boehm. Tel: (650) 857-3406. E-mail:
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15th Computer Security Foundation Workshop-CSFW-15, Keltic Lodge, Cape Breton, Nova Scotia, Canada, June 24-26 Contact: Iliano Cervesato. Tel: +1-202-404-4909. E-mail:
[email protected]. URL http:// www.csl.sri.com/csfw/csfwl5 ITiCSE’02: Innovation and Technology in Computer Science Education, Aarhus, Denmark, June 24-26 Contact: Daniel Joyce. Tel: 610-519-7344. E-mail:
[email protected] DIS 2002: Designing Interactive Systems 2002, London, UK, June 27-29 Contact: Alistair Sutcliffe. Tel: 44-161-200-3315. E-mail:
[email protected] IWPC 2002 – 10th International Workshop on Program Comprehension, Paris, France, June 27-29 Contact: Francoise Balmas. E-mail:
[email protected]. See also: http://www.ai.univ-paris8.fr/ iwpc02/
July 2002 SIACG’02: 1st Ibero-American Symposium on Computer Graphics, Portugal, July 2-3 Contact: Joaquim A. Jorge. Tel: +351-21 3100336. E-mail:
[email protected] ICDCS – 22nd International Conference on Distributed Computer Systems, Vienna, Austria, July 2-5 Contact: Luis Rodrigues. E-mail:
[email protected]. See also: http://icdcs2002.di.fc.u1.pt Second International Conference on Urban Regeneration and Sustainability: The Sustainable City 2002, Segovia, Spain, July 3-5 Contact: Wessex Institute of Technology. Tel: 44(0) 238 029 3223. E-mail: Isouthcott@wessex. Fax: 44 (0) 238 2853 7th International Conference-UK Systems Society, York University, York, UK, July 7-10 Contact: UK. Systems Society. E-mail:
[email protected] ASSETS’02: Fifth ACM SIGCAPH Conference on Assistive Technologies, Edinburgh, Scotland, UK, July 8-10 Contact: Andrew Sears. E-mail:
[email protected] ICTEM’02: Informatics Curricula teaching Methods and Best Practices, Florianopolis, Brazil, July 10-12 Contact: A. Joe Turner. Tel: 864-882-7509. E-mail:
[email protected] Special announcements REMINDER
UK Systems Society 7th International Conference At: York, U.K. – York University Dates: Sunday July 7th to Wednesday July 10th 2002 MESSAGE FROM THE ORGANISERS: The conference will build on the success of the previous conferences and will take the title of ‘‘Systems Theory and Practice for the Knowledge Age’’ – a title that embraces many pertinent issues for academics and practitioners. To date we have around 100 delegates already committed to preparing papers and making presentations. So, while all the space in the hard back proceedings is almost certainly taken, there are still opportunities for you to have a paper included in the soft bound proceedings; or in the UKSS publication entitled Systemist; or to present your paper without a publication. And, of course, you are welcome to attend without giving a paper. Fees are £450 for UKSS members – this is a very good value package that includes conference registration, a copy of the proceedings, all meals and refreshments, accommodation in a single en suite room and attendance at the banquet. We have a host of well known speakers lined up – it is early days but, to date, Professor Mike Jackson and Professor Max Boisot have confirmed their attendance. So, we do hope you will be able to join us – if you would like more information please don’t hesitate to contact us at ukssconference2002@ hotmail.com
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7th International Congress of Cybernetics, Computer Science, and System Theory At the Institution of Educational Technology, Faculty of Education, Constantine the Philosopher, University in NITRA, Slovakia Dates: 2–6 September 2002 The 7th Congress of TAXIS and the 16th Congress of AIC. The congresses are held under the auspice of the Rector: Doc. PaedDr. Eva Pola´kova´, PhD. (Sk) President Prof. Dr. Habil. Helmar G. Frank (D) Vicepresident A. Doc. Dr. Dan Maxwell (USA) Vicepresident T A K I S (world association for cybernetics, computer science and system theory) was founded in 1983 in Namur (Be) as a branch of International Association for Cybernetics (AIC) after the death its founder Prof. Dr. Ing. George R. Boulanger. These associations built a tradition of world congresses for cybernetics – AIC traditionally in Namur, TAKIS in different european countries. After the discontinuation of AIC, its function and aims were taken over by TAKIS. For this reason the congress in 2002 will continue the former tradition and will unite the 16th congress of AIC with the 7th congress of TAKIS. The Conference on Cybernetics, organized by TAKIS, will provide an opportunity to present the results of your scientific and pedagogical work in the following fields: 1. cybernetics of the humanities with a subsection anthropocybernetics 1.1 language cybernetics 1.2 psychocybernetics 1.3 educational cybernetics 1.4 aesthetic cybernetics 1.5 applied cybernetics of media use 2. general cybernetics 3. engineering cybernetics 4. biocybernetics Congress fee (including congress materials and magazine) is 70 EUR, 30 EUR for members of AIS, TAKIS, UNI-SK and students. CONTACT: U´stav didaktickej technolo´gie Pedagogicka´ fakulta UKF Tr. A. Hlinku 1 949 01 Nitra, Slovenska´ republika Tel./Fax: + 421 37 6511013 E-mail:
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Fifth European Systems Science Congress The fifth European Systems Science Congress will be held in Crete, Greece from 16th to 19th October 2002. The title of the Congress is ‘‘The System in its Context’’. An island of diverse and outstanding beauty, Crete has everything to inspire the Systems scientists in the framework of the birth-place of European civilization – with both Athena, the goddess of wisdom, and Europe, the wife of Zeus. The beautiful environment combining wonderful sea-side and sacred mountains is an excellent recipe for inspiration. The aim of the congress is to bring together Systems scientists from every discipline to exchange their experience, to present their approaches, and to discuss them in a very outstanding environment. President of the Congress: Nicholas Paritsis. Both hard and human sciences are dealing with complex systems, and are therefore involved in Systems Science. It is namely the case for: – social systems design – socio-cybernetics, – systems evolution – economy – environment, – systems management, – family and systems therapy, – cognitive science, – artificial intelligence – artificial life – biology and medicine, – mathematical models and theoretical approaches The central theme of the conference is ‘‘The System in its Context’’ and the aim is to bring together the general system theorists (generalists) and those who are concerned with the main systems applications such as family therapy, systems management, social systems design and development, social-cybernetics, and cognitive science. Papers, not restricted to the above area, are also welcome. You are kindly invited to send a paper and/or propose a workshop in the Systems Science framework. All submitted papers and workshops will be reviewed by the Congress scientific committee. Dead-line for abstracts and workshops proposal: 15 March, 02 Notification of acceptance: 15 May, 02 Papers, (4 sheets maximum) are to be submitted by 15 July **only by e-mail**, (instructions can be found in the web page) to: <
[email protected]> The Congress fees are 200 Euro, before 31 July and 250 Euro after. There will be a 50% reduction for students. A Satellite Conference on Family and Systems Therapy has also been arranged – Fees are 150 Euro before 31st July and 170 after. Accommodation can be at the Conference Hotel or at hotels in the surrounding area
Special announcements
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