Studies in Fuzziness and Soft Computing Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw, Poland E-mail: kacprzyk@ibspan.waw.pl

Vol. 3. A. Geyer-Schulz Fuzzy Rule-Based &pen Systems and Genetic Machine Learning, 2nd ed. 1996 ISBN 3-7908-0964-0

Vol. 4. T. Onisawa and J. Kacprzyk (Eds.) Relillbility and ~ety Allalyses under FuzziMss, 1995 ISBN 3-7908-0837-7

Vol. 5. P. Bose and J. Kacprzyk (Eds.) Fuu.iness in Database MQIIQgement Systems, 1995 ISBN 3-7908-0858-X

Vol. 6. B.S. Lee and Q. Zhu Fuzzy and Evidence ReQSOning, 1995 ISBN 3-7908-08~

Vol. 7. B.A. Juliano and W. Bandler Tracing Clwins-of-Thought, 1996 ISBN 3-7908-0922-5

Vol. 8. R Herrera and J.L. Verdegay (Eds.) Genetic Algorithms and Soft Computing, 1996, ISBN 3-7908-0956-X

Vol. 9. M. Sato et al. Fuzzy Clustering Models and Applications, 1997, ISBN 3-7908-1026-6

Vol. 10. L.C. Jain (Ed.) Soft Computing Techniques in Knowl­edge-based Intelligent Engineering Systems, 1997, ISBN 3-7908-1035-5

Vol. 11. W. Mielczarski (Ed.) Fuzzy Logic Techniques in Power Systems, 1998, ISBN 3-7908-1044-4

Vol. 12. B. Bouchon-Meunier (Ed.) Aggregation and Fusion of Imperfect Information, 1998 ISBN 3-7908-1048-7

Vol. 13. E. Ortowska (Ed.) Incomplete Information: Rough Set Analysis, 1998 ISBN 3-7908-1049-5

Vol. 14. E. Hisdal Logical Structures for Representation of Knowledge and Uncerti.Jinty, 1998 ISBN 3-7908-1056-8

Vol. 15. G.J. Klir and M.J. W'Jel'IIUIII Uncerti.Jinty-Based lriformation, 1998 ISBN 3-7908-1073-8

Vol. 16. D. Driankov and R. Palm (Eds.) Advances in Fuzzy Contro~ 1998 ISBN 3-7908-1090-8

Vol. 17. L. Rezoik, V. Dimitrov and J. Kacprzyk (Eds.) Fuzzy Systems Design, 1998 ISBN 3-7908-lll8-1

Vol. 18. L. Polkowski and A. Skowron (Eds.) Rough Sets in Knowledge Discovery 1, 1998, ISBN 3-7908-lll9-X

Vol. 19. L. Polkowski and A. Skowron (Eds.) Rough Sets in Knowledge Discovery 2, 1998, ISBN 3-7908-1120-3

Vol. 20. J.N. Mordeson and P.S. Nair Fuzzy Mathematics, 1998 ISBN 3-7908-1121-1

Vol. 21. L. C. Jain and T. Fukuda (Eds.) Soft Computing for Intelligent Robotic Systems, 1998 ISBN 3-7908-1147-5

Vol. 22. J.N. Mordeson and P.S. Nair Fuzzy Mathematics, 1998 ISBN 3-7908-1121-1

Vol. 23. P. S. Szczepaniak (Ed,) Computational Intelligence and Applications, 1999 ISBN 3-7908-1161-0

Vol. 24. E. Ortowska (Ed.) Logic at Work, 1999 ISBN 3-7908-1164-5

continued on page 424

Michael Zaus

Crisp and Soft Computing with Hypercubical Calculus

New Approaches to Modeling in Cognitive Science and Technology with Parity Logic, Fuzzy Logic, and Evolutionary Computing

With 104 Figures and 33 Tables

Springer-Verlag Berlin Heidelberg GmbH

Dr. Michael Zaus Institute for Cognitive Science University of Oldenburg D-26111 Oldenburg Gennany

e-mail: zaus @psychologie.uni-oldenburg.de

ISBN 978-3-662-11380-6

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Zaus, Michael: Crisp and soft computing with hypercubical calculus: new approaches to modeling in cognitive science and technology with parity logic, fuzzy logic and evolu­tionary computing I Michael Zaus.

(Studies in fuzziness and soft computing; Vol. 27) ISBN 978-3-662-11380-6 ISBN 978-3-7908-1879-6 (eBook) DOI 10.1007/978-3-7908-1879-6

This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of ttanslation, reprinting, reuse of illus­trations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its cwrent version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.

©Springer-Verlag Berlin Heidelberg 1999 Originally published by Physica-Verlag Heidelberg New York in 1999 Softcover reprint of the hardcover I st edition 1999 The use of general descriptive names, registered names, trademarks, etc. in this publica­tion does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Hardcover Design: Erich Kirchner, Heidelberg

SPIN 10696992 8812202-5 4 3 2 1 0 - Printed on acid-free paper

#\ In Memory of

Gerard A. Langlet CEA, Laboratoire d 'Informatique Theorique

C.E. Saclay, Gif sur Yvette, France

#\

Preface

This text grew out of a research project on the foundations of parity logic, fuzzy logic, and evolutionary computing at the Institute for Co­gnitive Science of the University of Oldenburg in Germany, and out of a series of seminars on evolutionary and fuzzy computing in cognitive science. What these seemingly diverse fields have in common from a conceptual and computational point of view is that all of them are ba­sed on hypercubical calculus. That was not apparent at the beginning, but arose gradually along intensive computational work. To provide an idea of what it implies, we list seven representative hypercubes together with their specific field of application.

1. Boolean Hypercube: Parity Logic & Evolutionary Computing

8" = {O,l}n = {x = (xl,X2 1 ... 1 Xn) E 'Rn : Xi E {0,1} for all!::::; i::::; n}

2. Trivalent Hypercube: Fuzzy Logic & Fuzzy Cognitive Maps {-1,0,1}" = {x = (x1,X2, ... ,xn) E 'R" : Xi E {-1,0,1} for all 1::::; i::::; n}

3. Bipolar Hypercube: Fuzzy Logic & Fuzzy Cognitive Maps [-l,lr={x=(xl,x2, ... ,xn)E'Rn: x;E(-l,l]for alll::=;i::=;n}

4. Unit Hypercube: Fuzzy Logic & Fuzzy Cognitive Maps In= (O,l]n = {x = (xl,X2 1 ••• 1 Xn) E 'Rn : X; E [0,1] for all 1::::; i::::; n}

5. Discrete Bipolar Hypercube: Crisp Signal State Spaces {-'l,l}n={x=(xl,X2, ... ,xn)E'Rn: x;E{-l,l}for alll::=;i::=;n}

6. Schemata- viz. Hyperplane Cube: Evolutionary Computing

Sn={s=(sl,s2, ... ,sn)E{{0,1}U{*}}n: s;E{1,0,*}for all1::=;i::=;n}

7. GRAY-Coded Hypercube: Parity Logic & Genetic Algorithms gn = {O,l}n = {x = (xl,X2 1 ••• 1 Xn) E 'Rn : x; E {0,1} for all 1 :::=; i :::=; n}

where Ln x; ffi y; = 1 and gn is pathwise Hamiltonian.

This is not the place to unpack their details, but it helps to guide the reader around the central topics of this book. The research project centered on formal models of connectionist information processing re­garding the emergence of meaning in the brain. By investigating the

viii Preface

formal foundations of emergent computing, and by being persistently faced with the target question of how meaning emerges in the brains of living systems, it turned out soon that this was not only one of the last questions science may try to answer, but also one that we are far from being able to treat in terms of traditional mathematics, at least with respect to explanatory models with high epistemic value. Rather than trying to model the formally intractable emergence of meaning pheno­menon, the author decided to search for more fundamental information generating algorithms which qualify for proper emergent computation.

The search for generic and most elementary information processing mechanisms led the author into the field of algorithmic compression, and by hitting upon Gerard Langlet's St. Petersburg paper on a Theory of Everything in terms of the programming language APL, the ice was broken ([LAN92]). However, neither Langlet's T.o.E. nor the incredible scope it covered was of primary interest to the author, but the mere fact that the cumulative n-bit parity function p : {0, 1}n -+ {0, 1}n constituted a minimal and irreducible algorithm for generating increas­ingly complex structures out of almost nothing by starting from the ele­mentary bit. It provided not only a unique tool for scientific modeling from scratch, but also the birth of parity logic, as the reader will learn from part I of the book. What followed from 1994 to 1996 was intensive computational research, partially rewarding due to successful progress with magnificient support and encouragings from Gerard Langlet, and partially frustrating because of an ignorant and denigrating academic environment. In 1996, the frustration turned into reward when the author introduced parity logic as an invited speaker to the audience of the APL96 conference held at Lancaster University, England. Then the big shock came by the end of 1996, when the author learned about the death of Gerard Langlet. My greatest debt is therefore acknowledged in the dedication. Whatever merits part I of this book possesses may truthfully be credited to his influence, support, and encouragement. I wished we had the chance for a bigger research project on the sub­ject matter, since so many things remain to be done, in particular with respect to multidimensional Langlet transforms, parallel data compres­sion, and binary dynamical systems. Efforts in this direction are now subject to further research by unifying Gerard Langlet's basic concepts of genitons, paritons, and fanions in mathematical group theory.

The second approach pursued in part II on fuzzy logic is at the very heart of hypercubical calculus. In that respect the work of Bart

Preface IX

Kosko ([KOS92], [KOS97]) is greatfully acknowledged. It is argued that fuzzy logic offers a paradigm shift in social and behavioral science by virtue of providing a sound framework for soft computing, for construc­ting nonlinear dynamical predictor systems, and for making knowledge engineering a prospering business. The current hands-off attitude of psychologists towards fuzzy logic is quite puzzling, for they exclude a logic in their practical and scientific activities which is common to any human individual, namely approximate reasoning. What comes next door to approximate reasoning is causal reasoning. Causal cognition abounds in problem solving, decision making, and in trying to predict future events. Causal modeling is a domain offuzzy logic, and it is best pursued with fuzzy cognitive maps, that is, knowledge projections par excellence. No knowledge, no map. No map, no cognitive guidance. No cognitive guidance, no intelligible behavior.

Causal knowledge is a benchmark for competence in almost any field of human activity, and the more complex the knowledge domain, the harder it is to achieve. Our structural modeling approach to fuzzy cognitive maps is precisely tailored to this task. Hypercubical cal­culus and fuzzy cognitive maps ease a gentle entrance into nonlinear modeling, because of their generic character that allows us to start small and grow in fuzzy knowledge engineering. It emphasizes the state-space approach, whereby state-space dynamics and the dyna­mics of interactions become apparent and tractable. It subscribes to the Ganzheitsproblem, i.e. the way how the whole is contained as a part in one of its own parts, thereby revealing the real nature of fuzzy mutual subsethood. And it admits a highly desired feature in research and practice, namely the growth of knowledge through cooperative learning processes in terms of aggregating individual fuzzy cognitive maps into an expanded reliable knowledge framework. The price psy­chologists have to pay is to accept the nature of natural thinking, that is, fuzzy thinking. That's presumably not too hard. A little harder is the acceptance of multi-valued logic by incorporating fuzzy logic into the curriculum of psychology. That precisely is the price, for otherwise it wouldn't be a paradigm shift. Readers on the sceptical side should notice that cognitive science gets currently more advanced outside psychology, particularly in biology, physics, engineering, and computer science, so something is going astray in psychology. It risks to loose another professional domain, as so many before, and that in view of the fact that fuzzy logic has become an international multi-

X Preface

million dollar business. If nothing happens, the real loosers will be the students, for there is no excuse when faced with reality by say­ing "Fuzzy logic? I thought that's only for machine learning!". It certainly is not, because fuzzy knowledge engineering is becoming a part of cognitive psychology, cognitive ergonomics in applied psycho­logy, and engineering psychology in general. Bart Kosko's pioneering work on fuzzy causality not only extends John Stuart Mill's method of concomitant variation to an algebra of causality and to differential equations, it also turns Bertrand Russell's -characterization of the "law of causality" into proper causal reasoning. To quote Russell:

"No doubt the reason why the old "law of causality" has so long continued to pervade the books of philosophers is simply that the idea of a function is unfamiliar to most of them, and therefore they seek an unduly simplified statement. There is no question of repetitions of the "same" cause producing the "same" effect; it is not in any sameness of causes and effects that the constancy of scientific law consists, but in sameness of relations. And even "sameness of relations" is too simple a phrase; "sameness of dif­ferential equations" is the only correct phrase. It is impossible to state this accurately in non-mathematical language; the nea­rest approach would be as follows: "There is a constant relation between the state of the universe at any instant and the rate of change in the rate at which any part of the universe is changing at that instant, and this relation is many-one, i.e. such that the rate of change in the rate of change is determinate when the state of the universe is given." If the "law of causality" is to be something actually discoverable in the practice of science, the above proposition has a better right to the name than any "law of causality" to be found in the books of philosophers." Rus­sell,B. 1929 On the Notion of Cause with Applications to the Free-Will Problem. In Feigl, H. & Brodbeck, M. 1953 Readings in the Philosophy of Science, 387-407, Appleton-Century-Crofts, New York

The potential of fuzzy logic, in particular that of fuzzy cognitive maps, will shed more light on the subtleness of causality. This will be the main theme in part II of the book.

The third and final approach pursued in part III on evolutionary computing is based equally well on hypercubical calculus inasmuch as the Boolean hypercube Bn serves as the search space in function optimization and multivariate feature analysis, while the schemata hypercube sn = {0, l,*}n provides an analytical frame of reference for hyperplane analysis in genetic and autogenetic algorithms. The purpose of part III is two-fold: First, to examine where and in what

Preface XI

respects parity logic affects the foundations of evolutionary compu­ting. Second, to outline a conceptually and computationally coherent approach to a new type of genetic algorithms, called autogenetic al­gorithms (AGAs). It is not the intention to develop another theory for genetic computing. Instead, the last chapter on AGAs will explore the possibility of reducing the algorithmic structure to a minimum of computational complexity. This is formally motivated by algorithmic compression.

As to the methodological aspect of evolutionary computing it is important to note that we are mostly interested in multivariate search in complex feature spaces. To make the respective fundamentals of AGAs as selfcontained as possible, we present at first their conceptual framework, secondly their theoretical foundations by examining the hypercube sn = {0, 1, * }n, thirdly by elaborating on their computa­tional foundations, and finally by discussing their applications to uni­and multivariate search tasks in cognitive science and technology.

This book adopts the transdisciplinary view of modeling in science wholeheartedly and is therefore aimed at a large audience in hard and soft computing. Part I on parity logic is certainly more on the side of crisp computing and of special importance to readers interested in scientific modeling from scratch in computer science, informatics, signal and image processing, mathematics, physics, biology, and psy­chology. This holds in particular for parity integration and its as­sociated Langlet- and Shegalkin transforms as binary competitors to Fourier- and Morlet- viz. wavelet transforms. Part II on fuzzy logic is definitely on the side of soft computing and of particular relevance to causal knowledge engineering in psychology, medicine, sociology, political science, ecology, and economics. Part III on evolutionary computing belongs as a model-free estimation approach both to crisp and soft computing. Search and optimization abound in any of the above fields, so part III covers a truly multidisciplinary approach and is as such highly adaptable to intradisciplinary target questions and research strategies.

Acknowledgements

This research was supported in parts by the German Science Foun­dation (DFG), Grant Sche 298/5-2 to the late Prof. Dr. Eckart Schee­rer from the "Interdisciplinary Research Group on Cognitive Science" of the Universities Bremen and Oldenburg, Germany. Moreover, by

Xll Preface

Prof. Dr. Hans Colonius, Director of the Institute for Cognitive Science, University of Oldenburg, for providing office room and facilities to complete the book. I also would like to thank my friends and collea­gues PD Dr. Adele Diederich, Prof. Dr. Gisela Szagun, PD Dr. Volker Zschorlich, and Prof. Dr. Robert Megnet for their support and encou­ragements.

Part I on parity logic owes its deepest debts to the late Dr. Gerard A. Langlet for his advice, support and encouragements in the years from 1994 to 1996. At that time G. Langlet was the head of the Laboratoire d 'Informatique Theorique of the Commissariat d 'Energie Atomique (CEA) at Saclay, France. Communicating and cooperating with him on the subject matter was not only a lot of fun, but also an unforgettable learning history. I am also greatly indebted toM. Sylvain Baron, president of the "Association Francophone pour la promotion du langage APL" (AFAPL), as both G. Langlet and S. Baron supported my work by publishing parts of it in French in "Les Nouvelles d'APL". As. to the connection of parity logic and APL, I thank Dieter Latter­mann from "APL-Germany" and Adrian Smith from the "British APL Association" for inviting me as a speaker to the APL96 conference at Lancaster University in the summer of 1996. The conference inspi­red more intensive work and current activities with respect to a joint venture for advancing hypercubical calculus, but the time hasn't yet come to draw major conclusions about its payoff. As to the promo­tion of the book in France and Russia, I would like to thank Madame L. Lemagnen, President of the Association Franco-Russe "Sciences et Cosmos", for publishing the introductory summary in "Les Nouvelles d'APL".

Regarding part II on fuzzy logic I thank particularly Prof. Dr. Bart Kosko from the University of Southern California for sending me a number of valuable papers on fuzzy logic and fuzzy cognitive maps. His work has influenced my views of fuzzy logic eminently and for­med the idea of synthesizing parity logic, fuzzy logic, and evolutionary computing into hypercubical calculus. I am not satisfied with the re­sults obtained so far, but the vision to make it a computational power tool is all the more motivating, since the "State-Space as Hypercube" paradigm is present in so many diverse fields that it cries for unifica­tion. Many thanks are also due to my students who went patiently through the methodology of concept mapping, mind mapping, fuzzy cognitive mapping, and a dozen of fuzzy cognitive maps in the seminars

Preface Xlll

of 1996/97 and 1997. They furnished the sunny side of this research, thanks to all of them. A forthcoming seminar in the winter-semester 1998/99 on "Theoretical and Applied Fuzzy Logic" at the University of Oldenburg will link fuzzy cognitive maps with Kurt Lewin's field theory, for the latter is "a method of analyzing causal relations and of building scientific constructs".

The work of part III on evolutionary computing has been influenced mostly by Prof. Dr. John Holland, Prof. Dr. Ingo Rechenberg, and Prof. Dr. Hans-Paul Schwefel, but in view of hundreds of different models I have developed my own ideas, guided by parity logic and the canons of scientific modeling from scratch.

Most of all I want to thank Prof. Dr. Dr. Janusz Kacprzyk for endorsing the book's publication by Physica-Verlag, and Prof. Dr. Alf Zimmer for his commitment of time and energy to reviewing the initial draft of the book. Finally, I would like to thank Dr. Martina Bihn, Springer-Verlag, and Gabriele Keidel, Physica-Verlag, for arranging its final publication.

Oldenburg-Hatten, Germany, July 1998

Mike Zaus

Contents

Preface

1 Introduction 1.1 Parity Logic . . . . . . . . 1.2 Fuzzy Logic . . . . . . . . 1.3 Evolutionary Computing .

I Parity Logic

2 Mathematical Foundations of Parity Logic 2.1 The Space Bn = {0, l}n ..... 2.2 Fundamental Properties of XOR ... . 2.3 Foundations of Generalized XOR ... . 2.4 Motions, Inner Products, and Genitons .

vii

1 1 5 7

11

13 13 17 21 28

3 Binary Signal Analysis in Parity Logic 33 3.1 Standard Function Systems . . . . . . . . . . . . . . . . 35 3.2 Towards the Binary Counterpart of Fourier Analysis

and the Role of Pari tons . . . . . . . . . . . . . . . . 39 3.3 Analytical Signal Representations . . . . . . . . . . 43

3.3.1 Taylor Expansions and Binary Differentials 44 3.3.2 Spectral Representations . . . 47 3.3.3 Representation by Sequences 50

3.4 Shegalkin- and Langlet Transforms 52 3.4.1 The Concept of Transforms 52 3.4.2 Shegalkin Transforms . 54 3.4.3 Langlet Transforms . 58 3.4.4 Conclusions . . . . . . . 62

xvi Contents

4 Modeling Perception and Action in Parity Logic 65

5

6

4.1 The Nature of Efficient Action . . . . . . 66 4.2 The Conjugacy of Perception and Action 69

4.2.1 Ecological Physics . . 70 4.2.2 Ecological Psychology . . . . . . . 76

4.3 Intrinsic Measurement Bases . . . . . . . 78 4.3.1 Fuel-Coins and Intrinsic Measurement 81 4.3.2 Cantor's Discontinuum and Fractal Rescalability 83 4.3.3 Parity Logic and the Ecological Action Potential 88

4.4 Conclusions . · . . . . . . . . . . . . . . . . . . . . . . . . 95

Parity Logic Engines and Excitable Media 99 5.1 From Feedback Machines to Parity Logic Engines . . 100 5.2 Parity Logic Engines . . . . . . . . . . . . . . . . . . 103

5.2.1 Input Sensitivity of Parity Logic Engines . 105 5.2.2 The Elementary Sequence and the Geniton . 107 5.2.3 From Genitons to Paritons . 109 5.~.4 From Paritons to Fanions .... . 114

5.3 Excitable Media and Paritons . . . . . . . 117 5.3.1 Paritons and Temporal Records . . 118 5.3.2 Reconsidering Parity Logic at a Glance . 121

5.4 Towards Artificial Retina Modeling with Fanions . 123 5.4.1 Topologies of Resistive Networks . 123 5.4.2 The Fanion's Network . 124

5.5 Conclusions . . . . . . . . . . . . . . . . . 126

Transdisciplinary Perspectives of Parity Logic 129 6.1 The Scope of Parity Integration . . . . 130 6.2 Perspectives of Applied Parity Logic ...... . 142

II Fuzzy Logic 149

7 Mathematical Foundations of Fuzzy Logic 151 7.1 The Space In= [0, 1]n ........... , . . . . . . 153 7.2 Conceptual and Computational Foundations . . . . . . 161 7.3 Emergent Meaning, Fuzzy Entropy, and Subsethood . 173

7.3.1 Subsethood, the Whole in Part, and Fuzzy XOR 181 7.3.2 Fuzzy Entropy, the Whole in Part, and the Yin-

Yang Equation ................... 185

Contents xvii

7.4 Generalized Fuzzy Inner- and Outer Products . 193 7.4.1 Generalized Inner Products . . 195 7.4.2 Generalized Outer Products . . . . . . . 198

8 Causal Modeling with Fuzzy Cognitive Maps 201 8.1 On the History of Cognitive Maps . . . . . . . 203 8.2 Fuzzy Cognitive Maps and Causal Reasoning . 208 8.3 Formal Properties of Causal Algebra . . 218 8.4 Constructing and Aggregating FCMs . . . 228

8.4.1 Interactive FCM Construction . . 229 8.4.2 Interpersonal FCM Aggregation . 236

8.5 Real and Virtual Worlds FCMs . . . . . . 242 8.6 Continuous FCMs and Methodological Issues . 262

8.6.1 Adaptive FCMs without Limit Cycles . 267 8.6.2 Evaluation, Limitation, and Implementations . 269

8. 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 275

III Evolutionary Computing 277

9 Foundations of Evolutionary Computing 279 9.1 Scientific Modeling from Scratch ............. 281 9.2 Parity Integration in Evolutionary Computation .... 284 9.3 Algorithmic Compression through Langlet Transforms . 287 9.4 N-fold Symmetry Operators and Autogenetic Growth . 291 9.5 Parity Logic Engines and Evolutionary Computing . 299 9.6 Conclusions ......................... 306

10 Fundamentals of Autogenetic Algorithms 309 10.1 A Conceptual Framework of Evolutionary Computing . 310

10.1.1 Defining Search Problems ............. 311 10.1.2 The Stages of Artificial Evolution ......... 313

10.2 Theoretical Foundations of Genetic and Autogenetic Al-gorithms ........................... 319 10.2.1 Hypercubical Calculus and Implicit Parallelism . 319 10.2.2 Genetic vs. Autogenetic Algorithms with Paper

and Pencil . . . . . . . . . . . . . . . . . . 327 10.3 Computational Foundations of GAs and AGAs . 337

10.3.1 Representation and Coding . 338 10.3.2 Evaluation and Scaling . . . . . . . . . . . 343

xviii Contents

10.3.3 Selection and Sampling . . . . . . . . . . . . . 347 10.3.4 Adaptive Mutation Access Modes and Rates . 351 10.3.5 Recombination and Crossover . . . . . . . . . 355 10.3.6 Parity Logic Tools for AGAs . . . . . . . . . 358

10.4 Uni- and Multivariate Search with GAs and AGAs . 365 10.4.1 Elementary Function Optimization . . . . . . 365 10.4.2 Pattern Search in High-dimensional Hypercubes 371 10.4.3 Search of Extrema in Response Surfaces ..... 377

10.5 Multivariate Search in Face Space ............ 383 10.5.1 Some Eigenface Image Data Technology Back-

ground ........................ 384 10.5.2 GAs and AGAs in Face Space Search . . . . . . 389 10.5.3 Extensions to Domain Specific Attribute Spaces 397

10.6 Conclusions ......................... 398

Bibliography 401

Index 417