Most of the material in this book has been presented in lectures at Brown University, either in courses taught in the Division of Applied Mathematics or in the author's Re search Seminar in Pattern Theory. I would like to thank the several members of the Division of Applied Mathematics that have participated in the discussions and in particular w. Freiberger, S. Geman, C.-R. Hwang, D. McClure and P. Thrift. I would also like to thank F. John, J. P. LaSalle, and L. Sirovich for accepting the manuscript for the Series Applied Mathematical Sciences published by Springer-Verlag. The research reported here has been supported by the National Science Foundation, Office of Naval Research and the Air Force Office of Scientific Research. I am grateful for the active interest and help given in various ways by Dr. Eamon Barrett, Dr. Kent Curtis, Dr. Robert Grafton and Dr. I. Shimi of these agencies. I also thank C.-R. Hwang and P. Thrift for help with proofreading. I am indebted to Mrs. E. Fonseca for her careful pre paration of the manuscript, to Miss E. Addison for helping me with the many diagrams, and to Mrs. K. MacDougall for the final typing of the manuscript. Ulf Grenander Providence, Rhode Island October 1980 v TABLE OF CONTENTS Page INTRODUCTION . . . . ."

Many persons have helped the author with comments and corrections, and I would like to mention D. E. McClure, I. Frolow, J. Silverstein, D. Town, and especially W. Freiberger for his helpful suggestions and encouragement. The work in Chapters 6 and 7 has been influenced and stimulated by discussions with other members of the Center for Neural Sciences, especially with L. Cooper and H. Kucera. I would like to thank F. John, J. P. LaSalle, L. Sirovich, and G. Whitham for accepting the manuscript for the series Applied Mathematical Sciences published by Springer-Verlag. This research project has been supported by the Division of Mathematical and Computer Sciences of the National Science Foundation and (the work on language abduction, pattern processors, and patterns in program behavior) by the Information Systems Program of the Office of Naval Research. I greatly appreciate the understanding and positive interest shown by John Pasta, Kent Curtiss, Bruce Barnes, Sally Sedelov vi PREFACE and Bob Agins of the Foundation, and by Marvin Denicoff of the Office of Naval Research. I am indebted to Mrs. E. Fonseca for her untiring and careful preparation of the manuscript, to Miss E. Addison for her skillful help with the many diagrams, and to S.V. Spinacci for the final typing. I gratefully acknowledge permission to reproduce figures, as mentioned in the text, from Cambridge University Press and from Hayden Book Company. Also, to Professor J. Carbury for permission to use his illustration on page 704.

This book consists of updated and considerably expanded lecture notes. I am grateful to F. John, J. P. LaSalle, L. Sirovich and G. Whitham for accepting this preliminary manu script for the series Applied Mathematical Sciences published by Springer-Verlag. Many colleagues and students have helped with their comments and suggestions. Among them are D. E. McClure, I. Frolow, J. Silverstein, R. A. Vitale, and I would like to thank especially W. Freiberger for all his encouragement and help in many different ways. The work has been supported by the Division of Mathematica and Computer Sciences of the National Science Foundation, and I appreciate the positive interest shown by John Pasta, Kent curtis, Bruce Barnes and other program directors, over the years. I am indebted to E. Fonseca for her untiring and careful preparation of the manuscript, to E. Addison for her skillful help with the many diagrams, and to K. H. Avery, D. P. Libutti, K. MacDougall and S. V. Spinacci for the final typing. I gratefully acknowledge permission to reproduce figures, as mentioned in the text, to: American Elsevier Publishing Company, Inc., A. J. Larkin, and M. Gardner. I dedicate this work to my wife, Emma-Stina. June, 1975 Ulf Grenander, Providence TABLE OF CONTENTS INTRODUCTION 1 CHAPTER 1. THE GENERATORS 6 l. Generators and Their Properties 6 2. Abstract Generators 10 3. Concrete Generators 13 4."