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Mathematical Aspects of Artificial Intelligence http://dx.doi.org/10.1090/psapm/055 Selected Titles in This Series 55 Frederick Hoffman, Editor, Mathematical aspects of artificial intelligence (Orlando, Florida, January 1996) 54 Renato Spigler and Stephanos Venakides, Editors, Recent advances in partial differential equations (Venice, Italy, June 1996) 53 David A. Cox and Bernd Sturmfels, Editors, Applications of computational algebraic geometry (San Diego, California, January 1997) 52 V. Mandrekar and P. R. Masani, Editors, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, Michigan, 1994) 51 Louis H. Kauffman, Editor, The interface of knots and physics (San Francisco, California, January 1995) 50 Robert Calderbank, Editor, Different aspects of coding theory (San Francisco, California, January 1995) 49 Robert L. Devaney, Editor, Complex dynamical systems: The mathematics behind the Mandlebrot and Julia sets (Cincinnati, Ohio, January 1994) 48 Walter Gautschi, Editor, Mathematics of Computation 1943-1993: A half century of computational mathematics (Vancouver, British Columbia, August 1993) 47 Ingrid Daubechies, Editor, Different perspectives on wavelets (San Antonio, Texas, January 1993) 46 Stefan A. Burr, Editor, The unreasonable effectiveness of number theory (Orono, Maine, August 1991) 45 De Witt L. Sumners, Editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992) 44 Bela Bollobas, Editor, Probabilistic combinatorics and its applications (San Francisco, California, January 1991) 43 Richard K. Guy, Editor, Combinatorial games (Columbus, Ohio, August 1990) 42 C. Pomerance, Editor, Cryptology and computational number theory (Boulder, Colorado, August 1989) 41 R. W. Brockett, Editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, Editor, Matrix theory and applications (Phoenix, Arizona, January 1989) 39 Robert L. Devaney and Linda Keen, Editors, Chaos and fractals: The mathematics behind the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis, Editor, Computational complexity theory (Atlanta, Georgia, January 1988) 37 Henry J. Landau, Editor, Moments in mathematics (San Antonio, Texas, January 1987) 36 Carl de Boor, Editor, Approximation theory (New Orleans, Louisiana, January 1986) 35 Harry H. Panjer, Editor, Actuarial mathematics (Laramie, Wyoming, August 1985) 34 Michael Anshel and William Gewirtz, Editors, Mathematics of information processing (Louisville, Kentucky, January 1984) 33 H. Peyton Young, Editor, Fair allocation (Anaheim, California, January 1985) 32 R. W. McKelvey, Editor, Environmental and natural resource mathematics (Eugene, Oregon, August 1984) 31 B. Gopinath, Editor, Computer communications (Denver, Colorado, January 1983) 30 Simon A. Levin, Editor, Population biology (Albany, New York, August 1983) 29 R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton, Applied cryptology, cryptographic protocols, and computer security models (San Francisco, California, January 1981) 28 R. Gnanadesikan, Editor, Statistical data analysis (Toronto, Ontario, August 1982) 27 L. A. Shepp, Editor, Computed tomography (Cincinnati, Ohio, January 1982) (Continued in the back of this publication) AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures published as a subseries of Proceedings of Symposia in Applied Mathematics Proceedings of Symposia in APPLIED MATHEMATICS Volume 55 Mathematical Aspects of Artificial Intelligence American Mathematical Society Short Course January 8-9, 1996 Orlando, Florida Frederick Hoffman Editor & American Mathematical Society " Providence, Rhode Island LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE MATHEMATICAL ASPECTS OF ARTIFICIAL INTELLIGENCE HELD IN ORLANDO, FLORIDA JANUARY 8-9, 1996 The AMS Short Course Series is sponsored by the Society's Program Committee for National Meetings. The series is under the direction of the Short Course Subcommittee of the Program Committee for National Meetings. 1991 Mathematics Subject Classification. Primary 68-xx; Secondary 03-xx, 05-xx, 51-xx, 60-xx, 90-xx. Library of Congress Cataloging-in-Publication Data Mathematical aspects of artificial intelligence : American Mathematical Society short course, January 8-9, 1996, Orlando, Florida / Frederick Hoffman, editor. p. cm. — (Proceedings of symposia in applied mathematics ; v. 55. AMS short course lecture notes) Includes bibliographical references and index. ISBN 0-8218-0611-4 (alk. paper) 1. Artificial intelligence—Mathematics—Congresses. I. Hoffman, Frederick, 1937- . II. American Mathematical Society. III. Series: Proceedings of symposia in applied mathemat• ics ; v. 55. IV. Series: Proceedings of symposia in applied mathematics. AMS short course lecture notes. Q335.M33756 1998 006.3'0151—dc21 98-4693 CIP Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 1998 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 03 02 01 00 99 98 Contents Preface ix Introduction and History FREDERICK HOFFMAN 1 Reasoning about Time MARTIN CHARLES GOLUMBIC 19 Orderings in Automated Theorem Proving HELENE KIRCHNER 55 Programming with Constraints: Some Aspects of the Mathematical Foundations CATHERINE LASSEZ 97 The Basis of Computer Vision VISHVJIT NALWA 139 Outsearching Kasparov MONTY NEWBORN 175 Mathematical Foundations for Probability and Causality GLENN SHAFER 207 Index 271 Preface Artificial Intelligence (AI) is an important and exciting field. It is an active research area and is considered to have enormous research opportunities and great potential for applications. At the same time, AI is highly controversial. There is a history of great expectations, and large investments, with some notable short• falls and memorable disappointments. We are, of course, most concerned with connections between AI and mathematics. In fact, one of the major controversies regarding AI is the issue of just how mathematical a field it is, or should be. The major research journal in the field, Artificial Intelligence, publishes a large number of papers with heavy mathematical content, although many authorities in the field question this emphasis. For one example, the currently hot AI topic of "data min• ing," obtaining information from incomplete or "noisy" sources, has a necessary mathematical component, and, in general, theoretical AI, like theoretical computer science, is at least arguably a mathematical science. No matter where we come down within the range of "just how mathematical is it?", there is a close enough tie to justify the AMS Short Course, and this volume. We feel that mathematics and mathematicians have a lot to contribute to AI, and that AI has excellent potential for fruitful applications to mathematics. The purpose of the course, and this book, is to introduce mathematicians and others to some of the more mathematical areas within AI, both for the intrinsic value of the material as well as with a view toward stimulating the interest of people who can contribute to the field or use it in their work. We must point out that the AMS has had special sessions and invited talks in the past on AI, so we are in no sense the first to try to draw the community's attention to the field. This volume begins with a brief introduction to the field of AI, to provide enough general information so that readers can place the remaining chapters in perspective. We provide the necessary definitions and a rather perfunctory outline, with a minimal amount of history. Emphasis within this chapter is somewhat driven by the topics of the remaining chapters. One of the best known, and most controversial topics of AI is computer chess. Early on in the history of the field, grandiose claims were made for the near-term success of computers as chess champions. The failure of the field to produce an artificial chess champion within the predicted time-line was used to attack AI and its practitioners unmercifully. In reality, while the time-line was unduly optimistic, it now seems that the initial claims have been met, and the attacks on the field because of failures of computer chess will be replaced by controversy about just how intelligent the artificial chess champions really are. In this volume, we present a chapter on computer chess by Monty
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