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 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 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 Newborn. Professor Newborn's comments were featured in news coverage of the famous match in February, 1996, between the program Deep Blue and the human chess champion, and he has written a

IX x PREFACE book on the match. Because of the volatility of the topic, this chapter is "frozen in time" immediately before the February match. The techniques discussed are highly mathematical, involving graph theory, combinatorics and probability and , among others. Glenn Shafer, whose seminal work on probabilistic reasoning has made him a household word in the AI applications area of expert systems, and in AI in general, continues, in this volume, his development of probability through causal probablity trees. The topic is related to an important practical issue in AI, especially in expert systems; that is, the extent to which causal information, rather than case-based, a posteriori data, should be used in making decisions. His chapter consists of new material, first presented by Professor Shafer in research papers and a 1996 book, with some results appearing here for the first time; at the same time, it has its origins in the early history of . We find it fascinating that the use of probability in AI has fostered exciting developments in the foundations of probability theory itself, especially since the new insights hark back to the classical beginnings of the field. The chapter is of great interest in its own right, as a contribution to the foundations of probability theory, and also serves as an example both of mathematics serving AI, and of mathematics being developed because of lessons learned from AI. One of the most difficult problems in reasoning in AI has to do with handling time. Temporal reasoning is part of the huge problem of planning actions to achieve objectives in a dynamic world. It ties into the famous philosophical bugbear of AI- the Frame Problem. Martin Golumbic's chapter explores the topic of temporal reasoning. This is a highly mathematical part of AI, with ties to logic as well as to combinatorics and graph theory. The real-world situations studied in both this chapter and the preceding one provide intriguing settings for the theoretical issues they develop. Mathematical AI is frequently associated with the "logicist" school within AI, and is heavily based on mathematical logic. The chapter by Golumbic is concerned in large part with logic as well as with graph theory. The next two chapters are even more heavily involved with logic. When an AI system reasons, it generally uses algorithms that are guaranteed to succeed, but require success within certain time frames to be of any real value. While providing a great deal of general information on logical reasoning in AI, Helene Kirchner explains techniques, using order rela• tions, to make deduction more efficient. These systems are highly mathematical in their strucure, and thus provide another example of synergy-the systems described here are closely related to those used by Larry Wos and his group at Argonne, who devote a lot of their attention to solving problems and proving theorems in pure mathematics. In fact, Dr. Wos and his work have been studied, and honored, by the AMS in the past, and have also been featured in national media regularly, including December, 1996 coverage of new mathematical results by the New York Times. In general, decision-making in AI involves optimization subject to constraints. In addition, constraints are added to reasoning systems to make them work more ef• ficiently. Once again, AI and mathematics each serve the other. Catherine Lassez's chapter deals with constraint logic programming. The applications of the methods described here range from pure mathematics to optimization in some very applied disciplines, like operations research and financial mathematics. It is unsurprising that the topic is intimately tied to linear programming. The nature of the relation- PREFACE XI ship, and the structure of the theory described here, are fascinating. In fact, the crucial theorems go back to Fourier, the unsung father of linear programming. Computer vision is obviously necessary for the creation of artificially intelligent beings to function in the real world, thinking robots, as it were. One could argue that, since lower, "unintelligent" animals can see, and blind humans can think and reason, that computer vision is not a legitimate part of the subject of AI. This view has much to commend it, since computer vision presents very difficult problems, and while it is a very mathematical field, the mathematics, at least in part, is both difficult and different from the mathematics used in other parts of AI. Because of the heavy mathematical component of vision, though, and because of image understanding, a part of vision, is part of AI, we have chosen to include a chapter on vision. Vishvjit Nalwa gently guides his readers on a tour of computer vision, giving brief exposure to various facets of the field, and tying it to several areas of mathematics, from combinatorics, probability, and geometry to partial differential equations. There is much more, and much more that is mathematical, to AI than we are able to touch on here. We sincerely hope that we have provided enough of the flavor of the field to have whetted the appetites of some of our audience, and we look forward to their contributing to and using the results of AI. We welcome comments from our readers. Index

1983 World Championship, 194 BUGGY, 7

A*-algorithm, 9 C. A. Gunter, 16 A. D. Sands, 16 calculus of , 2 Aaron, 7 canonical constraint set, 28 ACM, 192 Carl Hewett, 5 ACM CCC, 181 catalog, 231 ACM Computing Week '96, 175 Catherine Lassez, 4 ACSP, 38 causal logic, 269 after, 261 causality, 216 ALICE, 5 causally independent, 215, 221, 236 alignment, 249 causally uncorrelated, 215, 221, 236 aligns with, 247 "cause", 209 all consistent solutions problem, 38 center of projection, 150 Allen, James, 35 central projection, 150 allows, 226, 247 chance situations, 232 alpha-beta, 193 CHESS 3.0, 178 alpha-beta algorithm, 186 CHESS 4.0, 178, 193 alpha-beta search, 195, 199, 203 CHESS 4.5, 193 always foretells, 226, 248 CHESS 4.6, 178 always strictly foretells, 248 CHESS CHALLENGER, 192 AM, 6 chess circuitry, 193 Analysis of Variance, 212 chess circuits, 189 Anatoly Karpov, 176 CHESS GENIUS, 177 AND-OR graphs, 10 chess ratings 176, 177, 178 angle of emittance, 159 , 209 angle of incidence, 159 Church-Rosser property, 63 apparent contours, 154 clade, 226, 249 archaeology, 42 clash, 201 Atkin, 193 class rewrite system, 67 atomic relations, 37 coarsening, 237 AURA, 3, 5, 15 combinatorics, 19 automated theorem proving, 193 compatible, 246 autonomous agents, 45 completion procedure, 64 AWIT, 194 computational complexity, 22 Computer vision, 140 backtracking algorithm, 34 conditional ordered critical pair, 83 Banerji, 2 conditional ordered narrowing, 83 basic constructions, 257 conditional rewrite system, 72 before, 261 consistent scenerio, 24 BELLE, 177-181, 189, 195 constrained critical pair, 87 Bellman, 30 constrained equality, 86 Bellman-Ford algorithm, 32 constraint networks, 27, 34 Bender, 16 constraint propagation, 20, 26, 33 Boolean algebra, 260 constraint satisfaction problems, 34 brightness, 151 Convex hull, 122 brute force, 181 Correlation, 216 272 INDEX covariance, 211 expert systems, 2 Cray, 202 Experts, 176 CRAY BLITZ, 192, 202 extended search, 181, 193, 203 critical pair, 64 extension, 261 cusp, 158 extreme point method, 126 cut, 229, 249 Fail, 263 D. Subramanian, 16 Failure, 262 DAI, 45, 46 fair-bet catalog, 232 Danny Kopec, 177 FIDE, 176, 177 David Mumford, 16 Fischer, 178 decision situations, 232 Fix, 262 decomposes, 249 Flinders Petrie, 42 Decomposition, 256 Floyd-Warshall algorithm, 32-34 Deductive Databases, 46 focus of contraction (FOC), 168 DEEP BLUE, 2, 181, 189, 192, 194 focus of expansion (FOE), 168 DEEP THOUGHT, 189 fold, 158 Dempster-Shafer Theory, 3, 13 forbears, 235, 247, 257 DENDRAL, 6, 13 foreshortening, 162 depth-first minimax search, 186 foretells, 224, 247, 261 depth-first search, 182, 193 forward pruning, 194 determinate process, 231 Fourier's algorithm, 113 difference constraints, 30 Fourier's elimination, 112 diffraction, 151 fragment, 40, 41, 42 diffuse surface, 160 frame problem, 3, 14 disjoint, 226 FRITZ3, 177 distance matrix, 30 fuzzy logic, 14 distributed artificial intelligence, 26, 45 diverge, 257 Game trees, 10 divergent, 224 Garry Kasparov, 2, 175, 176 divergent clades, 256 Gata Kamsky, 176 diverges, 247 General Problem Solver, 5, 10 DNA, 19, 24, 39 general purpose, 147 Donald Johnson algorithm, 32 general-viewpoint assumption, 155 Doob catalog, 233 generalized linear program, 125 Douglas Lenat, 6 Genius 3.0, 177 DUCHESS, 178 Gentzen, 5 dynamic, 195 geometric stereo, 163 Gillogly, 195 Ebeling, 189 Ginsberg, 16 edge, 152 Glenn Shafer, 13 Edge Detection, 171 Golumbic, Martin C, 36, 39, 41 Elementary Refinements, 217 graph theoretic techniques, 36 Eliza, 5 Greenblatt MACHACK, 175 endpoint sequence problem, 38 EQP, 3, 15, 16 Helene Kirchner, 3, 11 equals, 265 Hantao Zhang, 15 equational constraint, 85 Hao Wang, 5 equipollent, 248 Harold Cohen, 7 ESP, 38 Hart, 9 event space, 208, 242, 249 hash code, 197, 200-203 event tree, 207, 209 hash tables, 196, 203 Event Trellises, 221 hashing error, 201 events, 210 hashing function, 202 Expand, 263 Hasse diagram, 223 expectation, 231, 233 head, 229 expected value, 211, 231 Herbert Simon, 1, 4 experiment, 235 heuristic search, 20, 33 expert, 193 Hill-climbing, 15 INDEX 273

HITECH, 189 Logic, 26 hits, 197 logic programming, 26, 44 Hsu, 189 Logic Theory Machine, 5 Huffman code, xx lower expectation, 233 Humean event, 229 lower expected value, 233 lower probability, 234 IBM, 6 IBM DEEP BLUE, 175 mandatory relations, 246 IGMs, 176, 177 Marion Tinsley, 2 IM, 177 Marsland, 194, 204 image compression, 141 Martin Golumbic, 3, 14, 19 image enhancement, 141 martingales, 208, 229 image irradiance, 151 Massachusetts State Championship, 175 Image processing, 141 Master, 176, 177 image restoration, 141 mathematical programming, 20 image understanding, 140 may align with, 246 implicit equalities, 117 may allow, 246 implies, 224, 247 may diverge from, 244 IMs, 176 may forbear, 244 inductive theorem, 74 may foretell, 246 influences, 221 may imply, 246 intelligent backtracking, 33 may require, 246 International Chess Federation, 176 may strictly allow, 246 International Grandmasters, 176 may strictly foretell, 244 International Masters, 176 may strictly require, 244 interval algebra, 26, 35, 36, 37, 42 Mephisto Genius 2.0, 177 interval graph sandwich problem, 39, 40 Merger, 255 interval realization, 38 merges, 244 Interval Satisfiability, 37 metric temporal constraint problem, 22, 27, interval satisfiability problem, 38 29, 33 ISAT, 38, 41, 42 minimal labeling problem, 38 Isomorphism, 265 minimax algorithm, 182-184, 186 iteratively-deepening depth-first search, 193 minimax search, 186 iteratively-deepening search, 179, 182, 193 Minnesota State Championship, 193 195 missionaries and cannibals, 8 Mitchell, 16 J. A. Robinson, 11 MLP, 38 Jean-Louis Lauriere, 5 Modal logic, 14 Jean-Michel Morel, 16 Moivrean event, 224 John McCarthy, 8 molecular biology, 19, 24, 39 John Sowa, 10 Monty Newborn, 2 motion parallax, 166 KAISSA, 178 move ordering, 195, 203 Kasparov, 175, 177, 181, 203 MTCP, 22, 27, 29, 33, 34, 37 Kautz, Henry, 42 multiprocessing, 201 Ken Thompson, 177 multiprocessing systems, 189 killer heuristic, 195 mutilated checkerboard, 8 MYCIN, 6, 13 Lambertian, 160 Larry Wos, 3, 15 Nokel, K., 42 latin squares, 15 neural networks, 8 , 237 Newborn, 204 Levy, 204 Newell, 5 limbs, 154 Nielssen, 9 line drawing, 155 Nilsson, 2 Line-Drawing Interpretation, 171 non-monotonic logics, 20 linear programming, 30 linear sign, 215, 221 opening book, 175, 203 LISP, 8 operations research, 20 274 INDEX

OPS, 14 rule of iterated expectation, 233 optional relations, 244 rule-based systems, 11 ordered completion, 76 ordered critical disequality, 77 sample space, 210, 224 ordered critical pair, 77 Samuel's checkers player, 2 ordering constraint, 85 saturation of a set of clauses, 83 OSTRICH, 178, 192 scaling, 162 Otter, 3, 5, 15, 16 scene radiance, 151 overlap, 244, 257 Schaeffer, 204 scope, 235 parallel search, 193 scoring function, 182, 183, 186, 189, 196, 203 parametric queries, 108 Senior Masters, 176 partial ordering, 221 sequenced, 222 path, 224 Sergio Solimini, 16 pattern classification, 141 seriation problem, 24, 42 Pattern recognition, 141 Seymour Benzer, 39 Paul Masson American Chess Classic, 193 shading, 158 perspective projection, 150 Shamir, Ron, 36, 39, 41 phase angle, 159 Shaw, 5 piece-square table, 202 shortest path problem, 32 pinhole camera, 149 SHRDLU, 5 Planner, 5 Shulz, 177 point algebra, 42 Simon, 5 precedence relation, 222 simple temporal problem, 29, 34 precedes, 248 simplex, 127 predecessor, 222 simplification ordering, 59 preimage, 148 Simulated annealing, 15 prerequisite, 226 singular extension heuristic, 194 principal variation splitting algorithm, 192 situation, 210 probability, 211, 231 situation calculus, 269 probability catalog, 231 Skolem, 11 probability tree, 208, 209 Slate, 193 process, 230 spatiotemporal, 166 Prolog, 3, 5, 8 Spatiotemporal Coherence Detection, 172 proof by refutation, 76 specular surface, 160 STAR SOCRATES, 192 qualitative relations, 24 STARTECH, 192 Quasi-dual, 123 static, 195 quasi-linear combination, 109 STEAMER, 7 Stereo, 162 Rl, 6 Stereoscopic Correspondence Establishment, Ramon Lull, 5 171 Raphael, 9 stochastic processes, 207 rating, 179, 180, 181 Stopping a Martingale, 232 Rebel 6.0, 177 STP, 29, 34 reduction ordering, 58 strictly allows, 247 refinement, 208, 216, 237, 266 strictly foretells, 247 refines, 247 strictly precedes, 222, 248 reflexive resolvent, 83 strictly requires, 247 regression coefficients, 215 strong solvability, 116 requires, 247 Subordination, 265 resolution, 11, 262 successor, 222 resolution refutation, 11 SUN PHOENIX, 192 resolution-based theorem provers, 15 Surface Representation, 172 Resolve, 263 Swedish Rating List, 177 restricted domain, 40-42 synthetic annealing, 3 rewrite system, 62 rewriting induction, 74 T-H Ngair, 16 Robbins Conjecture, 16 T-junction, 158 tail, 229 TECH, 195 temporal constraint network, 29 temporal databases, 26, 46 temporal logic, 26, 43, 44, 208, 269 temporal reasoning, 19-21, 37 terminating process, 231 Terry Winograd, 5 texture, 160 Thinking on the opponent's time, 203 Third Godesberg GM Tournament, 177 Thomas Mitchell, 14 Thompson, 178-180, 189 tolerates, 247 transposition table, 193, 197, 199-201 transposition table hits, 197 transposition tables, 195, 196, 198, 203 tree, 223 trellis, 222 triangulation, 163 type theory, 265, 269 ultrafilter, 266 uncorrelated, 235 unification algorithm, 11 United States Chess Federation, 176 upper expectation, 234 upper expected value, 234 upper probability, 234 USCF, 176, 178 value ordering, 35 vanBeek, Peter, 42 variable, 210, 224 variable ordering, 35 variance, 211 version spaces, 14, 16 Vic Nalwa, 16 viewpoint-dependent edge, 154 viewpoint-independent edge, 153 Villain, Mark, 42 Viswanathan Anand, 176 Vladimir Kramnik, 176

W. McCune, 16 Wang, 10 WAYCOOL, 192 Webber, A., 41 Weizenbaum, 5 windows, 195

XCon, 6

Yale Shooting Problem, 21, 44

Zobrist, 202 ZUGSWANG, 192 Selected Titles in This Series (Continued from the front of this publication)

26 S. A. Burr, Editor, The mathematics of networks (Pittsburgh, Pennsylvania, August 1981) 25 S. I. Gass, Editor, Operations research: mathematics and models (Duluth, Minnesota, August 1979) 24 W. F. Lucas, Editor, Game theory and its applications (Biloxi, Mississippi, January 1979) 23 R. V. Hogg, Editor, Modern statistics: Methods and applications (San Antonio, Texas, January 1980) 22 G. H. Golub and J. Oliger, Editors, Numerical analysis (Atlanta, Georgia, January 1978) 21 P. D. Lax, Editor, Mathematical aspects of production and distribution of energy (San Antonio, Texas, January 1976) 20 J. P. LaSalle, Editor, The influence of computing on mathematical research and education (University of Montana, August 1973) 19 J. T. Schwartz, Editor, Mathematical aspects of computer science (New York City, April 1966) 18 H. Grad, Editor, Magneto-fluid and plasma dynamics (New York City, April 1965) 17 R. Finn, Editor, Applications of nonlinear partial differential equations in mathematical physics (New York City, April 1964) 16 R. Bellman, Editor, Stochastic processes in mathematical physics and engineering (New York City, April 1963) 15 N. C. Metropolis, A. H. Taub, J. Todd, and C. B. Tompkins, Editors, Experimental arithmetic, high speed computing, and mathematics (Atlantic City and Chicago, April 1962) 14 R. Bellman, Editor, Mathematical problems in the biological sciences (New York City, April 1961) 13 R. Bellman, G. Birkhoff, and C. C. Lin, Editors, Hydrodynamic instability (New York City, April 1960) 12 R. Jakobson, Editor, Structure of language and its mathematical aspects (New York City, April 1960) 11 G. Birkhoff and E. P. Wigner, Editors, Nuclear reactor theory (New York City, April 1959) 10 R. Bellman and M. Hall, Jr., Editors, Combinatorial analysis (New York University, April 1957) 9 G. Birkhoff and R. E. Langer, Editors, Orbit theory (Columbia University, April 1958) 8 L. M. Graves, Editor, Calculus of variations and its applications (University of Chicago, April 1956) 7 L. A. MacColl, Editor, Applied probability (Polytechnic Institute of Brooklyn, April 1955) 6 J. H. Curtiss, Editor, Numerical analysis (Santa Monica City College, August 1953) (See the A MS Catalog for earlier titles.) ISBN 0-8218-0611-4

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