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 D e W i t t 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 Wil l iam Gewirtz , Editors, Mathematics of information

processing (Louisville, Kentucky, January 1984)

33 H. P e y t o n 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 S imon A. Levin, Editor, Population biology (Albany, New York, August 1983)

29 R. A. DeMil lo , 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)

http://dx.doi.org/10.1090/psapm/055

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

L E C T U R E N O T E S P R E P A R E D F O R T H E

A M E R I C A N M A T H E M A T I C A L S O C I E T Y S H O R T C O U R S E

M A T H E M A T I C A L A S P E C T S O F A R T I F I C I A L I N T E L L I G E N C E

H E L D IN O R L A N D O , F L O R I D A

J A N U A R Y 8 - 9 , 1996

The AMS Short Course Series is sponsored by the Society's Program Committee for National Meetings. T h e series is under the direction of the Short Course

Subcommittee of the Program Committee for National Meetings.

1991 Mathematics Subject Classification. P r i m a r y 68-xx; Secondary 03-xx , 05-xx, 51-xx, 60-xx, 90-xx.

Library of Congress Cataloging-in-Publicat ion D a t a

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 mathemat­ics ; 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.

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@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

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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 short­falls 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 min­ing," 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 statistics, 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 probability theory. 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 rela­tions, 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 ef­ficiently. 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

A*-algorithm, 9 A. D. Sands, 16 Aaron, 7 ACM, 192 ACM CCC, 181 ACM Computing Week '96, 175 ACSP, 38 after, 261 ALICE, 5 alignment, 249 aligns with, 247 all consistent solutions problem, 38 Allen, James, 35 allows, 226, 247 alpha-beta, 193 alpha-beta algorithm, 186 alpha-beta search, 195, 199, 203 always foretells, 226, 248 always strictly foretells, 248 AM, 6 Analysis of Variance, 212 Anatoly Karpov, 176 AND-OR graphs, 10 angle of emittance, 159 angle of incidence, 159 apparent contours, 154 archaeology, 42 Atkin, 193 atomic relations, 37 AURA, 3, 5, 15 automated theorem proving, 193 autonomous agents, 45 AWIT, 194

backtracking algorithm, 34 Banerji, 2 basic constructions, 257 before, 261 BELLE, 177-181, 189, 195 Bellman, 30 Bellman-Ford algorithm, 32 Bender, 16 Boolean algebra, 260 brightness, 151 brute force, 181

BUGGY, 7

C. A. Gunter, 16 calculus of probabilities, 2 canonical constraint set, 28 Carl Hewett, 5 catalog, 231 Catherine Lassez, 4 causal logic, 269 causality, 216 causally independent, 215, 221, 236 causally uncorrelated, 215, 221, 236 "cause", 209 center of projection, 150 central projection, 150 chance situations, 232 CHESS 3.0, 178 CHESS 4.0, 178, 193 CHESS 4.5, 193 CHESS 4.6, 178 CHESS CHALLENGER, 192 chess circuitry, 193 chess circuits, 189 CHESS GENIUS, 177 chess ratings 176, 177, 178 Christiaan Huygens, 209 Church-Rosser property, 63 clade, 226, 249 clash, 201 class rewrite system, 67 coarsening, 237 combinatorics, 19 compatible, 246 completion procedure, 64 computational complexity, 22 Computer vision, 140 conditional ordered critical pair, 83 conditional ordered narrowing, 83 conditional rewrite system, 72 consistent scenerio, 24 constrained critical pair, 87 constrained equality, 86 constraint networks, 27, 34 constraint propagation, 20, 26, 33 constraint satisfaction problems, 34 Convex hull, 122 Correlation, 216

272

covariance, 211

Cray, 202

CRAY BLITZ, 192, 202 critical pair, 64 cusp, 158 cut, 229, 249

D. Subramanian, 16 DAI, 45, 46 Danny Kopec, 177 David Mumford, 16 decision situations, 232 decomposes, 249 Decomposition, 256 Deductive Databases, 46 DEEP BLUE, 2, 181, 189, 192, 194 DEEP THOUGHT, 189 Dempster-Shafer Theory, 3, 13 DENDRAL, 6, 13 depth-first minimax search, 186 depth-first search, 182, 193 determinate process, 231 difference constraints, 30 diffraction, 151 diffuse surface, 160 disjoint, 226 distance matrix, 30 distributed artificial intelligence, 26, 45 diverge, 257 divergent, 224 divergent clades, 256 diverges, 247 DNA, 19, 24, 39 Donald Johnson algorithm, 32 Doob catalog, 233 Douglas Lenat, 6 DUCHESS, 178 dynamic, 195

Ebeling, 189 edge, 152 Edge Detection, 171 Elementary Refinements, 217 Eliza, 5 endpoint sequence problem, 38 EQP, 3, 15, 16 equals, 265 equational constraint, 85 equipollent, 248 ESP, 38 event space, 208, 242, 249 event tree, 207, 209 Event Trellises, 221 events, 210 Expand, 263 expectation, 231, 233 expected value, 211, 231 experiment, 235 expert, 193

INDEX

expert systems, 2 Experts, 176 extended search, 181, 193, 203 extension, 261 extreme point method, 126

Fail, 263 Failure, 262 fair-bet catalog, 232 FIDE, 176, 177 Fischer, 178 Fix, 262 Flinders Petrie, 42 Floyd-Warshall algorithm, 32-34 focus of contraction (FOC), 168 focus of expansion (FOE), 168 fold, 158 forbears, 235, 247, 257 foreshortening, 162 foretells, 224, 247, 261 forward pruning, 194 Fourier's algorithm, 113 Fourier's elimination, 112 fragment, 40, 41, 42 frame problem, 3, 14 FRITZ3, 177 fuzzy logic, 14

Game trees, 10 Garry Kasparov, 2, 175, 176 Gata Kamsky, 176 General Problem Solver, 5, 10 general purpose, 147 general-viewpoint assumption, 155 generalized linear program, 125 Genius 3.0, 177 Gentzen, 5 geometric stereo, 163 Gillogly, 195 Ginsberg, 16 Glenn Shafer, 13 Golumbic, Martin C , 36, 39, 41 graph theoretic techniques, 36 Greenblatt MACHACK, 175

Helene Kirchner, 3, 11 Hantao Zhang, 15 Hao Wang, 5 Harold Cohen, 7 Hart, 9 hash code, 197, 200-203 hash tables, 196, 203 hashing error, 201 hashing function, 202 Hasse diagram, 223 head, 229 Herbert Simon, 1, 4 heuristic search, 20, 33 Hill-climbing, 15

INDEX 273

HITECH, 189 hits, 197 Hsu, 189 Huffman code, xx Humean event, 229

IBM, 6 IBM DEEP BLUE, 175 IGMs, 176, 177 IM, 177 image compression, 141 image enhancement, 141 image irradiance, 151 Image processing, 141 image restoration, 141 image understanding, 140 implicit equalities, 117 implies, 224, 247 IMs, 176 inductive theorem, 74 influences, 221 intelligent backtracking, 33 International Chess Federation, 176 International Grandmasters, 176 International Masters, 176 interval algebra, 26, 35, 36, 37, 42 interval graph sandwich problem, 39, 40 interval realization, 38 Interval Satisfiability, 37 interval satisfiability problem, 38 ISAT, 38, 41, 42 Isomorphism, 265 iteratively-deepening depth-first search, 193 iteratively-deepening search, 179, 182, 193

195

J. A. Robinson, 11 Jean-Louis Lauriere, 5 Jean-Michel Morel, 16 John McCarthy, 8 John Sowa, 10

KAISSA, 178 Kasparov, 175, 177, 181, 203 Kautz, Henry, 42 Ken Thompson, 177 killer heuristic, 195

Lambertian, 160 Larry Wos, 3, 15 latin squares, 15 law of large numbers, 237 Levy, 204 limbs, 154 line drawing, 155 Line-Drawing Interpretation, 171 linear programming, 30 linear sign, 215, 221 LISP, 8

Logic, 26 logic programming, 26, 44 Logic Theory Machine, 5 lower expectation, 233 lower expected value, 233 lower probability, 234

mandatory relations, 246 Marion Tinsley, 2 Marsland, 194, 204 Martin Golumbic, 3, 14, 19 martingales, 208, 229 Massachusetts State Championship, 175 Master, 176, 177 mathematical programming, 20 may align with, 246 may allow, 246 may diverge from, 244 may forbear, 244 may foretell, 246 may imply, 246 may require, 246 may strictly allow, 246 may strictly foretell, 244 may strictly require, 244 Mephisto Genius 2.0, 177 Merger, 255 merges, 244 metric temporal constraint problem, 22, 27,

29, 33 minimal labeling problem, 38 minimax algorithm, 182-184, 186 minimax search, 186 Minnesota State Championship, 193 missionaries and cannibals, 8 Mitchell, 16 MLP, 38 Modal logic, 14 Moivrean event, 224 molecular biology, 19, 24, 39 Monty Newborn, 2 motion parallax, 166 move ordering, 195, 203 MTCP, 22, 27, 29, 33, 34, 37 multiprocessing, 201 multiprocessing systems, 189 mutilated checkerboard, 8 MYCIN, 6, 13

Nokel, K., 42 neural networks, 8 Newborn, 204 Newell, 5 Nielssen, 9 Nilsson, 2 non-monotonic logics, 20

opening book, 175, 203 operations research, 20

274

OPS, 14 optional relations, 244 ordered completion, 76 ordered critical disequality, 77 ordered critical pair, 77 ordering constraint, 85 OSTRICH, 178, 192 Otter, 3, 5, 15, 16 overlap, 244, 257

parallel search, 193 parametric queries, 108 partial ordering, 221 path, 224 pattern classification, 141 Pattern recognition, 141 Paul Masson American Chess Classic, 193 perspective projection, 150 phase angle, 159 piece-square table, 202 pinhole camera, 149 Planner, 5 point algebra, 42 precedence relation, 222 precedes, 248 predecessor, 222 preimage, 148 prerequisite, 226 principal variation splitting algorithm, 192 probability, 211, 231 probability catalog, 231 probability tree, 208, 209 process, 230 Prolog, 3, 5, 8 proof by refutation, 76

qualitative relations, 24 Quasi-dual, 123 quasi-linear combination, 109

R l , 6 Ramon Lull, 5 Raphael, 9 rating, 179, 180, 181 Rebel 6.0, 177 reduction ordering, 58 refinement, 208, 216, 237, 266 refines, 247 reflexive resolvent, 83 regression coefficients, 215 requires, 247 resolution, 11, 262 resolution refutation, 11 resolution-based theorem provers, 15 Resolve, 263 restricted domain, 40-42 rewrite system, 62 rewriting induction, 74 Robbins Conjecture, 16

INDEX

rule of iterated expectation, 233 rule-based systems, 11

sample space, 210, 224 Samuel's checkers player, 2 saturation of a set of clauses, 83 scaling, 162 scene radiance, 151 Schaeffer, 204 scope, 235 scoring function, 182, 183, 186, 189, 196, 203 Senior Masters, 176 sequenced, 222 Sergio Solimini, 16 seriation problem, 24, 42 Seymour Benzer, 39 shading, 158 Shamir, Ron, 36, 39, 41 Shaw, 5 shortest path problem, 32 SHRDLU, 5 Shulz, 177 Simon, 5 simple temporal problem, 29, 34 simplex, 127 simplification ordering, 59 Simulated annealing, 15 singular extension heuristic, 194 situation, 210 situation calculus, 269 Skolem, 11 Slate, 193 spatiotemporal, 166 Spatiotemporal Coherence Detection, 172 specular surface, 160 STAR SOCRATES, 192 STARTECH, 192 static, 195 STEAMER, 7 Stereo, 162 Stereoscopic Correspondence Establishment,

171 stochastic processes, 207 Stopping a Martingale, 232 STP, 29, 34 strictly allows, 247 strictly foretells, 247 strictly precedes, 222, 248 strictly requires, 247 strong solvability, 116 Subordination, 265 successor, 222 SUN PHOENIX, 192 Surface Representation, 172 Swedish Rating List, 177 synthetic annealing, 3

T-H Ngair, 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. Bel lman, Editor, Stochastic processes in mathematical physics and engineering (New

York City, April 1963)

15 N . C. Metropol is , 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. Bel lman, Editor, Mathematical problems in the biological sciences (New York City, April 1961)

13 R. Bel lman, 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. Be l lman 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

9