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55 Frederick Hoffman, Editor, Mathematical aspects of artificial intelligence (Orlando,
Florida, January 1996)
54 R e n a t o Spigler and S t e p h a n o s Venakides, Editors, Recent advances in partial
differential equations (Venice, Italy, June 1996)
53 D a v i d A. Cox and B e r n d 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
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51 Louis H. Kauffman, Editor, The interface of knots and physics (San Francisco,
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50 R o b e r t Calderbank, Editor, Different aspects of coding theory (San Francisco,
California, January 1995)
49 R o b e r t L. D e v a n e y , 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 D a u b e c h i e s , 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. S u m n e r s , Editor, New scientific applications of geometry and topology
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44 B e l a Bollobas, Editor, Probabilistic combinatorics and its applications (San Francisco,
California, January 1991)
43 Richard K. G u y , Editor, Combinatorial games (Columbus, Ohio, August 1990)
42 C. P o m e r a n c e , Editor, Cryptology and computational number theory (Boulder,
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41 R. W . B r o c k e t t , Editor, Robotics (Louisville, Kentucky, January 1990)
40 Charles R. J o h n s o n , Editor, Matrix theory and applications (Phoenix, Arizona,
January 1989)
39 R o b e r t L. D e v a n e y and Linda K e e n , Editors, Chaos and fractals: The mathematics
behind the computer graphics (Providence, Rhode Island, August 1988)
38 Juris H a r t m a n i s , Editor, Computational complexity theory (Atlanta, Georgia, January
1988)
37 H e n r y J. Landau, Editor, Moments in mathematics (San Antonio, Texas, January 1987)
36 Carl de B o o r , Editor, Approximation theory (New Orleans, Louisiana, January 1986)
35 Harry H. Panjer, Editor, Actuarial mathematics (Laramie, Wyoming, August 1985)
34 Michael A n s h e l and W i l l i a m G e w i r t z , Editors, Mathematics of information
processing (Louisville, Kentucky, January 1984)
33 H. P e y t o n Y o u n g , Editor, Fair allocation (Anaheim, California, January 1985)
32 R. W . M c K e l v e y , Editor, Environmental and natural resource mathematics (Eugene,
Oregon, August 1984)
31 B . G o p i n a t h , Editor, Computer communications (Denver, Colorado, January 1983)
30 S i m o n A . Levin, Editor, Population biology (Albany, New York, August 1983)
29 R. A. D e M i l l o , G. I. D a v i d a , D . P. D o b k i n , 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. S h e p p , Editor, Computed tomography (Cincinnati, Ohio, January 1982)
(Continued
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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
J a n u a r y 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 A S P E C T S OF ARTIFICIAL INTELLIGENCE
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
T h e A M S 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.
Primary 68-xx;
Secondary 0 3 - x x , 0 5 - x x , 5 1 - x x , 6 0 - x x , 9 0 - x x .
Library of C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n 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
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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
M O N T Y NEWBORN
175
Mathematical Foundations for Probability and Causality
GLENN SHAFER
Index
207
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 AIthe 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
BUGGY, 7
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
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
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
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
D E E P BLUE, 2, 181, 189, 192, 194
D E E P 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
HITECH, 189
hits, 197
Hsu, 189
Huffman code, xx
Humean event, 229
IBM, 6
IBM D E E P 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
27 3
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
27 4
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
Rl, 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. G a s s , 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 . H o g g , Editor, Modern statistics: Methods and applications (San Antonio, Texas,
January 1980)
22 G. H. G o l u b 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. F i n n , Editor, Applications of nonlinear partial differential equations in mathematical
physics (New York City, April 1964)
16 R. B e l l m a n , Editor, Stochastic processes in mathematical physics and engineering (New
York City, April 1963)
15 N . C . M e t r o p o l i s , A . H. Taub, J. T o d d , and C. B . T o m p k i n s , Editors,
Experimental arithmetic, high speed computing, and mathematics (Atlantic City and
Chicago, April 1962)
14 R. B e l l m a n , Editor, Mathematical problems in the biological sciences (New York City,
April 1961)
13 R. B e l l m a n , 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. W i g n e r , Editors, Nuclear reactor theory (New York City, April
1959)
10 R. B e l l m a n 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 . M a c C o l l , 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
ISBN 0-8218-0611-4
9
titles.)