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PROBLEM-SOLVING METHODS
IN
ARTIFICIAL INTELLIGENCE
Nils J. Nilsson
Artificial Intelligence Group
Stanford Research Institute
November 1969
(Preliminary
Draft, ® 1969 Nils J. Nilsson)
Bibliographical and reference sections of Chapters I thru VIII,
and combined reference list (revised February 1970)
.
(Chapter I)
Bibliographical and Historical Remarks
D.
The "intelligence" of
1.
Computers
The question of whether or not machines can (or ever will be able
to)
"think"
still provokes lively debate even among those who concede that
man is a machine.
Turing (1950) delightfully shatters many of the standard
arguments against intelligent machines.
To decide whether or not a
machine is intelligent, he proposes what has come to be called the
Turing
Test.
Selfridge and Kelly (1962) debate about the magnitude of the practical
problems in creating intelligent machines after agreeing that there are
Hubert Dreyfus (1965) thinks that digital
no known theoretical barriers.
are inherently incapable of such
computers
"necessities"
"fringe consciousness" and "perspicuous grouping." His
of intelligence as
arguments
are
systematically refuted by Papert (1968).
Overviews of Artificial Intelligence
2.
Attempts to
never been
wholly
"organize"
successful.
the field of Artificial
Intelligence have
The subtopics of Search, Pattern Recognition,
Learning, Problem-Solving, and Induction have been suggested by
in an important survey article.
Minsky (1961a)
This breakdown is still useful even though
its logical validity is challenged by the plethora of statements in the
literature that have the form:
of V
"
"The
problem of Xis basically a problem
where X and V could be any pair of subtopics.
More recently Minsky
(1968) has written another thoughtful article on the foundations of Artificial
Intelligence and concludes that a major problem is that of acquiring, maintain
ing and accessing a large
knowledge
116
base
3.
Ap pr oa c hes to Artificial Intelligence
In attempting to build intelligent machines one naturally asks
is the secret of animal intelligence?" People have had a variety
"what
of adventures pursuing this question but no one has yet found the
(1962) suggested brain models called perceptrons;
Rosenblatt
networks of
"artificial neurons"
and Pitts (1943)
.
secret.
these were
based on the neuron models of McCulloch
The study of perceptrons stimulated early pattern-
recognition research and led to some elegant mathematical results on
computational geometry by Minsky and Papert (1969).
of
"intelligence," however,
models
The complex processes
were beyond the power of these simple
percept ron
.
Another biologically-based strategy was the rather grandiose attempt
to simulate evolution itself.
in two billion years or
at
so, let
high speed in the computer.
Since evolution produced intelligent man
us simulate the processes of evolution
Fogel et al (1966) describe experiments
involving the production of many generations of finite state machines
using the strategies of mutation and selective survival.
Although this
technique may be capable of condensing the first few million years of
evolution
to a few days of computer
time, it seems that the important
middle and later stages of evolution involve structures already so complex
(though not yet
"intelligent") that their evolution cannot be
computer simulation.
Thus the
speeded up by
"artificial evolution" approach
did not
succeed in producing adequately complex machines either.
Another way to learn about intelligence from animals is to study their
behavior, particularly the problem-solving behavior of man.
Travis (1963,
1967) discusses the role of introspection in the design of problem-solvers.
117
Newell,
Shaw and Simon (1959) describe a
supposed to attack problems in much the
General Problem
Solver'
that is
same way that humans do
A rich source of ideas about how humans attack problems is found in
Polya (1957). -m
—^
1
Polya quotes the Greek mathematician Pappus who,
around 300
approach
A.D., gave an excellent description of the
Problem-reduction
':
"in analysis,
we start from what is required, we take it for granted,
and we draw consequences from
it, and consequences from the consequences,
till we reach a point that we can use as a starting point in synthesis.
For in analysis we assume what is required to be done as already done
(what
is sought as already found, what we have to prove as true).
We
inquire from what antecedent the desired result could be derived; then
we inquire again what could be the antecedent of that
on, until
passing from antecedent to
antecedent, we come eventually upon
something already known or admittedly true.
or solution backwards,
antecedent, and so
This procedure we call analysis,
or regressive reasoning.
*
In the problem-solving methods based on analysis of human behavior,
we find that trial and error search at some level plays a key role.
Campbell (1960) calls the unguided search process a
selective-survival process.
1.
blind-variation-and
He concludes:
A blind-variation-and-selective-survival process is fundamental
to all inductive
achievements, to all genuine increases in
knowledge, to all increases in fit of system to environment.
*
From page 142 of Polya (1957)
118
The processes which shortcut the full blind-variation-and-
2.
selective-survival process are
in themselves inductive
achieved
achievements containing wisdom about the environment
originally by a
blind-variation-and-selective
survival process.
in their own
In addition, such substitute processes contain
3.
process at
operation a blind -variation-and-selective-survival
some level.
primacy of search.
We agree with Campbell about the ultimate
The real
problem-solver is to search at
trick in designing an efficient automatic
the highest
level
permitted by the available information about the
problem and about how it might be solved.
Thus in this book we are
with how they can be made
primarily concerned with search techniques and
more efficient by using all available information.
4.
Problem-Solving
Methods
study problem-solving
There have been only a few attempts to
methods and
processes abstractly in order to catalog the different
In this book we identify three
deduce general properties of these methods.
major
to
probably not meet with universal
approaches although our division would
approval among researchers in this field.
A slightly different taxonomy of
(1969).
problem-solving methods is suggested by Newell
The organization of
by a series of difficult but
the present book has been much influenced
worthwhile papers
by
Aj^e^^^l^TJ;^ .
A paper by
Sandewall (1969)
ideas that we treat in this
formalizes some of the same problem-solving
book.
highly formal treatment of
A book by Baner.ii (1969) contains a
problem solving and game playing.
119
Problem-solving programs demand a heavy diet of puzzles and games
Some good general books of puzzles are
on which to sharpen techniques.
those of Martin Gardner (1959,
Scientific American.
1961) who edits a puzzle column in The
Also see the books of puzzles by Dudeney (1958,
who was a famous British puzzle inventor.
The 15 puzzle has a long history
and is discussed by Martin Gardner (1964, 1965a, b,c)
and by Ball (1931).
(1931) and in Gardner
1967)
,
by Schofield (1967),
The Tower of Hanoi problem is also discussed in Ball
(1959).
The state-space approach to problem-solving gets its name from the
use of "state-spaces" for similar purposes in control theory.
used extensively in operations research.
It is also
Some of the state -space search
methods that we shall be discussing later are identical to those called
branch and bound methods in operations research.
give a survey of branch and
Lawler and Wood (1966)
bound methods and their applications.
Our predilection to distinguish between state-space and problemreduction methods derives from the different search strategies used by
each method.
The distinction is precisely the same as that made by
Amarel (1967) between "production type" and
"reduction type"
methods-
(1963a) also finds the problem-reduction formulation useful in
Slagle
describing his program for
symbolic integration.
The General Problem
Solver (GPS) of Newell and his coworkers is thoroughly described in a
book by Ernst and Newell (1969).
of state-space methods,
Although GPS can be described in terms
it is our opinion that its operation can be under-
stood more clearly if it is described as a problem-reduction type problemsolver
.
120
The formal-logic based problem-solvers stem from tho
memoranda of
McCarthy (1958,1963).
Advice-Taker
The Advice Taker was to be a system
that used formal methods to deduce the solutions to problems from a large-
set of axioms representing the problem-solver's knowledge base.
be given
advice
merely by adding new axioms.
this idea was undertaken by Black (1964)
.
It could
Some early work related to
We shall mention some of the
later work in this field in Chapter VII.
Some excellent ideas on solving large combinatorial problems have been
suggested by Shen Lin (1965,1968). Although these ideas don't appear to fit
neatly into any of our three categories of problem-solving methods, they
deserve the attention of any serious student in the field.
5.
Applications of Problem-Solving Programs
It is fair to ask whether or not any oJ these methods that work
so well on
puzzles and games has been usefully employed on
State-space methods have been employed in the
real
problems.
solution of operations research
well-known
problems such as the
/traveling salesman problem.
The
"best'
exact solution
technique so far is a state-space method proposed in a Ph.D. dissertation
by Shapiro (1966) and discussed by Bellmore and Nemhauser (1968).
Although
the traveling salesman problem may appear as frivolous as do puzzles and games
it is a model for problems of economic importance in scheduling and production
design
.
Other applications of the state-space method have been made in the con-
trol of remote manipulators by Whitney (1969) and in sequential decoding by
Jelinek (1969)
.
Problem-reduction methods have been employed in a system
that performs symbolic integration (Slagle,
1963a)
and in a system that
analyzes mass spectrograph data (Buchanan, Sutherland, and Feigenbaum, 1969)
121
.
6.
I mportant Source Materials for Artificial In tell i g ence
Artificial Intelligence is a much-surveyed field, and there are plenty
of bibliographies around.
An early annotated one is that of Minsky (1961b)
.
Later
surveys by Feigenbaum (1963) and Solomonoff (1966) include many additional refer-
ences.
A recent survey by Feigenbaum (1968) lists more articles and also
speculates about the future of the field.
A vol umni, that is often referenced because it contains many of the early
papers is entitled Computers and Thought, edited by Feigenbaum and Feldman.
D. Michie and others have edited a series of books called Machine Intelligence
These contain papers delivered at the Machine Intelligence Workshops held annually
in Edinburgh.
Another important, book is called Semantic Information Processing
edited by Minsky; it contains complete versions of several Ph.D dissertations
dealing with language processing
and "understanding,"
Artificial Intelligence is a journal devoted exclusively to A.I;
publishing in 1970.
it began
Also, the Journal of the Association for Computing Machinery
occassionally publishes articles on artificial intelligence subjects.
In the United States, Artificial Intelligence activities are coordinated
through a Special Interest Group on Artificial Intelligence (SIGART) of the
Association for Computing Machinery
(ACM).
SIGART publishes' a
newsletter that
occassionally contains reference material not published elsewhere.
The Artificial Intelligence and Simulation of Behavior (AISB)
British
Computer Society publishes a European
group
of the
newsletter.
For completeness, we shall occassionally reference unpublished memoranda
and reports in this book.
The authors of such material will sometimes provide
copies upon request.
122
E.
References
Amarel, S.
(1965)
ACM
Solving Procedures for Efficient Syntactic Analysis"
Report
as
Scientific
(Also
printed
20th National Conference, 1965
1968).
No. 1, AFOSR Contract No. AF49(638)-1184,
"Problem
S. (1967)
Amarel,
in the
Approach to Heuristic Problem-Solving and Theorem Proving
Hart
and S.
Science,
J.
Propositional Calculus," Systems and Computer
Takasu (Eds.), Univ. of Toronto Press, 1967.
"An
(1969)
Amarel, S.
"On
the Representation of Problems and Goal Directed Procedures for
Computers," Communications of the American Society for Cybernetics,
Vol. 1, No. 2, 1969.
Ball, W.
(1931)
Mac Millan &Co
Mathematical Recreations and Essays, Tenth Edition, 1931,
(15 puzzle, pp. 224-228; Tower of Hanoi, pp. 228-229.)
London,
(1969)
Banerji, R.
Theory of Problem Solving: An Approach to Artificial Intelligence,
American Elsevier Publishing Co. Inc., New York, 1969.
Bellmore, M. and G. Nemhauser
(1968)
Problem: A Survey,"
Vol. 16, No. 3, May-June 1968, pp. 538-558.
"The
Black,
Traveling Salesman
Operations Research
(1964)
F.
Harvard,
Deductive Question-Answering System, "Ph.D. Dissertation,
Minsky (Ed)
M.
Processing,
June 1964. Reprinted in Semantic Information
MIT Press, Cambridge, Mass., 1968, pp. 354-402.
"a
Buchanan, 8.
,
G. Sutherland, and E. Feigenbaum (1969)
"Heuristic Dendral: A Program for
in Organic Chemistry." in Machine
Michie (Eds.), American Elsevier
1969, pp. 209-254.
Campbell, D.
Generating Explanatory Hypotheses
Intelligence 4, B. Meltzer and D.
Publishing Co. , Inc., New York,
(1960)
a General Strategy in
Survival as
"Blind Variation and Selective
Self-Organizing
Systems,
Knowledge-Processes," In
(Eds
j, Pergamon
Press,
New York 1960, pp. 205-231.
123
M. Yovits&S. Cameron
Collins, No and D. Michie (Eds.)
Machine
Intelligence
1, American Elsevier Publishing Co., Inc.,
1967.
New York,
Dale, E. and D. Michie (Eds.)
«.Phi BB intelligence
Co., Inc.,
2, American Elsevier Publishing
1968.
New York,
Dreyfus, H. (1965)
(AD
Corporation Paper
!^^^
625 719), December, 1965.
(1958)
Dudeney,
H.
Dudeney,
H. (1967)
York, 1958.
Dover Publications, Inc., New
The Canterbury Puzzles,
(Originally published in 1907.)
.
Sen bnc r s
Martin Gardner (Ed. ), Charles
536 Puzzles and Curious Problems,
on from two of Dudeney' books.
New York.
Problems. 1931).
£L
Modern
196/71X1^110^1
Puzzles
1926. and Puzzles and Curious
Ernst, G. and A. Newell (1969)
r.PS: A Case Study in Generalityjmd_Problem
Series, Academic Press, 1969.
Feigenbaum, E. (1963)
"Artificial
Vol.
Intelligence
Research,"
IT-9, No. 4, Oct., 1963, pp.
Solving
ACM Monograph
lEEE Trans, on Info. Theory,
248-261
Feigenbaum, E. (1968)
Decade,"
Inteligence: Themes in the Second
Also printed as Stanford
IMPS 68 Congress. Edinburgh, Aug. 1968.
Aug. 10, 1968.
Intelligence Project Memo No. 67,
Tj n
g2^£-
"Artificial
7ver77t7~ATtTFicial
124
(1963)
E. and Feldman, J.
Feigenbaum,
Computers and Thought, McGraw-Hill Book Company, Inc., New York, 1963
, A.
Fogel, L.
Owens and M. Walsh (1966)
Intelligence Through Simulated Evolution,
Artificial
John Wiley, New York,
1966.
Gardner,
(1959)
M.
The Scientific American Book of Mathematical Puzzlos and Divers tons,
Simon and Schuster, New York, 1959
.
Gardner,
_
(1961)
M.
Puzzles and Diversions
~
The
Second Scientific American Book "of Mathematical
X lit?
Simon and Schuster, New Y0rk, 1961
OOJJUIIU
Gardner,
OtJLCII L.
XX X\^
r*HK-
iJ.
voh
"^"^"
-«-
""-»
~»«^-""~
—
1965 a,b,c)
M. (1964,
Scientific American, Vol. 210, No. 2, Feb., 1964,
Vol. 212, No. 6,
212, No. 3, March,l96s, pp. 112-117;
pp 122-13p; Vol
1965,
pp. 222-236.
June, l96s, pp. 120-124; Vol. 213, No. 3, Sept.,
Games,
Mathematical
Jelinek, F.
"Fast
.
(1969)
Sequential Decoding Algorithm Using a
Stack,"
IBM J. of Res,
and Dcv
Vol. 13, No. 6, Nov, 1969, pp. 675-685.
Lawler, E.
and D. Wood (1966)
and Bound Methods: a Survey,"
pp. 699-719, July-August ,1966.
"Branch
Lin
,
14, No. 4,
Shen (1965)
"Computer Solutions of the
Traveling Salesman
Technical Journal, Vol. XLIV, No.
Lin,
Operations Research, Vol.
Problem,"
The Bell System
10, Dec, 1965.
Shen (1968).
"Heuristic
Computer,"
on a
Techniques for Solving Large Combinatorial Problems
by Computer,
Problem-Solving
and Non-Numerical
Formal Systems
Case Western Reserve Ufniversity,
125
Cleveland, Ohio,
Nov., 1968.
)
McCarthy, J. (1958)
Programs with Common Sense," in Mechanization of Thought Processes, Vol . I
pp. 77-84,. Proc Symposium, National Physical Laboratory,
London. Nov
24-27, 1958. Reprinted in Semantic Information Processing,
M. Minsky (Ed.)
MIT Press, Cambridge, Mass., 1968, pp.
"
403-4107
McCarthy, J. (1963)
"Situations,
Actions and Causal
Laws,"
Stanford University Artificial
Intelligence Project Memo. No. 2, 1963. Reprinted in Semantic Information Processing, M. Minsky, (Ed.) MIT Press, Cambridge, Ma55 .,1968,
pp. 410-418.
,
McCulloch, W. S. and W. Pitts (1943)
"A Logical Calculus of the Ideas Immanent in
Neural
Mathematical Biophysics, Vol. 5, pp. 115-137, 1943. Nets." Bulletin
~~"
Meltzer, B. and D. Michie (Eds.)
Machine Intelligence 4, American Elsevier Publishing Co., Inc.,
New York, 1969.
Michie, D. (Ed.)
Machine Intelligence 3, American Elsevier Publishing Co., Inc.,
New York, 1968.
Michie, D. (Ed.)
Machine Intelligence^, American
Elsevier Publishing to.,
Co
Tne
Inc.,
New York, 1970.
126
o1
Minsky, M.
(1961a)
"Steps Toward Artificial Intelligence, " Proc.
.,
of ire. Jnn
Reprinted in Computers and Thought , Feigenbaum
(Eds.), McGraw-Hill BookToT; New
York, 196^, pp. 406-450
pp. 8-30.
Minsky,
M.
19(U ,
F
Vol
r
nnd
49
r
,'h
(1961b)
"A Selected Descriptor -Indexed Bibliography to the Literature on
Artificial Intelligence," IRE Trans, on Human Factors in Electronics,
New York,
,
HFE-2, pp. 39-55. A Revision is reprinted in Computers
Feigenbaum, E. and J. Feldman (Eds.), McGraw-Hill Book Co.
1963, pp. 453-523.
March), 1961
and Thought,
M. (Ed.)
(1968)
Semantic Information Processing,
Minsky,
Mass.,
Cambridge,
The
MIT Press,
1968.
Minsky, M. and S. Papert (1969)
Perceptrons: An Introduction to Computational Geometry,
Cambridge, Mass. 1969.
Newell, A.
The MIT Press,
(1969)
"Heuristic
Programming:
Research,
J. Aronofsky
111-Structured Problems," Progress
(Ed.), Vol. 3, Wiley, 1969.
in
Operations
NewelliA.,J. Shaw and H. Simon (1959)
Int'l Con*,
on a General Problem-Solving Program," Proc of
256-264,
1969.
pp.
Paris,
on Information Processing, UNESCO House,
"Report
Papert, S. (1968).
"The Artificial Intelligence of Hubert L. Dreyfus. A Budget of
Fallacies,"
Massachusetts Institute of Technology Artificial Intelligence Memo #54,
Jan. , 1968.
Polya,
G. (1957)
How to Solve It
N. Y. ,1957.
Rosenblatt, F.
(Second Edition)
,
Doubleday & Co.,
Inc. Garden City
(1962)
Principles of Neurodynamics, Spartan Books, New York 1962.
127
"
Sandewall
(1969)
E.
"Concepts and Methods for Heuristic Search,"
Proc of the International
Joint Conference on Artificial Intelligence.
(1967)
Schofield, P.
"Complete Solution of the "Eight-Puzzle" in Machine Intelligence 1,
N. Collins and D. Michie (Eds), American Elsevier Publishing Co., Inc.,
New York, 1967, pp. 125—133.
0. and J. Kelly, Jr. (1962)
"Sophistication in Computers: a Disagreement,"
Selfridge,
Feb.
Theory,
IRE Trans, on Info.
1962.
(1966)
D.
"Algorithms for the Solution of the Optimal Cost Traveling Salesman
Problem," Sc.D. Thesis. Washington Univ., .it. Louis, M0. ,1966.
Shapiro,
J. (1963a)
Slagle,
"A
Heuristic Program that Solves Symbolic Integration Problems in
JACM, Vol. 10, No. 4, Oct. 1963, pp. 507-520
Also in Computers and Thought, Feigenbaum, E. and J. Feldman (Eds.)
McGraw-Hill Book Co., New York, 1963, pp. 191-203.
Freshman Calculus,
Solomonoff,
"Some
No.
(1966).
R.
Recent Work in Artificial Intelligence
112, Dec.
,
," Proc
.
lEEE , Vol. 54,
1966.
(1963)
Travis, L.
"The
"
Value of Introspection to the Designer of Mechanical ProblemBehav. Science Vol. 8, No. 3, pp. 227-233, July 1963.
Solvers,"
,
128
,
(1967)
Travis, L.
a Few New Machines, '
Conference Record 1967 Winter Convention on Aerospace and Electronic
Systems (WINCON) , Vol, VI, Feb. 1967, lEEE Publication No. 10-C-42.
"Psychology and Bionics: Many old Problems and
Turing A. M.
(1950)
Machinery and Intelligence," Mind ,
Oct. 1950, Vol. 59, pp
433-460. Reprinted in Computers and Thought, E. Feigenbaum and J.
Feldman (Eds.)
McGraw-Hill Book Co. , New York, 1963, pp. 11-35.
Whitney, D.
(1969)
Models of Remote Manipulation Tasks," in Proc
Int'l Joint Conf. on Artificial Intelligence, pp. 495-507.
"State Space
129
of the
((Chapter II)
D.
Bibliographical and Historical Remarks
I.
Elements of a State-Space Representation
The three elements of a state-space representation
—
descriptions, operators, and goal test
basic in automatic problem-solving.
—
state-
have long been recognized as
These notions are discussed in the
book on the General Problem Solving (GPS) program by Ernst and Newell
(1969).
A more abstract treatment of the problem of description of
states and operators may be found in Amarel (1967,1969).
An example of the use
of rewriting rules to represent operators can be found in the problem-solving
system of Quinlan and Hunt (1968)
.
This system uses tree structures for state
descriptions.
2.
Graph Theory
The basic vocabulary of graph theory (arcs, nodes, paths, and so
on) is often used to describe problem-solving processes.
It does not appear
that much more than this vocabulary is of use in automatic problem solving.
Perhaps this is because most of the theorems of graph theory refer to explicit
graphs while in problem solving we usually use the notion of an implicit graph
Classic books on graph theory are those of Berge
(1962)
and Ore (1962)
.
Ore
(1963) has also written a popular, elementary book illustrating applications
of graph theory to various combinatorial problems.
3.
Non-deterministic
The phrase
(1967a).
Programs
"non-deterministic
algorithm"
was proposed by Floyd
In these algorithms, Floyd allowed the use of a
"choice"
to simplify the description of exhaustive search strategies.
function
Later Manna
(1970) described a class of programs that allowed non-deterministic
assignment statements (similar to the choice function) as well as non-
deterministic branching operations.
He described techniques for proving
1133
.
the
"correctness"
of programs containing these new elements.
(In an
earlier paper, Manna (1969) considered the problem of proving the
correctness of ordinary, non-deterministic programs.)
Fikes (1970)
describes a complete problem-solving system in which problems are
stated in a procedural language that allows the use of non-deterministic
choice functions.
4
.
Background Literature on the Example Problems
The example problems that we have used to illustrate state-
space techniques come from diverse fields each with its own individual
tradition of specialized problem-solving methods.
Many of these methods
can be viewed as an application of the state-space approach.
It would be
instructive for the reader to carry out the exercise of translating some
of the descriptions of these specialized methods into the common language
of the state-space approach.
The traveling salesman problem has occupied an important place in
Operations Research.
For a review see Bellmore and Nemhauser (1968).
Problems of syntax analysis are common in language processing.
Feldman and Gries (1968) give a thorough survey of syntax analysis
techniques as used by translating systems for the formal languages of
computation.
Amarel (1965) discusses syntax analysis from the point of
view of automatic problem-solving and proposes a problem-reduction
procedure.
For a highly readable treatment of the use of symbol strings
and production systems as models of computation see the final chapters in
a book on computation by Minsky (1967).
1134
Many problems involving distribution, flow and queuing can be
solved
by similar techniques.
A discussion of these can be found in a book by
Ford and Fulkerson (1962).
Control theory is a large and specialized field with a variety of
problem
solution methods.
A book by De
Russo.
Roy and Close (1965) is a good
introduction to modern control theory.
5.
The Problem of Finding a Good Representation
The problem of finding good representations for problems has been
treated by only a few researchers notably by Amarel.
Amarel
(1968)
is a
classic paper on the subject; it takes the reader through a series of progressively better representations for the
"missionaries
and
cannibals"
problem.
The use of variables in state descriptions has been mentioned
as a
powerful representational technique.
use of "object
solving system.
schemas"
Later versions of GPS allowed the
with variables as did
Fikes'
A Paper by Newell (1965) examines some possible approaches
(and their limitations) toward making progress on
problem.
(1970) problem-
"
1135
"the
representation
E.
References
Amarel, S. (1965)
Procedures for Efficient Syntactic Analysis" ACM
20th National Conference, 1965 (Also printed as Scientific Report
No. 1, AFOSR Contract No. AF49(638)-1184, 1968).
"Problem
Amarel, S.
Solving
(1967)
"An
Approach to Heuristic Problem-Solving and Theorem Proving in the
Propositional Calculus," Systems and Computer Science, J. Hart and S.
Takasu (Eds.), Univ. of Toronto Press, 1967.
Amarel,
S. (1968)
of Problems of Reasoning About Actions,"
in Machine Intelligence 3, D. Michie (Ed), American Elsevier, New
"On Representations
York,
Amarel, S.
1968, pp. 131-171.
(1969)
"On the Representation of Problems and Goal Directed Procedures for
Computers," Communications of the American Society for Cybernetics,
Vol.
1, No.
2, 1969.
Bellmore, M. and G. Nemhauser
(1968)
Problem: A Survey,"
pp. 538-558.
Vol. 16, No. 3, May-June 1968,
"The
Berge,
C.
Traveling Salesman
pt^ej^ons^e^rch
(1962)
(Translated by Alison Doig)
The Theory of Graphs and its Applications,
(First published in France
John Wiley & Sons, Inc. New York, 1962.
in 1958 by Dunod, Paris).
De Russo,
P.
,
R. Roy and C. Close
(1965)
State Variables for Engineers, Wiley, New York,
Ernst, G.
1965
and A. Newell (1969)
GPS: A Case Study in Generality and Problem Solving
Series, Academic Press, 1969.
1136
ACM Monograph
1
Feldman,
J. and D. Gries (1968)
"Translator
Writing
Systems" Comm. of
77-113.
pp.
ACM,
Vol. 11, No. 2, Feb., 1968,
(1970)
Fikes, R.
"Ref-Arf: A
Artificial
System for Solving Problems Stated
Intelligence, Vol.l, No. l , 1970.
as Procedures,'
Floyd, R. (1967a)
J. of ACM, Vol. 14, No. 4, 0ct. ,1967,
"Nondeterministic Algorithms,"
636-644.
pp.
Ford, L.
Jr., and D.
Fulkerson (1962)
Flows in Networks, Princeton University Press,
"The
Correctness of
J. of Computer and Sytem Sciences, Vol
Programs,"
,1969.
May
Manna,
Z. (1970)
of Non-Deterministic
1, 1970.
"The Correctness
1, No.
Vol.
Minsky,
Programs," Artificial
Intelligence
M. (1967)
Computation: Finite and Infinite Machines, Prentice Hall, Inc.,
Englewood Cliffs, N.J. , 1967.
(1965)
Newell A.
the Current Stock of. Ideas about Problem Solving," in
Electronic Information Handling, A. Kent and 0. Taulbee (Eds. ), Spartan
Books, Wash. D.C., 1965.
"Limitations of
Ore,
1962
(1969)
Z.
Manna,
Princeton, New Jersey,
(1962)
0.
of Graphs, American Mathematical Society Colloquium Publication
XXXVIII, Providence, Rhode Island, 1962.
Theory
Vol.
1137
3
Ore,
0.
(1963)
Graphs and Their Uses, Random House,
Quinlan,
1963.
J. and E. Hunt. (1968)
"A
Formal Deductive Problem-Solving
No. 4, 0ct., 1968, pp. 625-646.
11-38
System,"
J. of ACM, Vol.
15.
(Chapter III)
D.
Bibliographical and Historical Remarks
1.
Shortest Path Algorithms
Efficient methods for finding the shortest (or least costly)
path between two nodes in a graph are of great interest in a variety of
disciplines.
The procedure that we have called the Uniform Cost Method
was first described by Dijkstra (1959).
A similar breadth-first search
technique was also proposed by Moore (1959).
programming
methods.
dynamic
algorithms of Bellman are essentially breadth-first search
For a thorough discussion of dynamic programming see the book
by Bellman and Dreyfus (1962).
called
Also, the
"backtrack programming"
and Baumert (1965).
Stuart
A depth-first search procedure, often
in computer science, is described by Golomb
Dreyfus (1969) presents a detailed survey of
some of these and other graph searching methods.
2.
Heuristic Search Techniques
The use of heuristic information
to increase search efficiency
has been studied both in Artificial Intelligence and in Operations Research
Probably the most well-known use of heuristic evaluation functions has been
in game-playing programs, notably the checker-playing program of Samuel
(1959, 1967).
The use of evaluation functions to direct the search of
state-space graphs has been proposed by Doran and Michie (1966) and further
discussed by Doran (1968).
Our examples using the 8-Puzzle are based on
those of Doran and Michie.
A general theory of the use of evaluation
functions to guide search was presented in a paper by
Raphael (1968);
Hart, Nilsson, and
our description of the Ordered Search Algorithm and its
properties is taken from that paper.
111-49
In the
branch and bound
methods of Operations Research we also see
For a description of
the use of evaluation functions to guide search.
these see the survey article by Lawler and Wood
(1966).
The branch and
bound technique proposed by Shapiro (1966) for the traveling salesman
problem can also be interpreted as a direct application of algorithm A
*
Our analysis supporting the importance of including g (as well as fi)
is taken from a dissertation by Pohl (1969).
(See also Pohl, 1970. )
Pohl (1969) considers also the problem of searching outward from both start
and goal nodes.
Particularly interesting here is his thorough discussion
of the more complex termination criterion needed for
"bidirectional search.'
Our proposal that the search process might be occasionally punctuated
by tree pruning operations (to
reclaim needed storage space) was also
suggested by Levy (1969).
3.
Measures of Performance
Doran and Michie (1966) proposed the
judging the efficiency of a given search.
another measure which they call the
factor" was motivated
penetrance
measure for
Slagle and Dixon (1969) propose
depth ratio.
by these earlier measures.
111-50
'
Our
effective branching
K.
References
Bellman, R. and S. Dreyfus (1962)
Applied Dynamic Programming, Princeton University Press,
Dijkstra, E.
(1959)
"A Note on Two Problems in Connection with
Mathematik, 1, 1959, pp. 269-271
Doran,
J.
"New
1962.
Graphs,"
Numerische
(1968)
Developments of the Graph Traverser," in Machine Intelligence 2
E. Dale and D. Michie (Eds), American Elsevier Publishing Co., Inc.,
New York, 1968, pp. 119-135.
Doran, J. and D. Michie (1966)
"Experiments with
the Graph Traverser Program,"
Royal Socitey, A, Vol. 294, pp. 235-259, 1966.
Dreyfus, S.
Proceedings of the
(1969)
TI
An Appraisal of Some Shortest Path Algorithms," Operations Research,
Vol. 17, No. 3, pp. 395-412, May-June 1969.
Golomb, L. and L.
"Backtrack
Baumert (1965)
Programming,"
524.
J. of ACM, Vol. 12, No. 4., 0ct., 1965, pp. 516
*
Hart, P., N. Nilsson and B. Raphael
"A Formal Basis for the
Paths," lEEE Trans, on
(1968)
Heuristic Determination of Minimum Cost
Systems Sciences and Cybernetics, Vol. SSC-4
No. 2, July, 1968, pp. 100-107.
Lawler,
E. and D. Wood (1966)
and Bound Methods: a Survey,"
pp. 699-719, July-August ,l966.
"Branch
111-51
Operations Research, Vol.
14
No. 4
(1969)
D.
Levy,
"A Comment on Some Tree Searching Methods Which is Designed to Optimize
15, April 1969, pp.
the Use of Storage," ACM SIGART Newsletter, No.
14-16.
Moore, E. (1959)
"The Shortest Path Through a Maze," Proc. Int'l Symposium on the Theory ol
Switching, Part 11, April 2-5, 1957, The Annals of the Computation
Harvard University Press ,1959.
Laboratory of Harvard University 30,
Pohl,
I.
(1969)
"Bi-Directional and Heuristic Search in Path Problems," Stanford
University Computer Science Dept. Ph.D. Dissertation, 1969. Also
printed as Stanford Linear Accelerator Center Report No. 104, May
1969.
Pohl, I.
(1970)
"First
Results on the Effect of Error in Heuristic Search,
Machine
Samuel,
Intelligence
5.
"
in
(1959)
A.
IBM
Studies in Machine Learning Using the Game of Checkers,"
Reprinted in Computers und Thought,
Journal, Vol. 3, pp. 211-229, 1959.
Feigenbaum, E. and J. Feldman (Eds.), McGraw-Hill Book Co., New York, 1963
pp. 71
105.
"Some
-
Samuel
, A.
(1967)
"Some Studies in Machine
Progress,"
IBM Journal
Nov. 1967, pp7 601-6177"
Shapiro, D.
Recent
Using the Game of Checkers 11.
11,
No.
6
of Research and Development, Vol.
Learning
(1966)
Optimal Cost Traveling Salesman
Sc.D. Thesis, Washington Univ., ot. Louis, M0. ,1966.
"Algorithms for the Solution of the
Problem,"
Slagle,
J and J. Dixon
(1969)
"Experiments with Some Programs that Search Game Trees,"
Vol.
16, No.
2, April 1969
,
pp.
111-52
189-207.
J. of ACM
(Chapter IV)
E.
Bibliographical and Historical Remarks
1.
Problem-Reduction Methods and AND/OR
Graphs
Some of the very earliest work in Artificial Intelligence used
the problem-reduction techniquef (also called
solve problems
.
"reasoning backwards")
to
Newell, Shaw, and Simon (1957) programmed a Logic Theory
Machine (LT) that proved theorems in the propositional calculus by
backwards from the theorem to be proved.
working
In the LT program, use was made
of a tree of subproblems to keep track of alternative chains of reasoning.
Another early program in which problem-reduction methods and subproblem trees were used was the Symbolic Integration (SAINT) program of
Slagle first called these trees
AND/OR trees; in
(1968) used the term Proposition Tree.
(1963a).
a later paper Slagle and Byrsky
Amarel (1967) also used special graph
structures in discussing problem- reduction methods.
examples of representing an implicit
AND/OR tree by
2.
Slagle
Manna (1970)
gives several
nondeterministic programs.
Example Programs Using the Problem-Reduction Approach
Our example symbolic integration system is based on the SAINT program
of Slagle (1963a).
tation (Slagle 1961).
This program is described in detail in Slagle's Ph.D disser-
A much more elaborate integration system (called SIN) was
later programmed by Moses (1967).
Moses' system embodied so many special criteria
for applying the various operators that most integrations are carried out with
little or no search.
* One
might speculate that most of the search effort that might have been needed
to perform integrations was carried out once and for all by Moses himself in
designing the program.
The results of this design search were the special rules
about.w hich operators to apply in all cases.
1143
The geometry
theorem-proving examples were based on the program by
Gele rnter et al (1959, 1960).
This program was never really completed;
nevertheless several of its features and proposed features represented important
and original
innovations.
The notion of key operators and differences and their use in problemsolving is due largely to Newell
1969)
.
and his co-workers on GPS.
and Newell
analysis" to refer
to the process of
He also distinguishes among three
kinds of goals in the GPS system: the transform object goal,
goal, and the apply operator goal.
"means-
Newell has used the phrase
extracting differences and matching
t hese differences with relevant operators.
the reduce difference
Our description does not acknowledge the
utility of these distinctions. Our interest in GPS is solely in its ability
as an automatic problem-solving
system whereas many of the GPS workers
interested in it as a psychological
model of human thought processes.
were also
This
difference in purpose undoubtedly contributed to part of the difference in its
description.
Stressing the problem-solving abilities of GPS,
Ernst (1969) inquired about
conditions under which GPS is guaranteed to find a solution.
Ernst's study
resulted in formalizing the notions of differences and ordering of differences.
3.
Games and the Problem-Reduction Approach
Slagle and his co-workers have stressed the essential similarities
between
AND/OR trees
and game trees (single 1970, Slagle and Bursky.
1968).
This similarity is particularly apparent in simple games that can be searched to
termination or in the
one
,
The reader should be cautioned that our explanation of these ideas
differs somewhat from that of Newell, et al.
ends
(See Ernst
"end-games" of more complex games such
as chess.
Indeed,
often thinks of chess-end game puzzles as problems of proof rather than as
games.
Our Grundy's game example/taken from an article by o' Beirne (1961)
1144
.
E.
References
Amarel, S.
(1967)
"An Approach to Heuristic Problem-Solving and Theorem Proving in the
Propositional Calculus," Systems and Computer Science, J. Hart and S.
Takasu (Eds.), Univ. of Toronto Press, 1967.
Ernst,
G.
(1969)
"Sufficient Conditions for the Success
No .4, o>l 1959, pp. o.'i 530
of GPS,"
Journal of ACM,
Vol.
Ernst, G. and A. Newell (1969)
GPS: A Case Study in Generality and Problem Solving
Series, Academic Press, 1969.
Gelernter, H.
ACM Monograph
(1959)
.
.
"Realization of a Geometry Theorem-Proving Machine," Proc of an Int 'l
Conf. on Info. Proc, UNESCO House, Paris, 1959, pp. 273-282. Reprinted
in Computers and Thought Feigenbaum, E. and J. Feldman (Eds.),
McGraw-Hill Book Co., New York, 1963, pp. 134-152.
,
Gelernter,
H. J. Hansen, and D. Loveland (1960)
of the Geometry Theorem Proving Machine," Proc
17, pp. 143-147. Reprinted
of the Western Joint Computer C0nf. ,1960, Vol.
Feigenbaum, E. and J. Feldman (Eds.), McGraw-Hill
in Computers and Thought ,
Book Co., New York, 1963, pp. 153-167.
"Empirical Explorations
Manna,
Z. (1970)
"The
Correctness of Non-Deterministic Programs,
Vol. 1, No. I , 1970.
"
Artificial
Intelligence,
Moses, J. (1967)
Symbolic Integration,
Massachusetts Institute of Technology, Project MAC,
Report MAC-TR-47 (Thesis),
Dec.
Newell, A., J. Shaw, and it. Simon
1967.
(i;>;>7)
.
ol Lhe Logic Theory Machine,"
Proc of tho
Western Joint Computer Conference, 1957, pp. 218-239. Reprinted in
Computers and Thought, Feigenbaum, E. and J. Feldman (Eds.), McGraw-Hill
Book Co., New York, 1963, pp. 109-133.
"Empirical Explorations
O'Beirne, T. H.
(1961)
Puzzles and Paradoxes, New Scientist, No.
August 3, 1961.
IV-44
245, July 27, 1961, and No. 246,
Slagle,
J.
(1961)
for
Computer Program/ Solving Problems in Freshman Calculus (SAINT),
Doctoral Dissertation, Mass. Inst, of Tech. Cambridge, Mass, 1961.
Also printed as Lincoln Laboratory Report SG-0001, May 10, 1961.
"A
Slagle, J. (1963a)
"A
Heuristic Program that Solves Symbolic Integration Problems in
Freshman Calculus," JACM, Vol. 10, No. 4, Oct. 1963, pp. 507-520.
Also in Computers and Thought, Feigenbaum, E. and J. Feldman (Eds.)
McGraw-Hill Book Co., New York, 1963, pp. 191-203.
Slagle, J.
(1970)
"Heuristic Search Programs,"
cfan Formal Systems and Non-Numerical
Problem Solving by Computer, M. Mesarovic and R. Banerji
Slagle, J. and P. Bursky,
(1968)
"Experiments with
J. of ACM,
(Eds.)
a Multipurpose, Theorem- Proving Heuristic
Vol. 15, No., 1, pp. 85-99, Jan 1968.
IV-45
Program
(Chapter V)
E.
Bibliographical and Historical Remarks
Development of
1.
AND/OR Graph
Searching Techniques
The search strategies of many of thet. early programs that generated
subproblem trees employed only rather simple ordering methods.
Early versions
of GPS used a depth-first strategy and a means for measuring problem difficulty;
backtracking occurred when a successor problem was judged more difficult than any
of its ancestors.
Slagle' s SAINT program used depth of function nesting as a
measure of problem difficulty and generally worked on the easiest problems
first.
Slagle and his co-workers experimented with many search strategies for game
trees and AND/OR trees during the 19605.
The game-tree strategies were sum-
marized in a paper by Slagle and Dixon (1969).
involved a
"dynamic ordering" of node
The most complex of these
expansions.
In their problem-solving
system called MULTIPLE (MULTlpurpose Program that LEarns)
,
Slagle and Bursky
(1968) incorporated a general strategy for searching AND/OR trees.
This strategy
uses the notion of the "probability that a proposition is true" and then defines
a
merit
function" over
the open nodes of the tree.
That node having greatest
effect on the probability that the original proposition is true is said to have
the largest
"merit";
this node is the one selected for expansion.
Amarel (1967) proposed an "attention control" strategy for ordering node
expansions
in an AND/OR tree.
solution.
The present author (Nilsson,
This strategy attempted to find a
"minimal
cost
1968) also suggested a cost-minimizing
strategy and later realized that it was essentially identical to Amarel 's.
In Nilsson (1968) a proof is given that the strategy does indeed find minimal
cost
solutions; this proof is based on a similar one for state-space graphs by
Hart, Nilsson and Raphael (1968).
The Amarel
Nilsson strategy
differs little from the dynamic ordering method of Slagle and Dixon. The exposition
of the ordered search algorithm in this Chapter is merely an attempt to describe
this basic strategy in a clear and general manner.
1147
2.
Development of Game Tree Searching Techniques
Claude Shannon (1950) discussed some of the problems inherent in
programming a machine to play a complex game such as chess.
He suggested a
minimax search procedure to be used in conjunction with a static evaluation
function.
Newell, Shaw and Simon (1958) used several of these ideas in construc-
ting an early chess-playing program;
this area of research.
with a
"Five
Year
Plan"
they also give an excellent discussion of
Additional discussion on chess-playing programs together
for automatic chess can be found in an article by Good
(1968) that also lists several references.
Later Samuel (1959) discribed a checker-playing program that incorporated
polynomial evaluation functions, minimax search methods and various '''learning"
strategies for improving play
.
Samuels' program plays an excellent checker
and beats all but the very best players.
game
It continues to be one of the out-
standing examples of the application of artificial intelligence techniques.
Later work on this program is described in Samuel (1967).
One of the features
of more recent versions of Samuel's program is a dynamic ordering search procedure somewhat similar to that of Slagle and Dixon (1969).
The a
-
0
procedure is usually thought to be a rather obvious elaboration
of the minimaxing technique and was thus
"discovered"
independently by many
workers. It is first described by Newell, Shaw and Simon (1958) and was the
subject of much investigation by McCarthy and his students (Edwards and Hart,
1963) at M. I.T.
* Samuel
0/
- 3
Few clear expositions of the method and its properties exist
later stated ( personal
communication) that his program also used the
procedure but that at the time he thought its use too straightforward to
merit discussion in the paper.
1148
1967) contains a good description as does
Samuel's second checkers paper (Samuel,
Hie paper by Slagle and Dixon (1969)
the
Ot
- |3
.
The results on the search efficiency of
procedures were first stated by Edwards and Hart (1963)
theorem that they attribute to Michael Levin.
give what they consider to
Later Slagle and Dixon (1969)
be the first published
proof of this theorem.
and Dixon discuss several variations of the a -(3 procedure
employing dynamic ordering.
based on a
Slagle
culminating with
one
The performance of these various strategies is
compared using the ancient game of Kalah.
Our discussion about improvements
on minimax-based methods is taken from
some proposals by Slagle (1963hQ and Slagle and Dixon (1970).
paper describes experiments with an
"M
& N Tree Searching
The latter
Program" that adds (or
subtracts) a bonus when backing up scores.
3.
Some Representative Game-Playing Programs
Chess
Kister,
et al (1957) describe
the earliest chess system programmed on
a computer (MANIAC lat Los Alamos).
It used a reduced board
(6 X 6) and played rather poorly.
BerjisteiiL,__£t_al (1958)
describe
a chess system programmed at IBM.
It also played rather poorly but on a full (8 X 8) board
Newell, Shaw and Simon (1958) present
another early chess program
under development at Carnegie
Kotok (1962) discussed an early M. I.T. program later taken to Stanford
by John McCarthy and modified slightly.
This one achieved the
level of mediocre play.
Adelson
-
Velskii, et al (no paper available) wrote a program at the
Institute for Theoretical and Applied Physics in Moscow.
gram beat the Kotok
-
This pro
McCarthy program in a tournament (See SICART
Newsletter, No. 4, June,
1967
1149
p.
11).
Greenblatt, et al (1967)
describe an M.I.T. program now called
"middle-
Mac Hack.
Its level of play can be described as
amateur.
It is an honorary member of the Massachusetts
chess society and has been given a class C rating.
For some
example games see the following SIGART Newsletters:
No. 6,
Oct. 1967, p. B (here, the computer beat H. Dreyfus who earlip r
doubted that a machine could beat even an amateur player) ;
No. 9, April 1968, pp. 9-10; No. 15, April 1969,
pp.
8-10,
and No. 16, June 1969, pp. 9-11
Checkers
Samuel (1959, 1967) continues to improve a program that plays
excellent checkers but can't quite beat world champions.
Kalah
Russell (1964) wrote an early Kalah program.
Slagle and Dixon (1969) describe experiments using the game of
Kalah.
Slagle and Dixon (1970) discuss more experiments using Kalah to
test the
"m
&
n"
(The Kalah programs
procedure.
are probably unbeatable by human players.)
Go
Zobrist (1969) has written a program to play this ancient and
difficult
game.
It plays rather poorly by human standards
and does no tree searching.
For an example of how
Zobrist's
program performs, see SIGART Newsletter No. 18, pp. 20-22,
October 1969.
1150
F.
References
S. (1967)
Amarel,
Approach to Heuristic Problem-Solving and Theorem Proving in the
Propositional Calculus," Systems and Computer Science, J. Hart and S.
Takasu (Eds.), Univ. of Toronto Press, 1967.
"An
Bernstein, A. et al.
(1958)
"A Chess-Playing Program for the IBM 704 Computer," Proc. of the Western
Joint Computer Conference,
pp. 157-159, 1958
.
and T. Hart
Edwards, D.
(1963)
"The a-g Heuristic,"
Massachusetts Institute of Technology Artificial
Intelligence Memo no. 30 (Revised), Oct. 28, 1963 (Originally printed
as "The Tree Prune (TP) Algorithm," Dec. 4, 1961).
Good,
(1968)
I.
Intelligence 2,
Plan for Automatic Chess," in Machine
Co., Inc.,
Publishing
E. Dale and D. Michie (Eds. ), American Elsevier
New York, 1968, pp. 89-118.
"A Five-Year
Greenblatt,
R. , et al (1967)
Greenblatt Chess Program," Proc. ol the AFIPS Fall Joint Computer
Conference, Anaheim, Call 1., 1967, pp. 801-810
"The
Hart, P.,
N. Nilsson and B. Raphael
"A Formal Basis for the
Paths," lEEE Trans, on
No. 2, July,
Kister,
J.
,
et al
1968, pp.
Kotok, A.
"A
Heuristic Determination of Minimum Cost
Systems Sciences and Cybernetics, Vol. SSC-4
,
100-107.
(1957)
"Experiments in Chess,"
April,
(1968)
J. of the ACM, Vol. 4, No.
2, pp.
174-177,
1957.
(1962)
Chess Playing Program for the IBM
7090,"
Massachusetts Institute of Technology, 1962.
Newell, A., J. Shaw and H. Simon
unpublished B.S. Thesis,
(1958)
'Chess Playing Programs and tho Problem ol Complexity," IBM Journal of
Research and Development, vol. 2, pp. 320-335, Oct., 1958. Reprinted in
Computers and Thought, Feigenbaum, E. and J. Feldman (Eds.) , McGraw-Hill
Book Co., New York, 1963, pp. 39-70.
V-52
Nilsson, N.
(1968)
"Searching Problem-Solving and Game-Playing Trees for Minimal Cost
Solutions," Proc. IFIP Cong. 68 , Edinburgh.
Russell, R.
(1964)
-
"Kalah The Game and The Program" Stanford University Artificial Intelligence
Project Memo No. 22, Sept. 3, 1964.
Samuel, A.
'
(1959)
I
MaChln Learnln Us
the Game of Checkers," IBM
JoZ f v^
° -229, 1959.
J"
'»«
pp . 211
*
Reprinted ln Computers and ThLht,
Vol^,
leigenbaum E. and J.
Eeldman
fl
Journal,
(Eds.), McGraw-Hill
, A.
Samuel
BoT^Cc ,
New
York"
1963
(1967)
"Some Studies in Machine Learning Using the Game of Checkers 11. Recent
Progress,"
IBM Journal of Research and Development, Vol. 11, No. 6,
Nov. 1967, pp. 601-617.
Shannon, C.
(1950)
"Programming
a Digital Computer for Playing Chess." Philosophy Magazine
Vol. 41, pp. 356-375, Reprinted in The World of
Mathematics, Newman (Ed.)
Vol. 4, Simon & Schuster, 1954, New York.
March 1950,
Slagle,
J.
(1963b)
"Game Trees, M
& N Minimaxing, and the M
Artificial Intelligence Group Report No. 3. UCRL-4671 , University' "
of
Calif. Lawrence Radiation Laboratory, Livermore,
Calif. Nov. ,1963.
& N Alpha-Beta Procedure
Slagle, J. and P. Bursky, (1968)
"Experiments with
a Multipurpose, Theorem-Proving Heuristic Program,
'
J. of ACM, Vol. 15, No., 1, pp. 85-99, Jan 1968.
Slagle,
J and J. Dixon
(1969)
M
Experiments with Some Programs that Search Game Trees,"
Vol. 16, No. 2, April 1969, pp. 189-207.
Slagle,
J. of ACM
J. and J. Dixon, (1970)
"Experiments with
Zobrist, A.
"A Model
the M& N Tree Searching Program,
" to
appear
(1969)
of Visual Organization for the Game of Go
Spring Joint Computer Conf.
pp. 103112, 1969.
,
V-53
in Proc.
AFIPS
(Chapter VI)
D.
Bibliographical and Historical Remarks
1.
Fundamentals of Logic
Our rather superficial treatment of mathematical logic can be
augmented by consulting
some of the
standard textbooks.
ones are those of Mendelson (1964) and Robbin (1969).
there is the classic by Alonso Church ( 1956 ).
"classical logic";
Two excellent
For the specialist
These treat what might be called
the resolution principle (or indeed any discussion of
automatic theorem-proving) has not yet found its way into textbooks on logic.
We have based this chapter on the first order predicate calculus with
resolution because of its apparent importance in automatic problem-solving.
A notable omission, however,
relation.
is any special discussion of the equality
It is still not clear how the equality relation (and other standard,
übiquitous relations) ought to be
"built-in"
to automatic theorem-provers.
A discussion of these complexities would have been beyond the scope of the
One scheme for building the equality relation into a resolution
book.
theorem-prover is discussed
by
Robinson and Wos (1969).
It also is becoming clear that complex, general problem-solvers will need
facilities for higher-order logics.
For a discussion of the application of
higher order logics to problem-solving,
see McCarthy and Hayes (1969).
Robbin's
book (1969) contains a section on second order logic, and some papers by J. A.
Robinson (1968a, 1969) discuss the general problem of proof procedures for higher
order logics.
The steps that we have outlined for converting any wff into clause form are
based on the procedure of Davis and Putnam (1960).
Clause form is also called
quantifier-free, conjunctive normal form.
2.
Herbrand Proof Procedures and Resolution
The resolution principle in automatic theorem-proving is based on the
1153
"semantic tableaux"
proof procedure of Herbrand (1930).
A direct implemen-
tation of Herbrand' s procedure would be grossly inefficient; improvements
due to Pra w itz (1960) and others ultimately led to the resolution principle of
J. A. Robinson (1965a). Our exposition of resolution proof methods, using the
notion of semantic trees,
is based on that of Kowalski and Hayes (1969).
also J. A. Robinson (1968b).)
(See
This semantic tree formulation makes obvious the
relationship between resolution and Herbrand methods and also justifies the use
of inference rules more general than simple resolution.
Our proof of the
completeness of resolution is a special case of a proof of the completeness of
a more
general
inference rule given by Kowalski
and Hayes (1969).
A clearly written, concise review of the resolution principle with proofs
of its soundness and completeness can be found in a
J. A. Robinson's initial paper (Robinson,
1965a)a150 contains
soundness and completeness of resolution
also contain proofs of the
"correctness"
paper by Luckham (1967).
proofs
of the
Both of these last mentioned papers
of the unification algorithm.
J. A.
Robinson (1970) has also written an excellent survey entitled "The Present
State of Mechanical Theorem-Proving".
1154
E.
References
(1956)
Church, A.
Introduction to Mathematical
1956.
Logic,
Princeton University Press,
Princeton, New Jersey,
Davis,
(1960)
M. and H. Putnam
"A
Computing Procedure for Quantification
No. 3, pp. 201-215, Ju1y, 1960.
Theory,"
Journal ACM,
Vol.
7,
(1930)
Herbrand, J.
"Recherches sur la theorie de la demonstration," Travaux de la Societe
des Sciences et dcs Lettres de Varsovie, Clas.se 111 Sciences Mathematiques
et Physiques, No. 33, 1930.
R.
Kowalski,
and P.
Hayes
(1969)
Trees in Automatic Theorem-Proving", in Machine Intelligence
,
B. Meltzer and D. Michie (Eds ) American Elsevier Publising Co. Inc.
New York ,1969, pp. 87-101.
"Semantic
Luckham, D.
. ,
1
.
(1967)
"
in Machine Intelligence 1,
Resolution Principle in Theorem-Proving,
Publishing Co., Inc.,
Elsevier
(Eds.
), American
N. Collins and D. Michie
1967, pp. 47-61.
"The
McCarthy,
J. and P. Hayes
(1969)
Intelligence
Problems from the Standpoint of Artificial
),
American
(Eds.
and D. Michie
in Machine Intelligence 4, B. Meltzer
ElseViTr~P"u¥]T.s~h7ng Co., Inc., New York, 1969, pp. 463-^O2.
"Some Philosophical
(1964)
Mendelson, E.
Introduction to Mathematical Logic.
1964.
Princeton, N.J.
,
Prawitz,
"An
Robbin,
D.
D. Van Nostrand Company,
Inc.
(1960)
Improved Proof
Procedure,"
Theoria,
J. (1969)
Mathematical
Logic:
Vol.
26, pp.
102-139, 1960.
,
1969.
A First Course, W. A. Benjamin, Inc. New Y0rk,
1155
Robinson,
G. and L. Wos
.
(1969)
and Theorem-Proving in First-Order Theories with Equality
B. Meltzer and D. Michie (Eds .) American
Machine Intelligence 4,
in
Elsevier Publishing Co., Inc., New York, 1969, pp. 135-150
"Paramodulation
(1965a)
Robinson, J. A.
"A Machine-Oriented Logic Based on the Resolution Principle,'
J. ACM, Vol. 12, No. 1, pp. 23-41, Jan., 1965.
Robinson,
J.A
(1968a)
in Mechanical Theorem Proving," Proc. of the IFIP Cong.
Edinburgh, 1968.
"New Directions
1968,
Robinson, J. A.
(1968b)
"The Generalized Resolution Principle, in Machine Intelligence 3,
D. Michie (Ed.), American Elsevier Publishing Co., Inc., New York,
pp. 77-93, 1968.
Robinson,
J. A.
(1969)
"Mechanizing Higher Order Logic," in Machine Intelligence 4, B. Meltzer
and D. Michie (Eds.), American Elsevier Publishing Co., Inc., New York,
1969.
Robinson,
J. A.
(1970)
"The
Present State of Mechanical Therem Proving," Formal Systems and
Non-Numerical Problem > Solving by Computer, M. Mesarovic and R. Banerji
(Eds.)
1156
(Chapter VII)
D.
Bibliographical and Historical Remarks
1
.
Development of
Formal
Logic Methods
We have already mentioned that work on techniques for solving
problems using formal logic methods was stimulated by the
memoranda of
McCarthy (1958, 1963)
a system was undertaken by Black (1964)
.
.
"advice taker"
Work toward implementing such
Cordell Green was the first to
develop a formal logic problem solver using a complete inference system
(resolution) for first-order logic.
Much of this work is described in his
dissertation (Green, 1969a) and in two papers (Green, 1969b, 1969c).
Professor McCarthy continued his
investigations on the re-
quirements for a general, formal problem-solving system.
He was particularly
concerned with the necessity for including higher (than first) order logic
features in order to formalize such concepts as situations, future operators,
actions, strategies, results of strategies, and knowledge.
An excellent
discussion of these ideas is contained in the paper by McCarthy and Hayes
(1969)
.
Many of the issues raised by McCarthy and Hayes are beyond
the scope of an introductory text
.
Our treatment of formal logic methods in
this chapter is based almost exclusively on the work of Green (1969a)
2
.
.
Question-Answering Systems
One of the tasks that could be given to a formal problem-solving
system is that of
"question answering." This
task involves a sophisticated
type of information retrieval in which the answers to queries require that
logical deductions be made from various facts stored
design of question-answering systems
in the data base.
also raises the question of how to
VII-31
The
translate between a natural language, such as English, and a formal language,
such as the predicate calculus, used by the deductive system
An
Raphael
(1964 a,
early general question-answering system
1964b)
.
was developed by
Raphael concentrated on the deductive and associative
mechanisms needed and largely ignored the issue of natural-language translation
On the other hand, Bobrow (1964a, 1964b) developed a system for solving simple
algebra problems stated in English.
His system could translate these into the
appropriate equations to be solved.
Another general question-answering system
called DEDUCOM (without English-logic translation abilities) was developed by
Slagle
(1965)
.
Green and Raphael (1968) collaborated in the development of a
question-answering system using resolution and first-order logic.
Coles
(1968)
has developed a limited English-logic translation program that has been added
to this question-answering system.
Two good surveys of work on natural-language question-answering
systems have been written by Simmons (1965,
3.
1970)
.
Answer-Extraction Processes
Although the subject of computing instances of existentially
quantified variables has been discussed in classical proof theory.
Green (1969b) was the first to propose a definite
The answer-extraction method discussed
procedure for resolution-based systems.
Green's technique
in this chapter is a generalization of
paper by Luckham and Nilsson (1970)
4
.
and is based on a
.
Example Applications of the Formal Logic Method
Our automatic program-writing example is an adaptation of one
discussed by Green (1969 a, Appendix C)
.
The problem of writing computer
VII-32
programs is related to that of proving them correct.
There is a substantial
body of work on this latter subject including papers by McCarthy (1962)
Floyd (1967b)
,
and Manna (1969)
.
London ( 196§ gives a good survey of work
on proving the correctness of programs.
resolution-based)
,
A somewhat different (but still
procedure for automatic program writing is described by
Waldinger and Lee (1969)
.
Applications of theorem-proving techniques to automatic problem
solving are well discussed by Green (1969 a,
1969c)
.
The state-variable method
described in this chapter was proposed by Green.
One of the most obvious applications of automatic theorem
provers (although it is not discussed in this chapter) is proving mathematical
theorems.
This application has been pursued by
by Guard, et al
.
(1969).
Robinson
and Wos (1969)
and
In particular, Guard's program (with some human
aid) has succeeded in finding the first proof for a conjecture in modular
lattice theory.
VII-33
E.
References
Black,
F.
(1964)
"A
Deductive Question-Answering System, "Ph.D. Dissertation, Harvard,
June 1964. Reprinted in Semantic Information Processing, M. Minsky (Ed)
MIT Press, Cambridge, Mass., 1968, pp. 354-402.
,
(1964a)
Bobrow, D.
"Natural Language Input for a Computer Problem-Solving System," Ph.D.
dissertation, MIT, Sept. 1964. Reprinted in Semantic Information
Processing, M. Minsky (Ed), MIT Press, Cambridge, Mass., 1968.
(1964b)
Bobrow, D.
Question-Answering System for High-School Algebra Word Problems,'
in Proc. of AFIPS Fall Joint Computer Conference, 1964, pp. 591-614.
"A
Coles, L.
S.
(1968)
An On-Line Question-Answering System with Natural Language and Pictorial
Input," Proc.
ACM 23rd Nat. Conf.,
N.J., pp. 157-167.
1968, Brandon Systems Press, Princeton,
Floyd, R. (1967b)
"Assigning
Meanings to
Programs," Proc. of Symposia in Applied
Mathematics, American Mathematical Society, Vol.
19, pp. 19-32,
19b/.
„
(1969a)
Ph.D
"The Application of Theorem-Proving to Question-Answering Systems,June,
Dept.,
Engineering
dissertation, Stanford University, Electrical
Project Memo
1969. Also printed as Stanford Artificial Intelligence
AI-96, June,l969.
Green, C.
Green. C.
(1969b)
"Theorem-Proving
by Resolution as a Basis for Question-Answering Systems,
in Machine Intelligence 4, B. Meltzer, and D. Michie (Eds. ), American
Elsevier Publishing Co., Inc., New Y0rk, 1969, pp. 183-205.
VII-34
Green, C.
(1969c)
"Application of Theorem-Proving to Problem Solving," Proc. Int'l Joint
Conference on Artificial Intelligence, Wash. D.C., May, 1969.
and B. Raphael
Green, C.
(1968)
The Use of Theorem-Proving Techniques in Question-Answering Systems,"
Brandon Systems Press, Princeton,
Proc. of 23rd Nat'l. Conf. ACM,,
N.J. , pp. 159-181.
Guard, J. et al,
(1969)
"Semi-Automated Mathematics,"
J
Jan, 1969.
.
oi ACM
, Vol.
16
,
No.
1, pp. 49-62,
(1969)
London, R.
on Proving the Correctness of Computer Programs," Tech.
June, l969.
University of Wisconsin, Computer Sciences Dept.
"Bibliography
Report #64,
,
Luckham, D. and N. Nilsson (1970)
"Extracting Information from Resolution Proof Trees," to appear.
(1969)
Manna, Z.
"The Correctness
May
of
Programs,"
J. of Computer and System Sciences, Vol.
3
,1969.
McCarthy,
J. (1958)
1
Thought Processes, Vol
London,
Nov.
Laboratory,
National
Physical
pp. 77-84, Proc. Symposium,
Minsky (Ed )
24-27, 1958. Reprinted in Semantic Information Processing, M.
MIT Press, Cambridge, Ma55. ,1968, pp. 403-410.
"Programs with Common Sense," in Mechanization of
McCarthy,
J. (1962)
a Mathematical Science of Computation, Proc.
North Holland Publishing Co. , Amsterdam, 1962.
"Towards
McCarthy,
IFIP Congress 62,
J. (1963)
'Situations, Actions and Causal Laws,"
Stanford University Artificial
Reprinted in Semantic InforIntelligence Project Memo. No. 2, 1963.
Cambridge, Ma55 .,1968
Press,
mation Processing, M. Minsky, (Ed.) MIT
pp. 410-418.
,
VII-35
McCarthy, J. and P. Hayes
(1969)
Problems from the Standpoint of Artificial Intelligence
in Machine Intelligence 4, B. Meltzer and D. Michie (Eds. ), American
Elsevier Publishing Co., Inc., New York, 1969, pp. 463-502.
"Some Philosophical
(1964a)
Raphael, B.
A Computer Program for Semantic Information Retrieval," Ph.D
MIT, June 1964. Reprinted in Semantic Information
MIT Press, Cambridge, Mass. 1968.
Processing, M. Minsky (Ed.)
"SIR:
dissertation,
,
Raphael, B. (1964b)
"A Computer Program Which 'Understands'",
Computer Conference, 1964, pp. 577-589.
G. and L. Wos
Robinson,
in Proc.
of AFIPS Fall Joint
.
(1969)
and Theorem-Proving in First-Order Theories with Equality.'
B. Meltzer and D. Michie (Eds. ) American
in
Machine Intelligence 4,
Co.,
Inc.,
New York, 1969, pp. 135-150
Elsevier Publishing
"Paramodulation
Simmons, R.
(1965)
"Answering English Questions by Computer: A Survey," Communications of
the ACM, Vol,
Simmons, R.
"Natural
8, Jan. 1965, pp. 53-70.
(1970)
Language Question Answering Systems:
Vol. 13, No. 1, pp. 15-30, January 1970.
1969,"
Comm. ACM,
'
(1965)
Slagle, J.
"Experiments with a Deductive-Question Answering Program,
the ACM,
Waldinger,
Vol.
8, pp. 792-798, Dec.
Communcations of
1965.
R. and R. Lee (1969)
"PROW:
A Step Toward Automatic Program-Writing," Proceedings of
Int'l Joint Conference on Artificial Intelligence , Wash D C
May,
. .
1969.
VI I -36
(Chapter VIII)
E
.
Bibliographical and Historical Remarks
1
.
Discussions of Search Strategies
An excellent discussion of the problem of developing
heuristically effective search strategies while maintaining logical
completeness is contained in a paper by J. A. Robinson (1967).
Robinson,
G. A
et al (1964) give a very clear exposition of several strategies
with many examples.
2
.
Simplification Strategies
The problem of the elimination of subsumed clauses during
a search for a proof is more subtle than it might appear.
Kowalski (1970a)
acknowledges inadequacies in the treatment of this subject by Kowalski and
Hayes (1969) and refers the reader to his dissertation (Kowalski,
1970b) for
a complete discussion.
3
.
Refinement Strategies
The Ancestry Filtered (AF) form strategy is the simplest of
a family of related strategies.
Our Theorem 8.1 states that the AF-form
strategy is logically complete; a proof of this theorem is given in Luckham
(1969)
.
Although we show the completeness of the set-of -support strategy as a
corollary of Theorem 8.1, it was in fact first developed and justified by
Wos. et al (1965)
.
Several elaborations of the AF-form strategy are possible
Some of these combine the AF-form strategy with a strategy proposed by
Andrews (1968) involving merges
.
Among the papers proving the completeness
of AF-form with merging are Kieburtz and Luckham (1970)
Hart (1970) and Anderson and Bledsoe (1970).
VIII-20
,
Yates, Raphael, and
The first of these contains
other results about the properties of AF-form proofs, while the last two
use rather novel methods of proving completeness that are of independent
interest
.
Anderson and Bledsoe use their method to establish the
completeness of other strategies as well.
A paper by Loveland (1968)
establishes the completeness of another restriction on the AF-form
strategy
.
The model strategies are elaborations of the P
proposed by J. A. Robinson (1965b).
deductions originally
Slagle (1967) proved the completeness
of some very general model strategies.
Our Theorem 8.2 is due to Luckham
(1969) and is an easier-understood, special case of one of Slagle 's theorems.
Meltzer (1966, 1968) provides some additional results about P -type deductions
.
4
.
Ordering Strategies
The unit preference strategy was proposed and justified in
a paper by Wos
incorporated
,
et al (1964)
as basic
.
It and the set of support strategy have been
parts of a number of automatic theorem-provers
.
The reader probably noticed that all of the search strategies
discussed in this chapter involved
"syntactic" rather than "semantic" rules
(that is, search restrictions and orderings were based on the form of the
clauses and possible deductions rather than on their meaning)
.
Semantic
guidance could be provided in a number of ways, but there have not yet
been many attempts in this direction.
possible to use an
One might ask whether it would be
"evaluation function" over
candidates to be resolved.
pairs of clause that are
Presumably this evaluation function could be
influenced by the available semantic information as well as by the forms
of the candidate clauses.
Some theoretical results about the properties
VIII-21
of strategies using evaluation-functions are contained in a paper by
a
Kowalski (1970
).
Kowalski' s results on searching "inference-graph"
structures parallel those of Hart, et al (1968) for state-space graph
structures
.
5.
Examples of Implementations
Several automatic theorem-proving programs have now been
written.
As examples we might mention those of Robinson and Wos (1969),
Allen and Luckham
(1970)
, Guard,
et al (1969) and the Green-Raphael-
Yates system recently modified by Garvey and Kling (1969).
VIII-22
References
F.
D.
Allen, J. and
Luckham (1970)
"An Interactive
Theorem-Proving
Program," in Machine
Intelligence 5
(1968)
Andrews, P.
with Merging," Journal of ACM, Vol. 15, No. 3, pp 367-381
July 1968.
(Also see corrections in JACM, Vol. 15, No. 4, Oct. 1968
p 720).
"Resolution
Anderson, R. and W. Bledsoe (1970)
"A Linear Format for Resolution with Merging and a New Technique for
Establishing Completeness ," Journal of ACM (to appear)
.
Garvey, T. and R. Kling
(1969)
.
"User's
Guide to QA3 5 Question-Answering System," Stanford Research
Institute Artificial Inetlligence Group Technical Note 15, Dec, 1969.
Guard,
J. et al,
(1969)
"Semi-Automatod Mathematics,"
J. ol ACM, Vol.
16, No.
1, pp. 49-62,
Jan, 1969.
Hart, P., N. Nilsson and B. Raphael
"A Formal Basis for the
Paths," lEEE Trans, on
No.
(1968)
Heuristic Determination of Minimum Cost
Systems Sciences and Cybernetics, Vol. SSC-4,
2, July, 1968, pp. 100-107.
Kieburtz, R.
and D. Luckham
(1970)
"Compatibility of Refinements of the Resolution Principle," to appear.
(1970a)
Kowalski, R.
"Search
Strategies for
Kowalski, R.
"Studies
Theorem-Proving" in Machine Intelligence 5.
(1970b)
in the Completeness and Efficiency of Theorem-Proving by
Ph.D. Thesis, Univers ity of Edinburgh.
Resolution,"
VIII-23
Kowalski, R.
and
P. Hayes
(1969)
"Semantic Trees in Automatic Theorem- Proving" in Machine Intelligence 4
B. Meltzer and D. Michie (Eds .) .American
Elsevier Publising Co. Inc. ,
New York ,1969, pp. 87-101.
,
(1968)
Loveland, D.
"A Linear Format for
Resolution,"
Carnegie-Mellon University Computer
Also to appear in Proc. IRIA 1968
Science Dept. Report, December 1968.
Symposium on Automatic Deduction.
Luckham, D.
(1969)
"Refinement Theorems in Resolution Theory,"
Stanford Artificial
Intelligence Project Memo AI-81, March 24, 1969. Also to appear in
Proc. IRIA 1968 Symposium on Automatic Deduction.
Me l l ze r , B
.
(1966)
"Theorem-Proving for Computers : Some Results on Resolution
Renaming," Computer Journal, Vol. 8, pp 341-343, 1966.
Meltzer,
and
(1968)
B.
Notes on Resolution Strategies," in Machine Inetlligence 3,
D. Michie (Ed.), American Elsevier Publishing Co., Inc., New York,
1968, pp. 71-75.
"Some
Robinson,
G. and L. Wos
.
(1969)
and Theorem-Proving in First-Order Theories with Equality
B. Meltzer and D. Michie (Eds. ) American
in
Machine Intelligence 4,
Elsevier Publishing Co., Inc., New York, 1969, pp. 135-150.
"Paramodulation
Robinson,
G. A., L.T. Wos and D. F. Carson (1964)
Theorem- Proving Strategies and Their Implementation,"
Nat'l Laboratories Technical Memorandum No. 72, 1964.
"Some
Robinson,
J. A.
(1965b)
"Automatic Deduction
Vol.
1,
pp.
Argonne
with Hyper-Resolution,"
227-234, 1965.
VI I I -24
Int'l J. of Comp. Math.
Robinson,
J. A.
(1967)
"Heuristic and Complete Processes in the Mechanization of TheoremProving, ' in Systems and Computer Science, J. T. Hart and S. Takasu
(Eds.), Univ. of Toronto Press, 1967,pp. 116-124.
Slagle,
(1967)
J.
Theorem-Proving with Renamable and Semantic
Vol. 14, No. 4, pp. 687-697, Oct., 1967.
of ACM,
"Automatic
J.
Wos, L. D.
Resolution,"
(1964)
Carson, and G. Robinson
"
"The
Proc. of the AFIPS
Unit Preference Strategy in Theorem Proving,
26,
616-621,
1964.
Conf.,
1964 Fall Joint Computer
Vol.
pp.
Wos, D., G. Robinson, and D. Carson
(1965)
"Efficiency and Completeness of the Set of Support Strategy in TheoremProving,"
Yates, R.,
JACM, Vol.
12, No. 4,
B. Raphael, and T. Hart
"Resolution Graphs," Stanford
pp. 536-541, Oct., 1965.
(1970)
Research Institute Artificial Intelligence
Memo.
VIII-25
COMBINED REFERENCE LIST
Allen, J. and D, Luckham (1970)
"An
Interactive Theorem-Proving
Anderson, R.
Program," in Machine Intelligence 5
and W. Bledsoe (1970)
A Linear Format for Resolution with Merging and a New Technique for
Establishing Completeness ," to appear in Journal of ACM.
Andrews, P.
(1968)
"Resolution
July 1968.
p 720).
Amarel, S.
with Merging," Journal of ACM, Vol. 15, No. 3, pp 367-381
(Also see corrections in JACM, Vol. 15, No. 4, Oct. 1968
(1965)
"Problem
Solving Procedures for Efficient Syntactic Analysis" ACM
20th National Conference, 1965 (Also printed as Scientific Report
No. 1, AFOSR Contract No. AF49(638)-1184, 1968).
Amarel, S. (1967)
"An Approach to Heuristic Problem-Solving and Theorem Proving in the
Propositional Calculus," Systems and Computer Science, J. Hart and S.
Takasu (Eds.), Univ. of Toronto Press, 1967.
Amarel, S.
(1968)
"On Representations
of Problems of Reasoning About Actions,"
in Machine Intelligence 3, D. Michie (Ed),
American Elsevier, New
York,
Amarel, S.
1968, pp.
131-171.
(1969)
"On the Representation of Problems and Goal Directed Procedures for
Computers," Communications of the American Society for Cybernetics,
Vol. 1, No. 2, 1969.
1169
Ball, W.
(1931)
Mathematical Recreations and Essays, Tenth Edition, 1931,
Mac Millan &Co
(15 puzzle, pp. 224-228; Tower of Hanoi, pp. 228-229.)
London,
Banerji, R.
(1969)
Theory of Problem Solving: An Approach to Artificial Intelligence,
American Elsevier Publishing Co. Inc., New York, 1969.
Bellman, R. and S. Dreyfus (1962)
Applied Dynamic Programming, Princeton University Press,
Bellmore, M. and G. Nemhauser
"The
Vol.
Berge,
C.
1962.
(1968)
Traveling Salesman Problem: A Survey,"
16, No. 3, May-June 1968, pp. 538-558.
Operations Research
(1962)
The Theory of Graphs and its Applications, (Translated by Alison Doig)
John Wiley & Sons, Inc. New York, 1962.
(First published in France
in 1958 by Dunod, Paris).
Bernstein, A.
et al
.
(1958)
"A Chess-Playing Program for the IBM 704 Computer,
pp. 157-159,
Joint Computer Conference,
1958.
Black, F.
,
"
Proc. of the Western
(1964)
A Deductive Question-Answering System, "Ph.D. Dissertation, Harvard,
June 1964. Reprinted in Semantic Information Processing, M. Minsky (Ed)
MIT Press, Cambridge, Mass., 1968, pp. 354-402.
1170
,
.
(1964a)
Bobrow, D.
Language Input for a Computer Problem-Solving System," Ph.D.
dissertation, MIT, Sept. 1964. Reprinted in Semantic Information
Processing, M. Minsky (Ed), MIT Press, Cambridge, Mass., 1968.
"Natural
(1964b)
Bobrow, D.
"A
Question-Answering System for High-School Algebra Word Problems,
in Proc. of AFIPS Fall Joint Computer Conference, 1964, pp. 591-614.
Buchanan, 8.
,
Feigenbaum (1969)
G. Sutherland, and E.
"Heuristic
Dendral: A Program for Generating Explanatory Hypotheses
in Organic Chemistry." in Machine Intelligence 4, B. Meltzer and D.
Michie (Eds.), American Elsevier Publishing Co. , Inc., New York,
1969, pp. 209-254.
N. and D. Michie (Eds.)
Machine Intelligence 1,
American Elsevier
Collins,
New York,
Campbell, D.
Publishing
Co., Inc.
1967.
(1960)
"Blind Variation and Selective Survival as a General Strategy in
Knowledge-Processes," in Self-Organizing Systems, M. Yovits & S. Cameron
(EdsJ.Pergamon Press, New York 1960, pp. 205-231.
(1956)
Church, A.
Introduction to Mathematical Logic,
Princeton, New Jersey, 1956.
Coles, L.
S.
Princeton University Press,
(1968)
An On-Line Quest ion-Answering System with Natural Language and
Pictorial Input," Proc. ACM 23rd Nat'l. Conf., 1968, Brandon Systems
Press, Princeton, N.J., pp. 157-167.
1171
and D. Michie (Eds)
Dale, E.
Machine Intelligence 2
York, 1968.
Davis, M. and H. Putnam
"A
American Elsevier Publishing Co., Inc., New
(1960)
Computing Procedure for Quantification Theory,
No. 3, pp. 201-215, Ju1y, 1960.
De Russo,
P.
,
R. Roy and C. Close
"A
Journal ACM,
Vol
(1965)
State Variables for Engineers, Wiley, New York,
Dijkstra, E.
"
1965
(1959)
Note on Two Problems in Connection with Graphs
1, 1959, pp. 269-271
,"
Numerische
Mathematik,
Doran,
J.
(1968)
"New
"
Developments of the Graph Traverser,
in Machine Intelligence 2,
Dale
and
D.
Michie
American
Elsevier
(Eds),
E.
Publishing Co. , Inc.,"
New York, 1968, pp. 119-135.
Doran,
J. and D. Michie (1966)
"Experiments with the Graph Traverser Program, "
Royal Socitey, A, Vol. 294, pp. 235-259, 1966.
Proceedings of the
Dreyfus, H. (1965)
Alchemy and Artificial Intelligence, Rand Corporation Paper P3244
(AD 625 719), December ,1965.
-R4-
7
Dreyfus,
S.
(1969)
Appraisal of Some Shortest Path Algorithms,"
Vol. 17, No. 3, pp. 395-412, May-June 1969.
"An
Dudeney,
H.
Operations Research,
(1958)
The Canterbury Puzzles,
Dover Publications, Inc., New York, 1958.
(Originally published in 1907.)
Dudeney, H. (1967)
536 Puzzles and Curious Problems, Martin Gardner (Ed. ), Charles Scribner
1967. (A collection from two of Dudeney' s books:
Modern Puzzles, 1926, and Puzzles and Curious Problems, 1931).
Sons, New York,
Edwards, D. and T. Hart
(1963)
"The a-|9 Heuristic,"
Massachusetts Institute of Technology Artificial
Intelligence Memo no. 30 (Revised), Oct. 28, 1963 (Originally printed
as "The Tree Prune (TP) Algorithm," Dec. 4, 1961).
Ernst, G.
(1969)
"Sufficient
Conditions for the Success of
16, No. 4, Oct. 1969, pp. 517-533.
GPS,"
Journal of ACM, Vol.
Ernst, G. and A. Newell (1969)
GPS: A Case Study in Generality and Problem Solving
Series, Academic Press, 1969.
1173
ACM Monograph
s
Feigenbaum,
E.
(1963)
"Artificial Intelligence Research,"
Vol. IT-9, No. 4, Oct., 1963,
Feigenbaum,
pp.
lEEE Trans, on Info. Theory,
248-261
E. (1968)
Themes in the Second Decade,"
Proc. of
IFIPS 68 Congress. Edinburgh, Aug. 1968. Also printed as Stanford
University Artificial Intelligence Project Memo No. 67, Aug. 15, 1968
"Artificial Inteligence:
Feigenbaum, E.
and J. Feldman (Eds.)
Computers and Thought, McGraw Hill Book Co., New Y0rk, 1963.
Feldman, J. and D. Gries (1968)
"Translator
pp.
Fikes, R.
Writing
Systems" Comm. of
ACM,
Vol. 11, No. 2, Feb., 1968,
77-113.
(1970)
"Ref-Arf: A System for Solving Problems Stated as
Artificial Intelligence, Vol.l, No.l, 1970.
Procedures,'
Floyd, R. (1967a)
"Nondeterministic Algorithms,"
pp.
Floyd, R.
J. of ACM, Vol. 14, No. 4, 0ct., 1967,
636-644.
(1967b)
"Assigning Meanings to Programs," Proc. of Symposia in Applied
Mathematics, American Mathematical Society, Vol. 19, pp. 19-32, 1967.
1174
Fogel, L.
A. Owens and M. Walsh
(1966)
Artificial Intelligence Through Simulated Evolution, John Wiley, New York,
~~
~~
"
"
1966.
Ford, L.
Jr., and D.
~
Fulkerson (1962)
Flows in Networks, Princeton University Press, Princeton,
Gardner,
M.
New Jersey,
1962
(1959)
_
The Scientific American Book of Mathematical
Puzzles
...„
„
.
and Diversions,
~
Simon and Schuster, New York, 1959
Gardner, M.
.—
...,-.,...v,u
11 lilt
J-'J.V^_XtJJ.»_»ll^)
(1961)
The Second Scientific American Book of Mathematical Puzzles and Diversions
Simon and Schuster, New Y0rk, 1961
Gardner, M. (1964, 1965 a,b,c)
Mathematical Games, Scientific American, Vol. 210, No. 2, Feb., 1964,
pp. 122-13p;Vol
212, No. 3, March, l96s, pp. 112-117;
Vol. 212,
6
June, l96s, pp. 120-124; Vol. 213, No. 3, Sept., 1965, pp. 222-236.
.
No!
Garvey, T. and R. Kling
(1969)
.
User's Guide to QA3 5 Question-Answering System," Stanford Research
Institute Artificial Inetlligence Group Technical Note 15, Dec, 1969.
Gelernter, H.
(1959)
'Realization
of a Geometry Theorem-Proving Machine," Proc. of an Int l
UNESCO House, Paris, 1959, pp. 273-282. Reprinted
in Computers and Thought , Feigenbaum, E. and J. Feldman (Eds.),
McGraw-Hill
Book Co., New York, 1963, pp. 134-152.
Conf. on Info. Proc
'
,
-H7-
Gelernter,
H. J. Hansen, and D. Loveland (1960)
of the Geometry Theorem Proving Machine," Proc
of the Western Joint Computer Conf. ,1960, Vol. 17, pp. 143-147. Reprinted
in Computers and Thought,
Feigenbaum, E. and J. Feldman (Eds.), McGraw-Hill
Book Co., New York, 1963, pp. 153-167.
"Empirical Explorations
Golomb, L.
and L. Baumert (1965)
"Backtrack
Programming,
"
J. of ACM, Vol. 12, No. 4., 0ct. ,1965, pp. 516-
524.
I.
Good,
(1968)
in Machine Intelligence 2,
"A Five-Year Plan for Automatic Chess
E. Dale and D. Michie (Eds. ), American Elsevier Publishing Co., Inc.,
New York, 1968, pp. 89-118.
(1969a)
"The Application of Theorem-Proving to Question-Answering Systems" Ph.D
dissertation, Stanford University, Electrical Engineering Dept., June
1969. Also printed as Stanford Artificial Intelligence Project Memo
AI-96, June, l969.
Green,
C.
Green,
C.
(1969b)
"Theorem-Proving by Resolution as a Basis for Question-Answering Systems,'
in Machine Intelligence 4, B. Meltzer, and D. Michie (Eds. ), American
Elsevier Publishing Co.,
Green,
C.
Inc., New Y0rk, 1969, pp.
(1969c)
"Application of
Theorem-Proving to Problem Solving," Proc. Int'l
Intelligence, Wash. D. C. May, 1969
,
Conference on Artificial
Green, C.
183-205.
and B. Raphael
.
Joint
(1968)
"The Use of Theorem-Proving Techniques in Question-Answering Systems,
Proc. of 23rd Nat' l. Conf. ACM, Brandon Systems Press, Princeton, N.J.,
pp. 169-181, 1968.
1176
Greenblatt, R.
,
et al (1967)
"The Greenblatt
Chess
Program,"
Conference, Anaheim, Calif.,
Proc. of the AFIPS Fall Joint Computer
1967, pp. 801-810
(1969)
Guard, J. et al,
"Semi-Automated Mathematics,"
J. of ACM, Vol. 16, No.
1, pp. 49-62,
Jan, 1969.
Hart, P., N. Nilsson and B. Raphael
"A Formal Basis for the
Paths," lEEE Trans, on
(1968)
Heuristic Determination of Minimum Cost
Systems Sciences and Cybernetics, Vol. SSC-4,
No. 2, July, 1968, pp. 100-107.
(1930)
Herbrand, J.
la theorie de la demonstration," Travaux de la Societe
dcs Sciences et dcs Lettres de Varsovie Classe 111 Sciences Mathematiques
et Physiques, No. 33, 1930.
"Recherches sur
Jelinek, F.
"Fast
Vol.
,
(1969)
Sequential Decoding Algorithm Using a
6, Nov, 1969, pp. 675-685.
Stack,"
IBM J. of Res, and Dev.
13, No.
Kieburtz, R.
and D. Luckham
(1970)
"Compatibility of Refinements of the Resolution Principle," to appear.
Kister,
J.
,
et al
"Experiments
(1957)
in
Chess,"
J. of the ACM, Vol. 4, No.
2, pp. 174-177,
April, 1957.
Kotok, A.
(1962)
Chess Playing Program for the IBM 7090," unpublished B.S. Thesis,
Massachusetts Institute of Technology, 1962.
"A
Kowalski, R.
"Search
(1970a)
Strategies for Theorem-Proving'
1177
in Machine Intelligence 5.
J
Kowalski, R.
(1970b)
in the Completeness and Efficiency of Theorem-Proving by
Ph.D. Thesis, Univers ity of Edinburgh.
"Studies
Resolution,"
Kowalski, R. and P. Hayes
(1969)
.
Trees in Automatic Theorem-Proving", in Machine Intelligence 4
B. Meltzer and D. Michie (Eds. ) American Elsevier Publising Co. Inc.
New York ,1969, pp. 87-101.
"Semantic
Lawler,
,
E. and D. Wood (1966)
and Bound Methods: a Survey,"
pp. 699-719, July-August ,l966.
"Branch
Levy, D.
Operations Research, Vol.
14, No
(1969)
on Some Tree Searching Methods Which is Designed to Optimize
the Use of Storage," ACM SIGART Newsletter, No. 15, April 1969, pp.
14-16.
"A Comment
Lin
,
Shen (1965)
"Computer Solutions of the Traveling Salesman Problem,"
Technical Journal, Vol. XLIV, No. 10, Dec, 1965.
The Bell System
Lin, Shen (1968).
"Heuristic Techniques for Solving Large Combinatorial Problems on a
Computer," Formal Systems and Non-Numerical Problem-Solving by Computer,
Case Western Reserve Uuniversity,
London, R.
Cleveland, Ohio, N0v. ,1968.
(1969)
"Bibliography on
Report #64,
Proving the Correctness of Computer
Programs," Tech.
University of Wisconsin, Computer Sciences Dept. f June, l969.
1178
"1
(1968)
Loveland, D.
"A Linear Format for Resolution," Carnegie-Mellon University Computer
Science Dept. Report, December 1968. Also to appear in Proc IRIA 1968
Symposium on Automatic Deduction.
(1967)
Luckham, D.
"The Resolution Principle in Theorem-Proving," in Machine Intelligence 1,
N. Collins and D. Michie (Eds. ), American Elsevier Publishing Co., Inc.,
1967, pp. 47-61.
(1969)
Luckham, D.
Theorems in Resolution Theory," Stanford Artificial
Intelligence Project Memo AI-81, March 24, 1969. Also to appear in
Proc. IRIA 1968 Symposium on Automatic Deduction.
"Refinement
Luckham, D.
and N. Nilsson (1970)
"Extracting Information from Resolution Proof
"The
Manna,
to appear.
(1969)
Manna, Z.
May
Trees,"
Correctness of Programs,"
J. of Computer and System Sciences, Vol. 3
,1969.
Z. (1970)
Correctness of Non-Deterministic Programs,"
Vol. 1, No. 1, 1970.
"The
1179
Artificial
Intelligence
McCarthy, J. (1958)
"Programs with Common Sense," in Mechanization of Thought
Processes, Vol. I,
London,
Nov.
pp. 77-84, Proc. Symposium, National Physical Laboratory,
24-27, 1958. Reprinted in Semantic Information Processing, M. Minsky (Ed.)
MIT Press, Cambridge, Mass., 1968, pp. 403-410.
,
McCarthy, J. (1962)
a Mathematical Science of Computation, Proc.
North Holland Publishing Co. , Amsterdam, 1962.
"Towards
McCarthy, J. and P. Hayes
IFIP Congress 62,
(1969)
"Some Philosophical
Problems from the Standpoint of Artificial Intelligence
in Machine Intelligence 4, B. Meltzer and D. Michie (Eds. ), American
Elsevier Publishing Co. , Inc., New York, 1969, pp. 463-502.
McCarthy, J. (1963)
" Stanford University Artificial
Intelligence Project Memo. No. 2, 1963. Reprinted in Semantic Information Processing, M. Minsky, (Ed.)
MIT Press, Cambridge, Ma55. ,1968,
"Situations, Actions and Causal Laws,
,
pp.
410-418.
McCulloch,
W. S. and W. Pitts (1943)
"A
Logical Calculus of the Ideas Immanent in Neural Nets,
Mathematical Biophysics, Vol. 5, ,ip. 115-137, 1943.
Meltzer, B.
Bulletin of
(1966)
"Theorem-Proving for Computers : Some Results
Renaming,"
"
Computer Journal, Vol.
-Rl2-
on Resolution
8, pp 341-343, 1966.
and
(1968)
Meltzer, B.
"Some Notes on Resolution Strategies," in Machine Inetlligence 3,
D. Michie (Ed.), American Elsevier Publishing Co., Inc., New York,
1968, pp. 71-75.
Meltzer, B. and D. Michie (Eds.)
Machine Intelligence 4, American Elsevier Publishing Co.,
1969.
Mendelson, E.
(1964)
Introduction to Mathematical Logic.
Princeton, N.J., 1964.
Michie, D.
D. Van Nostrand Company,
M.
,
Inc., New York
(Ed.)
Machine Intelligence 5, American Elsevier Publishing Co.,
1970.
Minsky,
Inc.
(Ed.)
Machine Intelligence 3, American Elsevier Publishing Co.,
1968.
Michie, D.
Inc., New York
Inc., New York
(1961a)
Steps Toward Artificial Intelligence," Proc. IRE, Vol. 49, pp. 8-30,
January 1961. Reprinted in Computers and Thought, Feigenbaum, E. and
J. Feldman (Eds.), pp. 406-450, McGraw-Hill Book Co., New York, 1963.
Minsky, M.
(1961b)
A Selected Descriptor-Indexed Bibliography to the Literature on
Artificial Intelligence," IRE Trans, on Human Factors in Electronics,
HFE-2, pp. 39-55, March 1961. A revision is reprinted in Computers
and Thought, Feigenbaum, E. and J. Feldman (Eds.), pp. 453-523,
McGraw-Hill Book Co., New York, 1963.
1181
Minsky, M. (1967)
Computation: Finite and Infinite Machines, Prentice Hall, Inc.,
Englewood Cliffs,
Minsky, M.
N.J.
, 19677
and S. Papert (1969)
An Introduction to Computational Geometry,
Cambridge, Mass. 1969.
Perceptrons:
Minsky, M. (Ed.) (1968)
Semantic Information Processing,
Cambridge, Mass., 1968.
The MIT Press,
The MIT Press,
Moore, E. (1959)
"The Shortest Path Through a Maze," Proc. Int'l Symposium on the Theory of
Switching, Part 11, April 2-5, 1957, The Annals of the Computation
aboratory of Harvard University 30,
Harvard University Press ,1959.
Moses,
J. (1967)
Symbolic Integration,
Massachusetts Institute of Technology, Project MAC,
Report MAC-TR-47 (Thesis), Dec. 1967.
Newell A.
(1965)
'
"Limitations of the Current Stock of Ideas ab ut Problem Solving, in
Electronic Information Handling, A. Kent and 0. Taulbee (Eds. ), Spartan
Books, Wash. D.C., 1965.
Newell, A.
(1969)
"Heuristic
Programming:
Research,
J. Aronofsky
111-Structured Problems," Progress in Operations
(Ed.), Vol. 3, Wiley, 1969.
1182
Newell, A., J. Shaw, and H. Simon
(1957)
"Empirical Explorations
of the Logic Theory Machine," Proc. of the
Western Joint Computer Conference, 1957, pp. 218-239. Reprinted in
Computers and Thought Feigenbaum, E. and J. Feldman (Eds.), McGraw-Hill
Book Co., New York, 1963, pp. 109-133.
,
Newell, A., J. Shaw and H. Simon
(1958)
"Chess Playing Programs
and the Problem of Complexity," IBM Journal of
Research and Development, vol. 2, pp 320-335, Oct., 1958. Reprinted in
Computers and Thought, Feigenbaum, E and J. Feldman (Eds.) , McGraw-Hill
Book Co., New York, 1963, pp. 39-70.
Newell^A^J. Shaw and H. Simon (1959)
"Report on a General Problem-Solving Program,'
Proc. of Int'l Conf.
on Information Processing, UNESCO House, Paris, pp. 256-2 64, 1969.
Nilsson, N.
(1968)
"Searching Problem-Solving and Game-Playing Trees for Minimal Cost
Solutions,"
O'Beirne,
T. H.
Proc.
IFIP Cong. 68
,
Edinburgh.
(1961)
Puzzles and Paradoxes, New Scientist, No. 245, July 27,
August 3, 1961.
1961, and No.
246,
(1962)
Ore, 0.
Theory of Graphs, American Mathematical Society Colloquium Publication,
Vol. XXXVIII, Providence, Rhode Island, 1962.
Ore,
(1963)
0.
Graphs and Their Uses, Random House, 1963.
Papert,
S. (1968).
"The
Artificial Intelligence of Hubert L. Dreyfus. A Budget of Fallacies,
Massachusetts Institute of Technology Artificial Intelligence Memo #54,
Jan. ,1968.
1183
Pohl ,
I
.
(1969)
"Bi-Directional and Heuristic Search in Path Problems," Stanford
University Computer Science Dept. Ph.D. Dissertation, 1969. Also
printed as Stanford Linear Accelerator Center Report No. 104, May
1969.
Pohl, I.
(1970)
"First
Results on the Effect of Error in Heuristic Search,"
Machine Intelligence 5.
in
Polya, G. (1957)
How to Solve It
N. Y. ,1957.
Quinlan,
,
Doubleday & Co., Inc. Garden City
(1960)
Prawitz, D.
"An
(Second Edition)
Improved Proof
Procedure,"
Theoria, Vol.
26, pp.
102-139, 1960.
J. and E. Hunt. (1968)
"A
Formal Deductive Problem-Solving
No. 4, Oct., 1968, pp. 625-646.
J. of ACM, Vol.
15,
(1964a)
Raphael, B.
"SIR: A
Computer Program for
dissertation,
Processing,
Raphael, B.
System,"
MIT, June 1964.
M. Minsky (Ed.)
,
Semantic Information Retrieval,"
Ph.D
Reprinted in Semantic Information
MIT Press, Cambridge, Mass. 1968.
(1964b)
"A Computer
Computer
Program Which
Conference,
'Understands'", in Proc. of AFIPS Fall Joint
1964, pp. 577-589.
1184
Robbin,
J. (1969)
Mathematical Logic:
A First Course, W. A. Benjamin, Inc., New Y0rk, 1969.
Robinson, J. A.
(1965a)
"A Machine-Oriented Logic Based on the Resolution Principle,'
J. ACM, Vol. 12, No. 1, pp. 23-41, Jan., 1965.
Robinson,
(1965b)
J. A
Deduction with Hyper-Resolution,"
1, pp. 227-234, 1965.
"Automatic
Vol.
Robinson,
Int'l J. of Comp. Math.
(1967)
J. A.
Heuristic and Complete Processes in the Mechanization of TheoremProving, " in Systems and Computer Science, J. T. Hart and S. Takasu
(Eds.), Univ. of Toronto Press, 1967, pp. 116-124.
Robinson, J. A
(1968a)
"New Directions
1968,
Robinson,
in Mechanical Theorem Proving," Proc of the IFIP Cong.
Edinburgh, 1968.
J. A.
.
(1968b)
"The Generalized Resolution Principle," in Machine Intelligence 3,
D. Michie (Ed. ) American Elsevier Publishing Co., Inc., New York,
1968, pp. 77-93.
Robinson, J. A.
(1969)
"Mechanizing
Higher Order Logic,"
in Machine Intelligence 4, B. Meltzer
and D. Michie (Eds.), American Elsevier Publishing Co., Inc., New York,
1969.
1185
Robinson,
J. A.
(1970)
"
Formal Systems and
Present State of Mechanical Therem Proving,
Mesarovic
and R. Banerji
M.
by
Computer,
Solving
Non-Numerical Problem i
"The
(Eds.)
Robinson,
G. A., L.T. Wos and D.F. Carson (1964)
"Some Theorem-Proving Strategies and Their Implementation,
Nat'l Laboratories Technical Memorandum No. 72, 1964.
Robinson,
G. and L. Wos
"
Argonne
.
(1969)
and Theorem-Proving in First-Order Theories with Equality,'
B. Meltzer and D. Michie (Eds. ) American
Machine Intelligence 4,
in
pp. 135-150
Elsevier Publishing Co., Inc., New York, 1969,
"Paramodulation
Rosenblatt, F.
(1962)
Principles of Neurodynamics
Project
Samuel,
Spartan Books, New York 1962.
(1964)
Russell, R.
"Kalah
,
The Game and The Program"
-Memo
No. 22, Sept. 3, 1964.
A.
Stanford University Artificial Intelligence
(1959)
IBM
Studies in Machine Learning Using the Game of Checkers,"
Thought,
and
Computers
1959.
in
Reprinted
3,
211-229,
pp.
Journal, Vol.
Feigenbaum, E. and J. Feldman (Eds.), McGraw-Hill Book Co., New York, 1963,
pp. 71 - 105.
"Some
Samuel
, A.
(1967)
Recent
Learning Using the Game of Checkers 11.
IBM Journal of Research and Development, Vol. 11, No. 6
Nov. 1967, pp. 601-617.
"Some Studies in Machine
Progress,"
1186
(1969)
Sandewall E.
"
"Concepts and Methods for Heuristic Search,
Joint Conference on Artificial Intelligence.
Proc of the International
(1967)
Schofield, P.
"Complete Solution of the "Eight-Puzzle" in Machine Intelligence 1,
N. Collins and D. Michie (Eds), American Elsevier Publishing Co., Inc.,
New York, 1967, pp. 125 133.
—
0. and J. Kelly, Jr. (1962)
"Sophistication in Computers: a Disagreement,"
Feb. 1962.
Theory,
Selfridge,
IRE Trans, on Info.
(1950)
Shannon, C.
"Programming
March 1950,
a Digital Computer for Playing Chess." Philosophy Magazine
of
The World of
Vol. 41, pp. 356-375, Reprinted in The
(Ed.)
Schuster,
1954,
New York.
Mathematics, Newman
Vol. 4, Simon &
(1966)
Shapiro, D.
"Algorithms for the Solution of the Optimal Cost Traveling Salesman
Problem,"
Sc.D. Thesis, Washington Univ., St. Louis, M0. ,1966.
(1965)
Simmons, R.
"Answering
the ACM,
Simmons, R.
English Questions by Computer: A
Vol, 8, Jan. 1965, pp. 53-70.
Survey,"
Communications of
(1970)
"Natural
Language Question Answering Systems:
Vol. 13, No. 1, pp. 15-30, January 1970.
1187
1969,
Comm. ACM,
Slagle,
J.
(1961)
ior
in Freshman Calculus (SAINT),'
Inst, of Tech. Cambridge, Mass, 1961.
Also printed as Lincoln Laboratory Report SG-0001, May 10, 1961.
"A Computer
Program / Sol ving Problems
Doctoral Dissertation, Mass.
Slagle,
J. (1963a)
"A
Heuristic Program that Solves Symbolic Integration Problems in
Freshman Calculus," JACM, Vol. 10, No. 4, Oct. 1963, pp. 507-520.
Also in Computers and Thought, Feigenbaum, E. and J. Feldman (Eds.),
McGraw-Hill Book Co., New York, 1963, pp. 191-203.
Slagle,
(1963b)
J.
Trees, M & N Minimaxing, and the M & N Alpha-Beta Procedure,'
Artificial Intelligence Group Report No. 3, UCRL-4671 , University of
Calif. Lawrence Radiation Laboratory, Livermore, Calif. N0v., 1963.
"Game
(1965)
"Experiments with a Deductive-Question Answering
the ACM, Vol. 8, pp. 792-798, Dec 1965.
Slagle,
Slagle,
J.
J. of ACM,
Slagle,
(1967)
J.
"Automatic
Program," Commune at ions of
Theorem-Proving with Renamable and Semantic
Vol. 14, No. 4, pp. 687-697, Oct., 1967.
J.
Resolution,"
(1970)
"Heuristic Search Programs," in Formal Systems and Non-Numerical
Problem Solving by Computer, M. Mesarovic and R. Banerji (Eds.)
Slagle,
J. and P.
Bursky,
(1968)
"Experiments with a Multipurpose, Theorem- Proving Heuristic Program,
J. of ACM, Vol. 15, No., 1, pp. 85-99, Jan 1968.
1188
J and J. Dixon
Slagle,
(1969)
"Experiments with Some
16, No.
Vol.
Programs that Search Game
2, April 1969, pp.
Slagle, J. and J. Dixon,
Trees,"
(1970,)
"Experiments with the M & N Tree Searching Program," to
"Some
Recent Work in Artificial Intelligence
112, Dec.
1966.
,
,"
Proc
.
lEEE , Vol. 54,
(1963)
Travis, L.
"The
appear.
(1966).
Solomonoff, R.
No.
J. of ACM
189-207.
Value of Introspection to the Designer of Mechanical ProblemBehav. Science, Vol. 8, No. 3, pp. 227-233, July 1963.
Solvers,"
Travis, L.
(1967)
"
"Psychology and Bionics: Many old Problems and a Few New Machines,
Conference Record 1967 Winter Convention on Aerospace and Electronic
Systems (WINCON) , Vol, VI, Feb. 1967, lEEE Publication No. 10-C-42.
Turing
A. M.
(1950)
vComputing Machinery
and Intelligence," Mind ,
Oct. 1950, Vol. 59, pp
433-460. Reprinted in Computers and Thought, E. Feigenbaum and J.
Feldman (Eds.)
McGraw-Hill Book Co., New York, 1963, pp. 11-35.
R. and R. Lee (1969)
Waldinger,
"PROW:
A Step Toward Automatic Program-Writing,
Int'l Joint Conference on Artificial
May,
1969.
1189
"
Intelligence,
Proceedings of
Wash. D.C.
,
(1969)
Whitney, D.
Models of Remote Manipulation Tasks," in Proc
Int'l Joint Conf. on Artificial Intelligence, pp. 495-507.
"State Space
Wos, L. D.
Carson,
of the
(1964)
and G. Robinson
Unit Preference Strategy in Theorem Proving," Proc of the AFIPS
Vol. 26, pp. 616-621, 1964.
1964 Fall Joint Computer Conf.
"The
,
Wos, L., G. Robinson,
and D. Carson
(1965)
"Efficiency and Completeness of the Set of Support Strategy in TheoremProving," JACM, Vol. 12, No. 4, pp. 536-541, Oct., 1965.
Yates, R.,
B. Raphael, and T. Hart
"Resolution Graphs,"
(1970)
Stanford Research Institute Artificial
Intelligence
Memo.
Zobrist, A.
(1969)
Model of Visual Organization for the Game of Go,"
pp. 103- 112, 1969.
Spring Joint Computer Conf.
"A
,
-R22-
in Proc. AFIPS