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CREATIVE SOLUTIONS TO PROBLEMS
John McCarthy
Computer Science Department
Stanford [email protected]
http://www-formal.stanford.edu/jmc/
started April 1, 1999; compiled May 18, 199
Almost all of my papers are on the web page.
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APPROACHES TO ARTIFICIAL INTELLIGE
biologicalHumans are intelligent; imitate huma
observe and imitate at either the psychological or
physiological level
engineeringAchieve goals in the worldso stu
world
1. Write programs using non-logical representatio2. represent facts about the world in logic and
what to do by logical inference
We aim at human level AI, and the key phenom
the common sense informatic situation.
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THE COMMON SENSE INFORMATIC SITUAT
Contrasts with the situation in a formal scientiory and most AI theories. Science is embed
common sense.
No limitation on what information may be re
Theories must be elaboration tolerant.
Needs non-monotonic reasoning.
Needs approximate entities.
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A LOGICAL ROAD TO HUMAN LEVEL A
Use Drosophilas that illustrate aspects of repr
tion and reasoning problems.
Concepts, context, circumscription, counterfa
consciousness, creativity, approximation
narrative, projection, planning
mental situation calculus
domain dependent control of reasoning
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IDENTIFYING CREATIVE SOLUTIONS T
PROBLEMS
Making creative programs will be hard.
Making a program that will recognize creativity
ier but still too hard for me now.
Distinguish the idea of a solution from the s
itself.
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IDENTIFYING CREATIVE SOLUTIONS T
PROBLEMS
I can identify, thinking by hand, creative solut
I can sometimes express the creative idea by a
formula.
The Drosophila for this research is the mutilate
board problem.
As much as possible, an idea should be one thi
it should be promising in itself.
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I know four creative solutions to the mutilated c
board problem, the standard solution, Marv
skys solution, Shmuel Winograds solution an
itri Stefanuks 17 similar solutions.
I tried to express each idea as concisely as wa
patible with leading a person to the solution.
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THE MUTILATED CHECKERBOARD
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DOMINO
MUTILATED CHECKERBOARD
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THE STANDARD SOLUTION
Whats the idea of the solution that a creative pe
program may come up with? This is distinct from
the detailed argument. English firstthen a form
Color the board as in a checkerboard.
A domino covers two squares of the opposite colo
Some people also need that the removed squares
the same color.
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THE COMMON FACTS IN SET THEORY
Board = Z8 Z8,
mutilated-board = Board {(0, 0), (7, 7)},
domino-on-board(x) (x Board) card(x) = 2(x1 x2)(x1 = x2 x1 x x2 x
adjacent(x1, x2))
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adjacent(x1, x2) |c(x1, 1) c(x2, 1)| = 1 c(x1, 2) = c(x2, 2)
|c(x1, 2) c(x2, 2)| = 1 c(x1, 1) = c(x2, 1).
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adjacent(x1, x2) |c(x1, 1) c(x2, 1)| + |c(x1, 2) c(x2, 2)| = 1,
partial-covering(z) (x)(x z domino-on-board(x))(x y)(x z y z x = y x y = {})
Theorem:
(
z)(partial-covering(z) z = mutilated-boar
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Makers of automatic or interactive theorem prov
ten dont like set theory because of the compre
principle. They had better get used to it.
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THE UNMATHEMATICAL REQUIRE MANY H
Take into account the colors.
What are the colors of the diagonally opposite
squares?
How many of each color does a domino cover?
How many of each do n dominoes cover? huh?
What about 7 dominoes?
How many of each do n dominoes cover? Equa
Two blacks left over, but maybe a more clever. . . .
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MARVIN MINSKYS IDEA
Starting with the two square diagonal next to
excluded square, successively compute how m
dominoes must project from each diagonal to
next diagonal.
Good enough hint for a horse doctor.
Not a sentence, but a program fragmentwit
termination condition.
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SHMUEL WINOGRADS IDEA
Note that an odd number of dominoes project fr
top row to the second and continue.
A movie showed a math teacher rejecting this id
Socratically leading the student to the standard so
Winograd showed the student was on the righ
but most people need something like
Starting with the top row, compute the parity
number of dominoes projecting down out of eac
Consider the parity of the sum. Repeat going h
tally. Compute the parity of the total number o
noes compared to the sum of the two parities.
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DIMITRI STEFANUKS IDEA
The idea seems to involve two program fragment
Label an arbitrary square 1, label its rectilin
neigbors 2, their neighbors 3, etc.
Starting with n = 1, successively compute hmany dominoes must project from the set of sq
labelled n to the squares labelled n + 1.
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FORMULAS FOR MINSKY SOLUTION
diag(n) = {x Board|c(x, 1) + c(x, 2) = n}
covering(u) partial-covering(u)
u = Boa
covering(u) 2 n 13dominoes-into(n, u) = {x u|x diag(n 1) = {} x diag(n) = {}}
card(dominoes-into(n, u)) + card(dominoes-into(n= card(diag(n))
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covering(u) card(dominoes-into(2, u)) = 2
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REALITY BEHIND APPEARANCE