Transcript Class 27: Modeling Computation CS200: Computer Science University of Virginia Computer Science David Evans http://www.cs.virginia.edu/evans Halting Problem Define a procedure halts? that takes a procedure and an input.

Class 27: Modeling
Computation
CS200: Computer Science
University of Virginia
Computer Science
David Evans
http://www.cs.virginia.edu/evans
Halting Problem
Define a procedure halts? that takes a
procedure and an input evaluates to #t if
the procedure would terminate on that
input, and to #f if would not terminate.
(define (halts? procedure input) … )
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Informal Proof
(define (contradict-halts x)
(if (halts? contradict-halts null)
(loop-forever)
#t))
If contradict-halts halts, the if test is true and
it evaluates to (loop-forever) - it doesn’t halt!
If contradict-halts doesn’t halt, the if test if false,
and it evaluates to #t. It halts!
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Proof by Contradiction
1. Show X is nonsensical.
2. Show that if you have A and B you can
make X.
3. Show that you can make A.
4. Therefore, B must not exist.
X = contradict-halts
A = a Scheme interpreter that follows the evaluation rules
B = halts?
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Virus Detection Problem
Problem 7. Melissa Problem
Input: A Word macro (like a program, but
embedded in an email message)
Output: true if the macro will forward the
message to people in your address book;
false otherwise.
How can we show it is undecidable?
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Proof by Contradiction
1. Show X is nonsensical.
2. Show that if you have A and B you can
make X.
3. Show that you can make A.
4. Therefore, B must not exist.
X = halts?
A = a Scheme interpreter that follows the evaluation rules
B = is-virus?
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Undecidability Proof
Suppose we could define is-virus? that
decides the Melissa problem. Then:
(define (halts? P input)
(if (is-virus? ‘(begin (P input)
virus-code))
it is a virus, we know virus-code was
#t Since
evaluated, and P must halt (assuming P wasn’t a virus).
#f))
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Its not a virus, so the virus-code never executed.
Hence, P must not halt.
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Undecidability Proof
Suppose we could define is-virus? that
decides the Melissa problem. Then:
(define (halts? P input)
(is-virus? ‘(begin ((vaccinate P) input)
virus-code))
Where (vaccinate P) evaluates to P with all mail
commands replaced with print commands (to
make sure (is-virus? P input) is false.
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Proof
•
•
•
•
If we had is-virus? we could define halts?
We know halts? is undecidable
Hence, we can’t have is-virus?
Thus, we know is-virus? is undecidable
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How convincing is our
Halting Problem proof?
(define (contradict-halts x)
(if (halts? contradict-halts null)
(loop-forever)
#t))
If contradict-halts halts, the if test is true and it evaluates to
(loop-forever) - it doesn’t halt!
If contradict-halts doesn’t halt, the if test if false, and it
evaluates to #t. It halts!
This “proof” assumes Scheme exists and is consistent!
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Modeling Computation
• For a more convincing proof, we need a
more precise (but simple) model of what a
computer can do
• Another reason we need a model:
Does (n) really make sense without this?
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How should we model a Computer?
Colossus (1944)
Cray-1 (1976)
Apollo Guidance
Computer (1969)
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IBM 5100 (1975)
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Modeling Computers
• Input
– Without it, we can’t describe a problem
• Output
– Without it, we can’t get an answer
• Processing
– Need some way of getting from the input to
the output
• Memory
– Need to keep track of what we are doing
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Modeling Input
Punch Cards
Altair BASIC Paper Tape, 1976
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Engelbart’s
mouse and keypad
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Simplest Input
• Non-interactive: like punch cards and
paper tape
• One-dimensional: just a single tape of
values, pointer to one square on tape
0
0
1
1
0
0
1
0
0
0
How long should the tape be?
Infinitely long! We are modeling a computer, not
building one. Our model should not have silly
practical limitations (like a real computer does).
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Modeling Output
• Blinking lights are
cool, but hard to
model
• Output is what is
written on the
tape at the end of
a computation
Connection Machine CM-5, 1993
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Charge
• PS6 due Monday
• Friday:
– Finite state machines with infinite tape
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