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David Evans CS200: Computer Science University of Virginia Computer Science Class 27: Modeling Computation

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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring Virus Detection Problem Problem 7. Melissa Problem Input: A Word macro (like a program, but embedded in an 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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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring 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)) #t #f)) Since it is a virus, we know virus-code was evaluated, and P must halt (assuming P wasn’t a virus). Its not a virus, so the virus-code never executed. Hence, P must not halt.

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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring How should we model a Computer? Apollo Guidance Computer (1969) Colossus (1944) IBM 5100 (1975) Cray-1 (1976)

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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring Modeling Input Engelbart’s mouse and keypad Punch Cards Altair BASIC Paper Tape, 1976

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26 March 2004CS 200 Spring Simplest Input Non-interactive: like punch cards and paper tape One-dimensional: just a single tape of values, pointer to one square on tape 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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26 March 2004CS 200 Spring 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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26 March 2004CS 200 Spring Charge PS6 due Monday Friday: –Finite state machines with infinite tape

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