David Evans cs.virginia/evans

Class 27: Modeling Computation David Evans http://www.cs.virginia.edu/evans CS200: Computer Science University of Virginia Computer Science

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) … ) CS 200 Spring 2004

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! CS 200 Spring 2004

Proof by Contradiction • Show X is nonsensical. • Show that if you have A and B you can make X. • Show that you can make A. • Therefore, B must not exist. X = contradict-halts A = a Scheme interpreter that follows the evaluation rules B = halts? CS 200 Spring 2004

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? CS 200 Spring 2004

Proof by Contradiction • Show X is nonsensical. • Show that if you have A and B you can make X. • Show that you can make A. • Therefore, B must not exist. X = halts? A = a Scheme interpreter that follows the evaluation rules B = is-virus? CS 200 Spring 2004

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. CS 200 Spring 2004

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. CS 200 Spring 2004

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 CS 200 Spring 2004

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! CS 200 Spring 2004

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? CS 200 Spring 2004

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

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 CS 200 Spring 2004

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

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). CS 200 Spring 2004

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 CS 200 Spring 2004

Charge • PS6 due Monday • Friday: • Finite state machines with infinite tape CS 200 Spring 2004

David Evans cs.virginia/evans