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Class 32: The Meaning of Truth. David Evans http://www.cs.virginia.edu/evans. CS200: Computer Science University of Virginia Computer Science. Menu. Lambda Calculus Review How to Prove Something is a Universal Computer Making “Primitives”. Lambda Calculus Review.
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Class 32: The Meaning of Truth David Evans http://www.cs.virginia.edu/evans CS200: Computer Science University of Virginia Computer Science
Menu • Lambda Calculus Review • How to Prove Something is a Universal Computer • Making “Primitives” CS 200 Spring 2004
Lambda Calculus Review term ::= variable |term term | (term)| variable .term ( f. (( x.f (xx))( x. f (xx)))) (z.z) Parens are just for grouping, not like in Scheme -reduction (renaming) y. M v. (M [yv]) where v does not occur in M. -reduction (substitution) (x. M)N M [ xN ] CS 200 Spring 2004
Alyssa P. Hacker’s Answer ( f. (( x.f (xx))( x. f (xx)))) (z.z) (x.(z.z)(xx))( x. (z.z)(xx)) (z.z) ( x.(z.z)(xx)) ( x.(z.z)(xx)) (x.(z.z)(xx)) ( x.(z.z)(xx)) (z.z) ( x.(z.z)(xx)) ( x.(z.z)(xx)) (x.(z.z)(xx)) ( x.(z.z)(xx)) ... CS 200 Spring 2004
Ben Bitdiddle’s Answer ( f. (( x.f (xx))( x. f (xx)))) (z.z) (x.(z.z)(xx))( x. (z.z)(xx)) (x.xx)(x.(z.z)(xx)) (x.xx)(x.xx) (x.xx)(x.xx) ... CS 200 Spring 2004
Be Very Afraid! • Some -calculus terms can be -reduced forever! • The order in which you choose to do the reductions might change the result! CS 200 Spring 2004
Take on Faith (until CS655) • All ways of choosing reductions that reduce a lambda expression to normal form will produce the same normal form (but some might never produce a normal form). • If we always apply the outermost lambda first, we will find the normal form if there is one. • This is normal order reduction – corresponds to normal order (lazy) evaluation CS 200 Spring 2004
Universal Language • Is Lambda Calculus a universal language? • Can we compute any computable algorithm using Lambda Calculus? • To prove it isn’t: • Find some Turing Machine that cannot be simulated with Lambda Calculus • To prove it is: • Show you can simulate every Turing Machine using Lambda Calculus CS 200 Spring 2004
Simulating Every Turing Machine • A Universal Turing Machine can simulate every Turing Machine • So, to show Lambda Calculus can simulate every Turing Machine, all we need to do is show it can simulate a Universal Turing Machine CS 200 Spring 2004
Simulating Computation z z z z z z z z z z z z z z z z z z z z • Lambda expression corresponds to a computation: input on the tape is transformed into a lambda expression • Normal form is that value of that computation: output is the normal form • How do we simulate the FSM? ), X, L ), #, R (, #, L 2: look for ( 1 Start (, X, R HALT #, 0, - #, 1, - Finite State Machine CS 200 Spring 2004
Simulating Computation z z z z z z z z z z z z z z z z z z z z Read/Write Infinite Tape Mutable Lists Finite State Machine Numbers to keep track of state Processing Way of making decisions (if) Way to keep going ), X, L ), #, R (, #, L 2: look for ( 1 Start (, X, R HALT #, 0, - #, 1, - Finite State Machine CS 200 Spring 2004
Making “Primitives”from Only Glue () CS 200 Spring 2004
In search of thetruth? • What does true mean? • True is something that when used as the first operand of if, makes the value of the if the value of its second operand: ifTMN M CS 200 Spring 2004
Charge • Remember: if you have team requests for PS8, I must receive them before midnight tonight! • Friday: finish discovering truth (and proving lambda calculus is as powerful as a Turing Machine) • Monday: Review session for Exam 2 CS 200 Spring 2004
