Exploring the Lambda Calculus and the Meaning of Truth

1. Class 32: The Meaning of Truth David Evans http://www.cs.virginia.edu/evans CS200: Computer Science University of Virginia Computer Science

2. Menu • Lambda Calculus Review • How to Prove Something is a Universal Computer • Making “Primitives” CS 200 Spring 2003

3. 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 2003

4. 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 2003

5. 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 2003

6. 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 2003

7. 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 is there is one. • This is normal order reduction – corresponds to normal order (lazy) evaluation CS 200 Spring 2003

8. 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 2003

9. 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 2003

10. 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 2003

11. 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 2003

12. Making “Primitives”from Only Glue () CS 200 Spring 2003

13. 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 2003

14. Don’t search for T, search for if T  x (y. x)  xy. x F  x ( y. y)) if pca . pca CS 200 Spring 2003

15. Finding the Truth T  x . (y. x) F  x . (y. y) if p . (c . (a . pca))) if T M N ((pca . pca) (xy. x)) M N  (ca . (x.(y. x)) ca)) M N   (x.(y. x))M N  (y. M ))N  M Is the if necessary? CS 200 Spring 2003

16. and and or? and x (y. ifxyF)) or x (y. ifxTy)) CS 200 Spring 2003

17. Meaning of Life • Okay, so we know the meaning of truth. • Can we make the meaning of life also? CS 200 Spring 2003

18. What is 42? 42 forty-two XXXXII CS 200 Spring 2003

19. Meaning of Numbers • “42-ness” is something who’s successor is “43-ness” • “42-ness” is something who’s predecessor is “41-ness” • “Zero” is special. It has a successor “one-ness”, but no predecessor. CS 200 Spring 2003

20. Meaning of Numbers pred (succ N)  N succ (pred N)  N succ (pred (succ N))  succ N CS 200 Spring 2003

21. Meaning of Zero zero? zero  T zero? (succ zero)  F zero? (pred (succ zero))  T CS 200 Spring 2003

22. Is this enough? • Can we define add with pred, succ, zero?and zero? add  xy.if(zero?x) y (add(predx)(succy)) CS 200 Spring 2003

23. Can we define lambda terms that behave likezero, zero?, pred and succ? Hint: what if we had cons, car and cdr? CS 200 Spring 2003

24. Numbers are Lists... zero?  null? pred  cdr succ   x . cons F x CS 200 Spring 2003

25. Making Pairs • Remember PS2… (define (make-point x y) (lambda (selector) (if selector x y))) (define (x-of-point point) (point #t)) (define (y-of-point point) (point #f)) CS 200 Spring 2003

26. cons and car cons x.y.z.zxy cons M N = (x.y.z.zxy)M N   (y.z.zMy)N   z.zMN carp.p T car (cons M N)  car(z.zMN)  (p.p T) (z.zMN)   (z.zMN) T   TMN   (x . y. x)MN   (y. M)N   M T  x . y. x CS 200 Spring 2003

27. cdr too! cons xyz.zxy carp.p T cdrp.p F cdr consMN cdrz.zMN = (p.p F) z.zMN   (z.zMN) F   FMN   N CS 200 Spring 2003

28. Null and null? nullx.T null?x.(x y.z.F) null? null  x.(x y.z.F) (x. T)   (x. T)(y.z.F)   T CS 200 Spring 2003

29. Null and null? nullx.T null?x.(x y.z.F) null? (cons M N)  x.(x y.z.F) z.zMN   (z.z MN)(y.z.F)   (y.z.F)MN   F CS 200 Spring 2003

30. Counting 0 null 1 cons F 0 2 cons F 1 3 cons F 2 ... succ  x.consFx pred  x.cdrx CS 200 Spring 2003

31. 42 = xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y x.T CS 200 Spring 2003

32. Arithmetic zero? null? succ  x. cons F x pred x.x F pred 1 = (x.x F) cons Fnull   (cons Fnull) F  (xyz.zxy Fnull) F   (z.z Fnull) F  FFnull   null  0 CS 200 Spring 2003

33. Lambda Calculus is a Universal Computer z z z z z z z z z z z z z z z z z z z z ), X, L ), #, R (, #, L 2: look for ( 1 • 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 Start (, X, R HALT #, 0, - #, 1, - Finite State Machine We have this, but we cheated using  to make recursive definitions! CS 200 Spring 2003

34. Charge • Wednesday: show we can make recursion with Lambda Calculus • Friday: chance to ask questions before Exam 2 is handed out • The real meaning of life lecture is April 25 • Exam 2 covers through today • Office hours this week: • Wednesday, after class - 3:45 • Thursday 10am-10:45am; 4-5pm. CS 200 Spring 2003