david evans http www cs virginia edu evans n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
David Evans http://www.cs.virginia.edu/evans PowerPoint Presentation
Download Presentation
David Evans http://www.cs.virginia.edu/evans

Loading in 2 Seconds...

  share
play fullscreen
1 / 34
ojal

David Evans http://www.cs.virginia.edu/evans - PowerPoint PPT Presentation

815 Views
Download Presentation
David Evans http://www.cs.virginia.edu/evans
An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

  1. Class 32: Computability in Theory and Practice David Evans http://www.cs.virginia.edu/evans CS150: Computer Science University of Virginia Computer Science

  2. Menu • Lambda Calculus Review • Computability in Theory and Practice • Learning to Count

  3. Universal 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 To prove Lambda Calculus is as powerful as a UTM, we must show we can make everything we need to simulate any TM.

  4. Don’t search for T, search for if T  x (y. x)  xy. x F  x ( y. y)) if pca . pca

  5. 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?

  6. and and or? and x (y. ifxyF)) or x (y. ifxTy))

  7. 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

  8. Computability in Theory and Practice(Intellectual Computability Discussion on TV Video)

  9. Ali G Multiplication Problem • Input: a list of 2 numbers with up to d digits each • Output: the product of the 2 numbers Is it decidable? Yes – a straightforward algorithm solves it. Is it tractable? (how much work?) Yes – it using elementary multiplication techniques it is O(d2) Can real computers solve it?

  10. What about C++? int main (void) { int alig = 999999999; printf ("Value: %d\n", alig); alig = alig * 99; printf ("Value: %d\n", alig); alig = alig * 99; printf ("Value: %d\n", alig); alig = alig * 99; printf ("Value: %d\n", alig); } Results from SunOS 5.8: Value: 999999999 Value: 215752093 Value: -115379273 Value: 1462353861

  11. Ali G was Right! • Theory assumes ideal computers: • Unlimited, perfect memory • Unlimited (finite) time • Real computers have: • Limited memory, time, power outages, flaky programming languages, etc. • There are many decidable problems we cannot solve with real computer: the numbers do matter

  12. 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

  13. What is 42? 42 forty-two XLII cuarenta y dos

  14. 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.

  15. Meaning of Numbers pred (succ N)  N succ (pred N)  N succ (pred (succ N))  succ N

  16. Meaning of Zero zero? zero  T zero? (succ zero)  F zero? (pred (succ zero))  T

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

  18. Can we define lambda terms that behave likezero, zero?, pred and succ? Hint: what if we had cons, car and cdr?

  19. Numbers are Lists... zero?  null? pred  cdr succ   x . cons F x

  20. Making Pairs (define (make-pair x y) (lambda (selector) (if selector x y))) (define (car-of-pair p) (p #t)) (define (cdr-of-pair p) (p #f))

  21. 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

  22. 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

  23. 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

  24. 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

  25. Counting 0 null 1 cons F 0 2 cons F 1 3 cons F 2 ... succ  x.consFx pred  x.cdrx

  26. 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

  27. 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

  28. 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!

  29. Way to Keep Going ( 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)) ... This should give you some belief that we might be able to do it. We won’t cover the details of why this works in this class. (CS655 sometimes does.)

  30. 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

  31. Universal Computer • Lambda Calculus can simulate a Turing Machine • Everytime a Turing Machine can compute, Lambda Calculus can compute also • Turing Machine can simulate Lambda Calculus (we didn’t prove this) • Everything Lambda Calculus can compute, a Turing Machine can compute also • Church-Turing Thesis: this is true for any other mechanical computer also

  32. Charge • Exam 2 out Friday • Covers through today • Links to example exams on the web • Review session Wednesday, 7pm • PS8 Project Ideas due tomorrow (11:59pm) • Short email is fine, just explain who your team is and what you plan to do