1. David Evans http://www.cs.virginia.edu/evans
Class 33:
Learning to Count
CS200: Computer Science
University of Virginia
Computer Science
David Evans

2. Universal Computationz z z z z z z z z z z z z z z z z z z z
), X, L
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
), #, R
(, #, L 1
2: look for (
Start
(, X, R
HALT
#, 1, -
#, 0, -
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.
23 January 2004
CS 200 Spring 2004

3. Making “Primitives” from Only Glue ()
23 January 2004
CS 200 Spring 2004

4. In search of the truth? 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:
if T M N  M
23 January 2004
CS 200 Spring 2004

5. Don’t search for T, search for if
T  x (y. x)
 xy. x
F  x ( y. y))
if  pca . pca
23 January 2004
CS 200 Spring 2004

6. Finding the Truth if T M N 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?
23 January 2004
CS 200 Spring 2004

7. and and or? and  x (y. if x y F)) or  x (y. if x T y))
23 January 2004
CS 200 Spring 2004

8. 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 1
2: look for (
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
#, 1, -
#, 0, -
Finite State Machine
23 January 2004
CS 200 Spring 2004

9. 42 forty-two XLII What is 42? cuarenta y dos 23 January 2004
CS 200 Spring 2004

10. 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.
23 January 2004
CS 200 Spring 2004

11. Meaning of Numbers pred (succ N)  N succ (pred N)
succ (pred (succ N))
 succ N
23 January 2004
CS 200 Spring 2004

12. Meaning of Zero zero? zero  T zero? (succ zero)  F
zero? (pred (succ zero))
23 January 2004
CS 200 Spring 2004

13. Is this enough? add  xy.if (zero? x) y (add (pred x) (succ y))
Can we define add with pred, succ, zero? and zero?
add  xy.if (zero? x) y
(add (pred x) (succ y))
23 January 2004
CS 200 Spring 2004

14. Can we define lambda terms that behave like zero, zero?, pred and succ?
Hint: what if we had cons, car and cdr?
23 January 2004
CS 200 Spring 2004

15. zero?  null? pred  cdr succ   x . cons F x Numbers are Lists...
23 January 2004
CS 200 Spring 2004

16. 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))
23 January 2004
CS 200 Spring 2004

17. cons and car cons  x.y.z.zxy car  p.p T T  x . y. x
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
23 January 2004
CS 200 Spring 2004

18. cdr too! cons  xyz.zxy car  p.p T cdr  p.p F cdr cons M N
cdr z.zMN = (p.p F) z.zMN
  (z.zMN) F
  FMN
  N
23 January 2004
CS 200 Spring 2004

19. 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
23 January 2004
CS 200 Spring 2004

20. 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
23 January 2004
CS 200 Spring 2004

21. Counting 0  null 1  cons F 0 2  cons F 1 3  cons F 2 ...
succ  x.cons F x
pred  x.cdr x
23 January 2004
CS 200 Spring 2004

22. 42 = xy. (z. z xy) xy. y xy. (z. z xy) xy. y xy. (z. z xy) xy
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
23 January 2004
CS 200 Spring 2004

23. Arithmetic zero?  null? succ  x. cons F x pred  x.x F
pred 1 = (x.x F) cons F null
  (cons F null) F
 (xyz.zxy F null) F
  (z.z F null) F
  F F null
  null
 0
23 January 2004
CS 200 Spring 2004

24. Lambda Calculus is a Universal Computerz z z z z z z z z z z z z z z z z z z z
), X, L
), #, R
(, #, L 1
2: look for (
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
#, 1, -
#, 0, -
Finite State Machine
We have this, but
we cheated using 
to make recursive definitions!
23 January 2004
CS 200 Spring 2004

25. 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))
 ...
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.)
23 January 2004
CS 200 Spring 2004

26. Charge Monday: Review session for Exam 2 Tuesday Office Hours
Exam 2 covers through today
There will definitely be a question that requires understanding the answer to question 5 on the PS7 comments!
Tuesday Office Hours
1-2pm and 4-5pm
Exam 2 out Wednesday
23 January 2004
CS 200 Spring 2004