1. David Evans http://www.cs.virginia.edu/~evans
Class 37:
Making Lists, Numbers and Recursion from Glue Alone
CS200: Computer Science
University of Virginia
Computer Science
David Evans

2. Menu Review Making Numbers and Lists
Lambda Calculus
How to Prove Something is a Universal Computer
Last time: The Meaning of Truth
Making Numbers and Lists
Making Recursive Definitions without define
22 April 2002
CS 200 Spring 2002

3. Lambda Calculus -reduction (renaming) y. M  v. (M [y v])
term ::= variable |term term | (term)|  variable . term
-reduction (renaming)
y. M  v. (M [y v])
where v does not occur in M.
-reduction (substitution)
(x. M)N   M [ x N ]
22 April 2002
CS 200 Spring 2002

4. Church-Turing Thesis
Any mechanical computation can be performed by Lambda Calculus reduction rules
There is a Lambda Calculus term equivalent to every Turing Machine
22 April 2002
CS 200 Spring 2002

5. What do we need? Read/Write Infinite Tape Mutable Listsz 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
22 April 2002
CS 200 Spring 2002

6. Making “Primitives”
22 April 2002
CS 200 Spring 2002

7. The Meaning of 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?
22 April 2002
CS 200 Spring 2002

8. and and or? and  xy. if x y F or  xy. if x T y 22 April 2002
CS 200 Spring 2002

9. Meaning of Life Okay, so we know the meaning of truth.
Can we make the meaning of life also?
22 April 2002
CS 200 Spring 2002

10. What is 42? 42
forty-two
XXXXII
22 April 2002
CS 200 Spring 2002

11. 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.
22 April 2002
CS 200 Spring 2002

12. Meaning of Numbers pred (succ N)  N succ (pred N)
succ (pred (succ N))
 succ N
22 April 2002
CS 200 Spring 2002

13. Meaning of Zero zero? zero  T zero? (succ zero)  F
zero? (pred (succ zero))
22 April 2002
CS 200 Spring 2002

14. 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))
22 April 2002
CS 200 Spring 2002

15. Can we define lambda terms that behave like zero, zero?, pred and succ?
Hint: what if we had cons, car and cdr?
22 April 2002
CS 200 Spring 2002

16. zero? x = null? x pred x = cdr x succ x = cons F x
Numbers are Lists...
zero? x = null? x
pred x = cdr x
succ x = cons F x
22 April 2002
CS 200 Spring 2002

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 z.zMN = (p.p T) z.zMN
  (z.zMN) T
  TMN
  (x . y. x) MN
  (y. M)N
  M
T  x . y. x
22 April 2002
CS 200 Spring 2002

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
  (x . y. y) MN
  (y. y)N
  N
22 April 2002
CS 200 Spring 2002

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
22 April 2002
CS 200 Spring 2002

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
22 April 2002
CS 200 Spring 2002

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
22 April 2002
CS 200 Spring 2002

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
22 April 2002
CS 200 Spring 2002

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
22 April 2002
CS 200 Spring 2002

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!
22 April 2002
CS 200 Spring 2002

25. Charge Progress Meetings today and tomorrow Wednesday: Friday: review
Demonstrate and explain what you have done so far
Explain your plans for finishing
Wednesday:
show we can make recursive definitions with Lambda Calculus
computing and biology
Friday: review
22 April 2002
CS 200 Spring 2002