1. David Evans http://www.cs.virginia.edu/evans
Class 33:
Making Recursion
M.C. Escher, Ascending and Descending
also known as
f. (( x.f (xx))
( x. f (xx))
CS200: Computer Science
University of Virginia
Computer Science
David Evans

2. Menu Recap: Proving Lambda Calculus is Universal
How to make recursive definitions without define
Fixed points
16 April 2003
CS 200 Spring 2003

3. Lambda Calculus Review
term ::= variable |term term | (term)|  variable . term
Parens are just for grouping, not like in Scheme
-reduction (renaming)
y. M  v. (M [y v])
where v does not occur in M.
-reduction (substitution)
(x. M)N   M [ x N ]
( f. (( x.f (xx)) ( x. f (xx)))) (z.z)
16 April 2003
CS 200 Spring 2003

4. 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
16 April 2003
CS 200 Spring 2003

5. Simulating Computationz 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 1
2: look for (
Start
(, X, R
HALT
#, 1, -
#, 0, -
Finite State Machine
16 April 2003
CS 200 Spring 2003

6. 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!
16 April 2003
CS 200 Spring 2003

7. 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))
 ...
16 April 2003
CS 200 Spring 2003

8. Recursive Definitions
Which of these make you uncomfortable?
x = 1 + x
x = 4 – x
x = 9 / x
x = x
x = 1 / (16x3)
No solutions over integers
1 integer solution, x = 2
2 integer solutions, x = 3, x = -3
Infinitely many integer solutions
No integer solutions,
two rational solutions (1/2 and –1/2),
four complex solutions (1/2, -1/2, i/2, -i/2)
16 April 2003
CS 200 Spring 2003

9. f  x. sub 4 x Equations  Functions Find an x such that (f x) = x.
Same as solve for x.
sub  x . y. if (zero? y) x
(sub (pred x) (pred y))
16 April 2003
CS 200 Spring 2003

10. What’s a function? f  x. sub 4 x f: Nat  Nat
{ <0, 4>, <1, 3>, <2, 2>,
<3, 1>, <4, 0>,
<5, ???>, ... }
16 April 2003
CS 200 Spring 2003

11. Domains Set: unordered collection of values
Nat = { 0, 2, 1, 4, 3, 6, 5, 8, 7, ... }
Domains: like a set, but has structure
Nat = { 0, 1, 2, 3, 4, 5, 6, 7, 8, ... }
where 0 is the least element, and there is an order of the elements.
16 April 2003
CS 200 Spring 2003

12. Making Domains Primitive domains can be combined to make new domains
Product domain: D1 x D2
Pairs of values
Nat x Nat =
{ (tupleNat,Nat 0 0), (tupleNat,Nat 0 1),
(tupleNat,Nat 1 0), (tupleNat,Nat 1 1),
(tupleNat,Nat 0 2), ... }
16 April 2003
CS 200 Spring 2003

13. Functions A set of input-output pairs
The inputs (Di) and outputs (Do) are elements of a domain
The function is an element of the domain Di  Do.
It is a (completely defined) function if and only if for every element d  Di, the set of input-output pairs has one member whose first member matches d.
16 April 2003
CS 200 Spring 2003

14. Functions? f: Nat  Nat  z. subNat  Nat 4 z
f: Nat  Int  z. subNat  Int 4 z
f: Nat  Nat  z. addNat  Nat z 1
Not a function since there is no value in output domain for z > 4.
16 April 2003
CS 200 Spring 2003

15. Functions f: (Nat  Nat)   n. add 1 ( f n))
f: (Nat  Nat)   n. f (add 1 n))
No solutions (over natural numbers) –
would require x = 1 + x.
Infinitely many solutions – e.g., f(x) = x. 3
{ <0, 3>, <1, 3>, <2, 3>, ... }
16 April 2003
CS 200 Spring 2003

16. Fixed Points
A fixed point of a function f: (D  D) is an element d D such that
(f d) = d.
Examples:
f: Nat  Int  z. sub 4 z
f: Nat  Nat  z. mul 2 z
f: Nat  Nat  z.z 2
infinitely many
16 April 2003
CS 200 Spring 2003

17. Generating Functions f: (Nat  Nat)   n. (add 1 ( f n))
Any recursive definition can be encoded with a (non-recursive) generating function
f: (Nat  Nat)   n. (add 1 ( f n))
 g: (Nat  Nat)  (Nat  Nat) 
 f.  n. (add 1 ( f n))
Solution to a recursive definition is a fixed point of its associated generating function
16 April 2003
CS 200 Spring 2003

18. Example fact: (Nat  Nat)  n. if (= n 0) 1 (mul n (fact (sub n 1)))
gfact: (Nat  Nat)  (Nat  Nat)
 f.n. if (= n 0) 1
(mul (n (f (sub n 1)))
16 April 2003
CS 200 Spring 2003

19. gfact I gfact (z.z)  n. if (= n 0) 1 (mul (n ((z.z) (sub n 1)))
gfact: (Nat  Nat)  (Nat  Nat)
 f.n. if (= n 0) 1
(mul (n (f (sub n 1)))
gfact (z.z) 
n. if (= n 0) 1
(mul (n ((z.z) (sub n 1)))
What is (gfact I) 5?
16 April 2003
CS 200 Spring 2003

20. gfact fact gfact fact  n. if (= n 0) 1 (mul n (fact (sub n 1)))
gfact: (Nat  Nat)  (Nat  Nat)
 f.n. if (= n 0) 1
(mul (n (f (sub n 1)))
fact : (Nat  Nat)
 n. if (= n 0) 1
(mul (n (fact (sub n 1)))
gfact fact
gfact fact 
n. if (= n 0) 1
(mul n (fact (sub n 1)))
 fact
fact is a fixed point of gfact
16 April 2003
CS 200 Spring 2003

21. Unsettling Questions
Is the factorial function the only fixed point of gfact?
Given an arbitrary function, how does one find a fixed point?
If there is more than one fixed point, how do you know which is the right one?
16 April 2003
CS 200 Spring 2003

22. Iterative Fixed Point Technique
To find a fixed point of g: D  D
Start with some element d D
Calculate g(d), g(g(d)), g(g(g(d))), ... until you get
g(g(v)) = v
then v is a fixed point.
16 April 2003
CS 200 Spring 2003

23. Where to start?
If you start with   D you get the least fixed point (which is the “best” one)
 (pronounced “bottom”) is the element of D such that for any element d D,  d.
means “has less information than” or “is weaker than”
Not all domains have a .
16 April 2003
CS 200 Spring 2003

24. Pointed Partial Order
A partial order (D, ) is pointed if it has a bottom element u  D such that u d for all elements d  D.
Bottom of (Nat, <=)?
Bottom of (Nat, =)?
Bottom of (Int, <=)?
Bottom of ({Nat}, <=)?
Not a pointed partial order
Not a pointed partial order {}
16 April 2003
CS 200 Spring 2003

25. Getting to the  of things
Think of bottom as the element with the least information, or the “worst” possible approximation.
Bottom of Nat  Nat is a function that is undefined for all inputs. That is, the function with the graph {}.
To find the least fixed point in a function domain, start with the function whose graph is {} and iterate.
16 April 2003
CS 200 Spring 2003

26. Least Fixed Point of gfact
gfact: (Nat  Nat)  (Nat  Nat) 
 f. n. if (= n 0) 1
mul n ( f (sub n 1)))
gfactn (function with graph {})
= fact as n approaches infinity.
16 April 2003
CS 200 Spring 2003

27. Fixed Point Theorem Do all -calculus terms have a fixed point?
16 April 2003
CS 200 Spring 2003

28. Fixed Point Theorem  F ,  X  such that FX = X Proof:
Let W =  x.F(xx) and X = WW.
X = WW = ( x.F(xx))W
  F (WW) = FX
16 April 2003
CS 200 Spring 2003

29. Why of Y? Y is  f. WW: Y calculates a fixed point of any lambda term!
Y   f. ( x.f (xx)) ( x. f (xx))
Y calculates a fixed point of any lambda term!
Hence: we don’t need define to do recursion!
Works in Scheme too - check the “lecture” from the Adventure Game
16 April 2003
CS 200 Spring 2003

30. Fixed Point
The fixed point of our Turing Machine simulator on some input is the result of running the TM on that input. If there is no fixed point, the TM doesn’t halt!
fixed-point TM input
  result of running TM on input
16 April 2003
CS 200 Spring 2003

31. 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
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
(, #, L 1
2: look for (
Start
(, X, R
HALT
#, 1, -
#, 0, -
Finite State Machine
16 April 2003
CS 200 Spring 2003

32. Mystery Function (Teaser)
p  xy.  pca.pca (x.x xy.x) x) y
(p ((x.x xy.y) x) (x. z.z (xy.y) y)
m  xy.  pca.pca (x.x xy.x) x) x.x
(p y (m ((x.x xy.y) x) y))
f  x.
 pca.pca ((x.x xy.x) x)
(z.z (xy.y) (x.x))
(m x (f ((x.x xy.y) x)))
16 April 2003
CS 200 Spring 2003

33. QED Lambda calculus is a Universal Programming language
All you need is beta-reduction and you can compute anything
Integers, booleans, if, while, +, *, =, <, classes, define, inheritance, etc. are for wimps! Real programmers only use  and beta-reduction.
16 April 2003
CS 200 Spring 2003

34. Charge Friday: chance to ask questions before Exam 2 is handed out
questions before if you want to make sure they are covered
Today’s lecture is not covered by Exam 2
Office hours this week:
Wednesday, after class - 3:45
Thursday 10am-10:45am; 4-5pm.
16 April 2003
CS 200 Spring 2003