1. David Evans http://www.cs.virginia.edu/~evans
Lecture 15:
Crazy Eddie and the Fixed Points
Background
just got here last week
finished degree at MIT week before
Philosophy of advising students
don’t come to grad school to implement someone else’s idea
can get paid more to do that in industry
learn to be a researcher
important part of that is deciding what problems and ideas are worth spending time on
grad students should have their own project
looking for students who can come up with their own ideas for research
will take good students interested in things I’m interested in – systems, programming languages & compilers, security
rest of talk – give you a flavor of the kinds of things I am interested in
meant to give you ideas (hopefully even inspiration!) but not meant to suggest what you should work on
CS655: Programming Languages
University of Virginia
Computer Science
David Evans

2. University of Virginia CS 655
Menu
Billy Bob and the Beta Reducers
Survey Results
Fixed Points
15 May 2019
University of Virginia CS 655

3. University of Virginia CS 655
-term Evaluation
Y   f. (( x.f (xx)) ( x. f (xx)))
What is (Y I)? Recall I = z.z
Evaluation rule:
-reduction (substitution)
(x. M)N   M [ x := N]
Substitute N for x in M.
15 May 2019
University of Virginia CS 655

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))
  ...
15 May 2019
University of Virginia CS 655

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)
  ...
15 May 2019
University of Virginia CS 655

6. University of Virginia CS 655
Be Very Afraid!
Some -calculus terms can be -reduced forever!
The order in which you choose to do the reductions might change the result!
15 May 2019
University of Virginia CS 655

7. University of Virginia CS 655
Survey Results
Only 2 really conclusive results:
Less Readings (6 way too much, 15 too much, 4 just right)
More Pizza (1 for, none against)
15 May 2019
University of Virginia CS 655

8. University of Virginia CS 655
Survey Results
Lectures
majority too fast, but not overwhelming
Programming
12 just right, 9 not enough, 1 too much
Topics
6 fewer topics/more depth, 10 more topics/less depth, 2 fewer topics/less depth, 2 more topics/more depth (5 write in like current mix)
Grading
13 prefer unbounded, 8 prefer traditional, 4 don’t care
15 May 2019
University of Virginia CS 655

9. University of Virginia CS 655
Elevator Speeches
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University of Virginia CS 655

10. 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)
15 May 2019
University of Virginia CS 655

11. More interesting for functions
Functions are just sets of input/output pairs.
f: (Nat  Nat) =  n. (1 + ( f n))
f: (Nat  Nat) =  n. ( f (1 + n))
No solutions (over natural numbers) –
would require x = 1 + x.
Infinitely many solutions – e.g., f(x) = 3.
15 May 2019
University of Virginia CS 655

12. University of Virginia CS 655
Generating Functions
Any recursive definition can be encoded with a (non-recursive) generating function
f: (Nat  Nat) =  n. (1 + ( f n)) 
g: (Nat  Nat)  (Nat  Nat) =
 f.  n. (1 + ( f n))
15 May 2019
University of Virginia CS 655

13. University of Virginia CS 655
Fixed Points
A fixed point of a function f: (D  D) is an element d D such that (f d) = d.
Solution to a recursive definition is a fixed point of its associated generating function.
15 May 2019
University of Virginia CS 655

14. University of Virginia CS 655
Example
fact: (Nat  Nat) =
 n. if (n = 0) then 1
else (n * ( fact (n - 1)))
gfact: (Nat  Nat)  (Nat  Nat) =
 f. n. if (n = 0) then 1
else (n * ( f (n - 1)))
15 May 2019
University of Virginia CS 655

15. University of Virginia CS 655
(gfact I)
(gfact (z.z)) =
n. if (n = 0) then 1
else (n * ((z.z) (n - 1)))
Function that returns 1 for 0, 0 for 1, 2 for 2, ..., n * n-1 for n ~= 0.
15 May 2019
University of Virginia CS 655

16. University of Virginia CS 655
(gfact factorial)
(gfact factorial) =
n. if (n = 0) then 1
else (n * (factorial (n - 1)))
= factorial
factorial is a fixed point of gfact
15 May 2019
University of Virginia CS 655

17. University of Virginia CS 655
Unsettling Questions
Is a 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?
15 May 2019
University of Virginia CS 655

18. University of Virginia CS 655
An odd example
odds = { 1 }  { x + 2 | x  odds }
godds = ({Nat}  {Nat}) =
 s. { 1 }  { x + 2 | x  s}
godds ({}) = {1}
godds (godds ({}) ) = godds{1} = {1 , 3}
goddsn ({}) = {1, 3, ..., (n * 2) + 1}
OOPS - this is broken
15 May 2019
University of Virginia CS 655

19. University of Virginia CS 655
Oddities of Odds
What is a fixed point of godds?
All odd numbers: { 1, 3, 5, ... }
What about: { 0, 1, 2, 3, 4, ... } ?
What about: { 1, 3, 4, 5, 6, 7, ... } ?
godds has infinitely many fixed points, the set of all odd numbers is the least fixed point.
OOPS - this is broken
15 May 2019
University of Virginia CS 655

20. Iterative Fixed Point Technique
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.
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 .
15 May 2019
University of Virginia CS 655

21. University of Virginia CS 655
Partial Order
A partial order is a pair (D, ) of a domain D and a relation that is
for all a, b, c  D
reflexive
transitive
and anti-symmetric
a a
a b and b c imply a c
a b and b a imply a = b
15 May 2019
University of Virginia CS 655

22. University of Virginia CS 655
Partial Orders?
(Nat, <=)
(Nat, =)
({ burger, fries, coke }, yummier)
where burger yummier fries, burger yummier coke, fries yummier coke
({ burger, fries, beer }, yummier)
where burger yummier fries, fries yummier beer, beer yummier burger
15 May 2019
University of Virginia CS 655

23. University of Virginia CS 655
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 {}
15 May 2019
University of Virginia CS 655

24. Partial Order of Functions
((D, D)  (E, E), DE) is a partial order using:
f DE g if for all d  D, (f d) E (g d).
double: Nat  Nat =
n. if (n = 0) then 0 else (2 + (double (n – 1))
Is double fact under (Nat, <=)?
No, (double 1) is not <= (fact 1)
15 May 2019
University of Virginia CS 655

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.
15 May 2019
University of Virginia CS 655

26. Least Fixed Point of gfact
gfact: (Nat  Nat)  (Nat  Nat) =
 f. n. if (n = 0) then 1
else (n * ( f (n - 1)))
(gfactn (function with graph {})) = fact
as n approaches infinity.
15 May 2019
University of Virginia CS 655

27. University of Virginia CS 655
Fixed Point Theorem
Do all -calculus terms have a fixed point?
 F ,  X  such that FX = X
Proof:
Let W =  x.F(xx) and X = WW.
X = WW = ( x.F(xx))W
  F (WW) = FX
15 May 2019
University of Virginia CS 655

28. University of Virginia CS 655
Why of Y?
Y is  f. WW:
Y   f. (( x.f (xx)) ( x. f (xx)))
Y calculates a fixed point (but not necessarily the least fixed point) of any lambda term!
If you’re not convinced, try calculating ((Y fact) n).
15 May 2019
University of Virginia CS 655

29. University of Virginia CS 655
Still Uncomfortable?
15 May 2019
University of Virginia CS 655

30. Remember the problem reducing Y
Different reduction orders produce different results
Reduction may never terminate
Normal Form means no more ß-reductions can be done
There are no subterms of the form (x. M)N
Not all -terms can be written in normal form
15 May 2019
University of Virginia CS 655

31. Church-Rosser Theorem
If a -term has a normal form, any reduction sequence that produces a normal form always the same normal form
Computation is deterministic
Some orders of evaluation might not terminate though
Evaluating left-to-right finds the normal form if there is one.
Proof by lack of space/time.
15 May 2019
University of Virginia CS 655

32. University of Virginia CS 655
Charge
Next time: pragmatic issues of functional programming
Rewrite rules
Type inference
Depositions due today, will be sent to opposing attorneys (including my comments if any) by Thursday
PS3 is harder than PS2
15 May 2019
University of Virginia CS 655