Lecture 10:
Fixed Points ad
Infinitum
M.C. Escher,
Moebius Ants
CS655: Programming Languages
University of Virginia
Computer Science
David Evans
http://www.cs.virginia.edu/~evans
Goal: Understand the
Least Fixed Point Theorem
If D is a pointed complete partial order,
then a continuous function f: D  D
has a least fixed point (fixD f) defined by
n )|n0}
{
(f
D
D
20 Feb 2000
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Last Time
• A domain is a structured set of values
• A function domain is constructed from
two primitive domains, D1  D2 by
associating an element of D2 with each
element of D1.
• A fixed point of a function f: D1  D2 is
an element d D such that f d = d.
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Last Time, cont.
• Any recursive definition can be encoded
with a (non-recursive) generating
function by abstracting out the thing that
is defined.
• A fixed point of a generating function is a
solution of its associated recursive
definition.
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Last Time, cont.
• (D, ) is a partial order if is:
– reflexive: a a
– transitive: a b and b
– anti-symmetric:
c imply a
c
a b and b a imply a = b
• (D, ) is a pointed partial order if it has
a bottom element u  D such that u d
for all elements d  D.
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Ordered Product Domains
If <D, D > and <E, E > is are POs,
<D x E , D x E > is a partial order,
ordered by:
<d1, e1> D x E <d2, e2>
if d1 D d2 and e1 E e2
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Ordered Product Example
• What is << Nat,  > x < Nat,  >>?
<1, 73>
<3, 3>
<0, 2>
<2, 0>
<0, 1>
<1, 0>
<0, 0>
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(Hasse diagram)
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Ordered Function Domains
If <D, D > and <E, E > is are POs,
<D  E , D  E > is a partial order,
ordered by: f  D  E
f1 D  E f2
if for all d  D,(f1 d)
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(f2 d)
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Ordered Function Example
< Bool,  > = { false, true }
false  true
true  false
What is << Bool,  >  < Bool,  >>?
<{false, true}, {true, true}>
<{false, false}, {true, true}>
<{false, true}, {true, false}>
<{false, false}, {true, false}>
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T-Shirt Exercise
• What is the order of the domain:
<<Bool,  > x < Bool,  >  < Bool,  >>
(includes xor, and, or, implies, ...)
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The Domain Nat
0
1
2
3
4
...
Nat
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Ordered Function Bottom
• What is the bottom of
<Nat, >  <Nat, > ?
{ <0, >, <1, >, <2, >, ... }
= { <x, > }
={}
If a function map has no entry for x,
it maps x to .
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Least Upper Bounds
• The least upper bound of a subset X of
a domain D is the weakest element of X
that is at least as strong as every other
element of X.
l  X = DX
if for every x  X, x l
and for every m  X such that x m x  X,
l m
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Least Upper Bounds in Nat
Nat
Nat
0
1
{ 0, 2, 4, 6, ... } = any element of { 0, 2, 4, 6, ... }
{Nat, 3, 17} = 3 or 17
2
3
4
...
Nat
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Complete Partial Orders
• A partial order D is complete if every
chain in D has a least upper bound in
D.
– Any upward chain through a Hasse
diagram converges to a limit
– All finite partial orders are complete
– Most sensible partial orders (including
Nat) are complete (see Gifford’s notes for
some incomplete POs.)
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Monotonic Functions
• f  D  E is monotonic if d1
implies (f d1)
D d2
E (f d2).
• Is not over << Bool,  >  < Bool,  >>
monotonic?
• Is { <x, x * 2>} over << Nat,  >  < Nat,
 >> monotonic?
• What functions are monotonic over
Nat  Nat?
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Continuous Functions
• f  D  E is continuous if, for all chains
C in D, f applied to the least element of
the chain over D is the least element of
(f c) for cC over E.
• Continuity is like monotonicity, but it
works for limits of infinite chains also.
• If the CPO is finite, monotonicity implies
continuity.
• Continuity always implies monotonicity
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Monotonic/Continuous
Functions in Domain Nat
0
1
2
3
4
...
Nat
What is a monotonic function in Nat?
What is a continuous function in Nat?
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Least Fixed Point Theorem
• If D is a pointed complete partial order,
then a continuous function f: D  D has
a least fixed point (fixD f) defined by
n )|n0}
{
(f
D
D
The least upper bound of applying f any
number of times, starting with D.
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InSanity Check
g: (Nat  Nat)  (Nat  Nat) =
 f. n. if (n = 0) then 1
else (n * ( f (n - 1)))
What is (fixNat  Nat g)?
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Sanity Check
g: (Nat  Nat)  (Nat  Nat) =
 f. n. if (n = 0) then 1
else (n * ( f (n - 1)))
What is (fixNat  Nat g)?
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What is the bottom of
Nat  Nat?
{ <x, > | x  Nat }
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What is (g { <x, > | x  Nat })?
g =  f. n. if (n = 0) then 1
else (n * ( f (n - 1)))
g { <x, > | x  Nat } =
{<0, 1>, <x, > | x > 0 and x  Nat }


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What is (g (g { <x, > | x  Nat }))?
g =  f. n. if (n = 0) then 1
else (n * ( f (n - 1)))
g { <0, 1>, <x, > | x > 0 and x  Nat } =
{ <0, 1>, <1, 1>,

<x, > | x > 1 and x  Nat }
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What is LUB
n
(g
{ <x, > | x  Nat }))?
g =  f. n. if (n = 0) then 1
else (n * ( f (n - 1)))
g { <0, 1>, <x, > | x > 0 and x  Nat } =
{ <0, 1>, <1, 1>, <2, 2>, <3, 6>, ...}
= {<x, x!> | x  Nat }


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Getting to the  of things
• Think of bottom as the element with
the least information, or the “worst”
possible approximation.
• To find the least fixed point in a
function domain, start with the
bottom of the function domain and
iterate...
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Fixed Point Theorem
• Do all -calculus terms have a
fixed point?
• (Smullyan: Is there a Sage
bird?)
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Finding the Sage Bird
•  F ,  X  such that FX = X
• Proof:
Let W =  x.F(xx) and X = WW.
X = WW = ( x.F(xx))W
  F (WW) = FX
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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). (PS1, 1e)
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Still Uncomfortable?
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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
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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 leftmost first finds the normal form
if there is one.
• Proof by trust the theory people, but don’t
become one.
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Charge
• PS2 is due Thursday
• Domains are like types in programming
languages, we will see them again
soon...
• Next time: Intro to Language Design
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