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Com S 541
Fixed Points
Overview:^!^
Recursion and the fixed-point combinator! Fixed-point semantics of objects! The typed lambda-calculus! The polymorpic lambda calculus References:^!^
H.P. Barendregt, “The Lambda Calculus – Its Syntax and Semantics”, North-Holland, 1984! David A. Schmidt, “Denotational Semantics – A Methodology for LanguageDevelopment”, Allyn and Bacon Inc., 1986! Joseph E. Stoy, “Denotational Semantics – The Scott-Strachey Approach toProgramming Language Theory”, MIT Press, 1992! William R. Cook, “A Denotational Semantics of Inheritance”, Ph.D., BrownUniversity, 1989
Com S 541
Recursion
!^ Suppose we want to define arithmetic operations on our lambda-encodednumbers (see “Representing Numbers”).In Haskell we can program:
plus n m
| n == 0 = m| otherwise = plus (n-1) (m+1) so we might try to define:plus
≡ λ^ n m. iszero n m (plus (pred n) (succ m)) !^ Unfortunately this is not a definition, since we are trying to use “plus” beforeit is defined. Although recursion is fundamental to functional programming, itis not primitive in the lambda calculus, so we must find a way to “program” it!
Com S 541
Fixed Points
!^ In gener
al,a fixed poi
ntofa f
unction i
s a value i
n the funct
ion’s dom ai
n,
w hich i
s m apped t
o itselfby t
he funct
ion.Ther
efore,a
fixed point of a
function
fis a value
p^ such that
f p = p
!^ Examples:
fact 1 = 1fact 2 = 2fib 0 = 0fib 1 = 1 λx. 27 = 27 λx. 6 – x = 3 !^ However, not all functions have exactly one fixed point: “succ n = n + 1”has none, while “id” has infinitely many.
Com S 541
Fixed Point Theorem
Fixed point Theorem:^ For every
F^ there exists a fixed point
X^ such that
F X^ ↔
X.
Proof:^ Let
Y^ ≡ λf. (
λx. f (x x)) (
λx. f (x x))
Now consider:X^
≡^ Y F^ "
(λx. F (x x)) (
λx. F (x x)) "^ F ((λ
x. F (x x)) (
λx. F (x x)))
"^ F X Therefore, the “Y combinator” can always be used to find a fixed point ofan arbitrary lambda expression, if such a fixed point exists.
Com S 541
Recursive Functions Are Fixed Points
!^ We cannot write:
plus^ ≡ λ
n m. iszero n m (plus (pred n) (succ m)) because plus is unbound in the “definition”.! We can, however, abstract over plus:rplus
≡ λ^ plus n m. iszero n m (plus (pred n) (succ m)) !^ Now we seek a lambda expression plus, such that:
rplus plus
↔^ plus !^ I. e., plus is a fixed point of rplus. By the fixed point theorem, we can take:
plus^ ≡^
Y rplus
Com S 541
Unfolding Recursive Lambda Expressions
!^ Consider:plus 1 1
= (Y rplus) 1 1^ "
rplus plus 1 1 " iszero 1 1 (plus (pred 1) (succ 1)) " False 1 (plus (pred 1) (succ 1)) " plus (pred 1) (succ 1) " rplus plus (pred 1) (succ 1) " iszero (pred 1) (succ 1) (plus (pred (pred 1)) (succ (succ 1))) " iszero 0 (succ 1) (...) " True (succ 1) (...) " succ 1 " 2
Com S 541
Fixed-point Semantics
!^ Fixed-point semantics of recursive programs provides the mathematicalsetting for an inheritance model for an object-oriented language.!^ Consider the following definition of the factorial function:
fact =^
λn.^ if^
n = 1^ then
^1 else
n *fact (n – 1)
The identifier fact is equated with a function definition, in which “fact”appears. Thus the definition is self-referential. The use of the name fact torepresent the self-reference is however just a syntactic convention. Inobject-oriented languages we commonly use the special symbol
self :
fact =^
λn.^ if^
n = 1^ then
^1 else
n * self (n – 1)
Com S 541
Function Transformation
!^ By using fixed-point technique, recursive definitions can by transformedinto a non-recursive form:
FACT =
λself.
λn.^ if
n = 1^
then^^1
else^ n * self (n – 1)
!^ FACT is a functional, or a mapping from functions to functions. The formalparameter “self” represents the function to call in order to compute thefactorial function. The original definition of
fact can now be given in terms
ofFACT:
fact =
FACT(fact)
But now
fact is defined as a value that is unchanged when
FACT is applied.
Such value is called a “fixed point” of
FACT. Under certain conditions, it is
possible to compute a unique fixed point, the least fixed point, of anyfunction by using the fixed-point function “fix” that has the followingproperty: if
f= fix(
F), then
F(f)= f
Com S 541
Generators
Definition:A function intended to specify a fixed point whose formal parameterrepresents self-reference is called generator. Thus a generator has theform
G =^ λ^ self. body
where self may occur free in body. Intuitively, self-reference is “unbound”in a generator, while self-reference in its fixed point is connected back tothe generator.
client
O Fixed point
O Generator
self
Com S 541
A Record Generator
!^ Consider the following record generator representing a pair ofvalues where the second depends upon the first:G =
λ^ self. [ base
"^ 7, square
"^ self.base * self.base ]
The fixed point of this generator is the recordm = fix(G) = [ base
"^ 7, square
"^ 49 ]
Com S 541
Wrappers
Definition:A wrapper is a function designed to modify a self-referential structure in aself-referential way; it has two parameters, one representing self-referenceand the other representing the superstructure being modified. Thus awrapper is a function of the form
W =^ λ^
self.^ λ
super. Body
where self and super may occur free in body.! The application of a wrapper to a generator involves binding together self-reference in the wrapper and the generator, and then applying thewrapper modification to the value of the generator:W
⊕^ G =
λ^ self. W(self)(G(self)), where
⊕^ =^ λ
a.^ λb.^
λs. a(s)(b(s))
Com S 541
Applying Wrappers
!^ The result of applying a wrapper to a generator is a generator.Self-references in in the wrapper W and the generator G arebound together though the variable self, signified by thejoining of the arrows out of W and G. The arrow form W to Grepresents the application of W to G:
client^
W^
G super self
self
