Golfed fixed point combinator

Question

Write a fixed point combinator in as few characters as possible, in the language of your choice.

• free form (i.e., whatever's shortest): whole program, actual function, code snippet
• you may not use your standard library's if it has one
• you may however extract it from other high-level functions it you'd rather do that than construct it from the bases

Please include a recursive factorial or Fibonacci that uses it as a demo.

In this question, self-reference is acceptable, the aim is solely to remove it from the recursive function it will apply to.

Answer 1

Haskell: 10 characters

y f=f$y f

Example of use to create recursive definitions of factorial or nth-Fibonacci:

> map ( y(\f n->if n <= 1 then 1 else n*f(n-1)) ) [1..10]
[1,2,6,24,120,720,5040,40320,362880,3628800]

> map ( y(\f n->if n <= 1 then 1 else f(n-1)+f(n-2)) ) [0..10]
[1,1,2,3,5,8,13,21,34,55,89]

Though, a more common way to use y would be to generate these sequences directly, rather than as functions:

> take 10 $ y(\p->1:zipWith (*) [2..] p)
[1,2,6,24,120,720,5040,40320,362880,3628800]

> take 11 $ y(\f->1:1:zipWith (+) f (tail f))
[1,1,2,3,5,8,13,21,34,55,89]

Of course, with Haskell, this is a bit like shooting fish in a barrel! The Data.Function library has this function, called fix, though implemented somewhat more verbosely.

Answer 2

Perl, 37

sub f{my$s=$_[0];sub{$s->(f($s),@_)}}

Factorial demonstration:

sub fact {
  my ($r, $n) = @_;
  return 1 if $n < 2;
  return $n * $r->($n-1);
}
print "Factorial $_ is ", f(\&fact)->($_), "\n" for 0..10;

Fibonacci demonstration:

sub fib {
  my ($r, $n) = @_;
  return 1 if $n < 2;
  return $r->($n-1) + $r->($n-2);
}
print "Fibonacci number $_ is ", f(\&fib)->($_), "\n" for 0..10;

Answer 3

GNU C - 89 chars

#define fix(f,z)({typeof(f)__f=(f);typeof(__f(0,z))x(typeof(z)n){return __f(x,n);}x(z);})

Example:

#define lambda2(arg1, arg2, expr) ({arg1;arg2;typeof(expr)__f(arg1,arg2){return(expr);};__f;})

int main(void)
{
    /* 3628800 */
    printf("%ld\n", fix(lambda2(
        long factorial(int n), int n, 
            n < 2 ? 1 : n * factorial(n-1)
        ), 10));

    /* 89 */
    printf("%ld\n", fix(lambda2(
        long fibonacci(int n), int n, 
            n < 2 ? 1 : fibonacci(n-1) + fibonacci(n-2)
        ), 10));

    return 0;
}

Answer 4

k2, 12 char

The obvious self-referential implementation is the shortest. This is a sign of good language design. Unfortunately, K isn't lazy, so we can only manage call-by-value.

Y:{x[Y[x]]y}

This definition should also work in k4 and q without trouble, though I assume k2 for the examples below.

  Y:{x[Y[x]]y}
  fac: {[f;arg] :[arg>0; arg*f[arg-1]; 1]}
  Y[fac] 5
120
  fib: {[f;arg] :[arg>1; f[arg-1] + f[arg-2]; arg]}
  Y[fib]' !10
0 1 1 2 3 5 8 13 21 34

A more modest 18 characters lets us exactly transcribe (λx. x x) (λxyz. y (x x y) z) into K.

{x[x]}{y[x[x;y]]z}

Maybe someday (k7?), this could look like Y:{x Y x}.

Answer 5

Python 3, 30 Bytes

Y=lambda f:lambda a:f(Y(f))(a)

Demo :

Y=lambda f:lambda a:f(Y(f))(a)
quicksort = Y(
lambda f:
    lambda x: (
        f([item for item in x if item < x[0]])
        + [y for y in x if x[0] == y]
        + f([item for item in x if item > x[0]])
    ) if x
    else []
)
print(quicksort([1, 3, 5, 4, 1, 3, 2]))

Credits : https://gist.github.com/WoLpH/17552c9508753044e44f