Y combinator

From Rosetta Code
Task
Y combinator
You are encouraged to solve this task according to the task description, using any language you may know.

In strict functional programming and the lambda calculus, functions (lambda expressions) don't have state and are only allowed to refer to arguments of enclosing functions. This rules out the usual definition of a recursive function wherein a function is associated with the state of a variable and this variable's state is used in the body of the function.

The Y combinator is itself a stateless function that, when applied to another stateless function, returns a recursive version of the function. The Y combinator is the simplest of the class of such functions, called fixed-point combinators.


Task

Define the stateless Y combinator and use it to compute factorials and Fibonacci numbers from other stateless functions or lambda expressions.


Cf



ALGOL 68[edit]

Translation of: Python
Note: This specimen retains the original Python coding style.
Works with: ALGOL 68S version from Amsterdam Compiler Kit ( Guido van Rossum's teething ring) with runtime scope checking turned off.
BEGIN
MODE F = PROC(INT)INT;
MODE Y = PROC(Y)F;
 
# compare python Y = lambda f: (lambda x: x(x)) (lambda y: f( lambda *args: y(y)(*args)))#
PROC y = (PROC(F)F f)F: ( (Y x)F: x(x)) ( (Y z)F: f((INT arg )INT: z(z)( arg )));
 
PROC fib = (F f)F: (INT n)INT: CASE n IN n,n OUT f(n-1) + f(n-2) ESAC;
 
FOR i TO 10 DO print(y(fib)(i)) OD
END

AppleScript[edit]

AppleScript is not particularly "functional" friendly. It can, however, support the Y combinator.

AppleScript does not have anonymous functions, but it does have anonymous objects. The code below implements the latter with the former (using a handler (i.e. function) named 'lambda' in each anonymous object).

Unfortunately, an anonymous object can only be created in its own statement ('script'...'end script' can not be in an expression). Thus, we have to apply Y to the automatic 'result' variable that holds the value of the previous statement.

The identifier used for Y uses "pipe quoting" to make it obviously distinct from the y used inside the definition.

-- Y COMBINATOR ---------------------------------------------------------------
 
on |Y|(f)
script
on |λ|(y)
script
on |λ|(x)
y's |λ|(y)'s |λ|(x)
end |λ|
end script
 
f's |λ|(result)
end |λ|
end script
 
result's |λ|(result)
end |Y|
 
 
-- TEST -----------------------------------------------------------------------
on run
 
-- Factorial
script fact
on |λ|(f)
script
on |λ|(n)
if n = 0 then return 1
n * (f's |λ|(n - 1))
end |λ|
end script
end |λ|
end script
 
 
-- Fibonacci
script fib
on |λ|(f)
script
on |λ|(n)
if n = 0 then return 0
if n = 1 then return 1
(f's |λ|(n - 2)) + (f's |λ|(n - 1))
end |λ|
end script
end |λ|
end script
 
{facts:map(|Y|(fact), enumFromTo(0, 11)), fibs:map(|Y|(fib), enumFromTo(0, 20))}
 
--> {facts:{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800},
 
--> fibs:{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,
-- 1597, 2584, 4181, 6765}}
 
end run
 
 
-- GENERIC FUNCTIONS FOR TEST -------------------------------------------------
 
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
 
-- enumFromTo :: Int -> Int -> [Int]
on enumFromTo(m, n)
if n < m then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end enumFromTo
 
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
Output:
{facts:{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800}, 
fibs:{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765}}

ATS[edit]

 
(* ****** ****** *)
//
#include "share/atspre_staload.hats"
//
(* ****** ****** *)
//
fun
myfix
{a:type}
(
f: lazy(a) -<cloref1> a
) : lazy(a) = $delay(f(myfix(f)))
//
val
fact =
myfix{int-<cloref1>int}
(
lam(ff) => lam(x) => if x > 0 then x * !ff(x-1) else 1
)
(* ****** ****** *)
//
implement main0 () = println! ("fact(10) = ", !fact(10))
//
(* ****** ****** *)
 

BlitzMax[edit]

BlitzMax doesn't support anonymous functions or classes, so everything needs to be explicitly named.

SuperStrict
 
'Boxed type so we can just use object arrays for argument lists
Type Integer
Field val:Int
Function Make:Integer(_val:Int)
Local i:Integer = New Integer
i.val = _val
Return i
End Function
End Type
 
 
'Higher-order function type - just a procedure attached to a scope
Type Func Abstract
Method apply:Object(args:Object[]) Abstract
End Type
 
'Function definitions - extend with fields as locals and implement apply as body
Type Scope Extends Func Abstract
Field env:Scope
 
'Constructor - bind an environment to a procedure
Function lambda:Scope(env:Scope) Abstract
 
Method _init:Scope(_env:Scope) 'Helper to keep constructors small
env = _env ; Return Self
End Method
End Type
 
 
'Based on the following definition:
'(define (Y f)
' (let ((_r (lambda (r) (f (lambda a (apply (r r) a))))))
' (_r _r)))
 
'Y (outer)
Type Y Extends Scope
Field f:Func 'Parameter - gets closed over
 
Function lambda:Scope(env:Scope) 'Necessary due to highly limited constructor syntax
Return (New Y)._init(env)
End Function
 
Method apply:Func(args:Object[])
f = Func(args[0])
Local _r:Func = YInner1.lambda(Self)
Return Func(_r.apply([_r]))
End Method
End Type
 
'First lambda within Y
Type YInner1 Extends Scope
Field r:Func 'Parameter - gets closed over
 
Function lambda:Scope(env:Scope)
Return (New YInner1)._init(env)
End Function
 
Method apply:Func(args:Object[])
r = Func(args[0])
Return Func(Y(env).f.apply([YInner2.lambda(Self)]))
End Method
End Type
 
'Second lambda within Y
Type YInner2 Extends Scope
Field a:Object[] 'Parameter - not really needed, but good for clarity
 
Function lambda:Scope(env:Scope)
Return (New YInner2)._init(env)
End Function
 
Method apply:Object(args:Object[])
a = args
Local r:Func = YInner1(env).r
Return Func(r.apply([r])).apply(a)
End Method
End Type
 
 
'Based on the following definition:
'(define fac (Y (lambda (f)
' (lambda (x)
' (if (<= x 0) 1 (* x (f (- x 1)))))))
 
Type FacL1 Extends Scope
Field f:Func 'Parameter - gets closed over
 
Function lambda:Scope(env:Scope)
Return (New FacL1)._init(env)
End Function
 
Method apply:Object(args:Object[])
f = Func(args[0])
Return FacL2.lambda(Self)
End Method
End Type
 
Type FacL2 Extends Scope
Function lambda:Scope(env:Scope)
Return (New FacL2)._init(env)
End Function
 
Method apply:Object(args:Object[])
Local x:Int = Integer(args[0]).val
If x <= 0 Then Return Integer.Make(1) ; Else Return Integer.Make(x * Integer(FacL1(env).f.apply([Integer.Make(x - 1)])).val)
End Method
End Type
 
 
'Based on the following definition:
'(define fib (Y (lambda (f)
' (lambda (x)
' (if (< x 2) x (+ (f (- x 1)) (f (- x 2)))))))
 
Type FibL1 Extends Scope
Field f:Func 'Parameter - gets closed over
 
Function lambda:Scope(env:Scope)
Return (New FibL1)._init(env)
End Function
 
Method apply:Object(args:Object[])
f = Func(args[0])
Return FibL2.lambda(Self)
End Method
End Type
 
Type FibL2 Extends Scope
Function lambda:Scope(env:Scope)
Return (New FibL2)._init(env)
End Function
 
Method apply:Object(args:Object[])
Local x:Int = Integer(args[0]).val
If x < 2
Return Integer.Make(x)
Else
Local f:Func = FibL1(env).f
Local x1:Int = Integer(f.apply([Integer.Make(x - 1)])).val
Local x2:Int = Integer(f.apply([Integer.Make(x - 2)])).val
Return Integer.Make(x1 + x2)
EndIf
End Method
End Type
 
 
'Now test
Local _Y:Func = Y.lambda(Null)
 
Local fac:Func = Func(_Y.apply([FacL1.lambda(Null)]))
Print Integer(fac.apply([Integer.Make(10)])).val
 
Local fib:Func = Func(_Y.apply([FibL1.lambda(Null)]))
Print Integer(fib.apply([Integer.Make(10)])).val

Bracmat[edit]

The lambda abstraction

 (λx.x)y

translates to

 /('(x.$x))$y

in Bracmat code. Likewise, the fixed point combinator

Y := λg.(λx.g (x x)) (λx.g (x x))

the factorial

G := λr. λn.(1, if n = 0; else n × (r (n−1)))

the Fibonacci function

H := λr. λn.(1, if n = 1 or n = 2; else (r (n−1)) + (r (n−2)))

and the calls

(Y G) i

and

(Y H) i

where i varies between 1 and 10, are translated into Bracmat as shown below

(   ( Y
= /(
' ( g
. /('(x.$g'($x'$x)))
$ /('(x.$g'($x'$x)))
)
)
)
& ( G
= /(
' ( r
. /(
' ( n
. $n:~>0&1
| $n*($r)$($n+-1)
)
)
)
)
)
& ( H
= /(
' ( r
. /(
' ( n
. $n:(1|2)&1
| ($r)$($n+-1)+($r)$($n+-2)
)
)
)
)
)
& 0:?i
& whl
' ( 1+!i:~>10:?i
& out$(str$(!i "!=" (!Y$!G)$!i))
)
& 0:?i
& whl
' ( 1+!i:~>10:?i
& out$(str$("fib(" !i ")=" (!Y$!H)$!i))
)
&
)
Output:
1!=1
2!=2
3!=6
4!=24
5!=120
6!=720
7!=5040
8!=40320
9!=362880
10!=3628800
fib(1)=1
fib(2)=1
fib(3)=2
fib(4)=3
fib(5)=5
fib(6)=8
fib(7)=13
fib(8)=21
fib(9)=34
fib(10)=55

C[edit]

C doesn't have first class functions, so we demote everything to second class to match.
#include <stdio.h>
#include <stdlib.h>
 
/* func: our one and only data type; it holds either a pointer to
a function call, or an integer. Also carry a func pointer to
a potential parameter, to simulate closure */

typedef struct func_t *func;
typedef struct func_t {
func (*fn) (func, func);
func _;
int num;
} func_t;
 
func new(func(*f)(func, func), func _) {
func x = malloc(sizeof(func_t));
x->fn = f;
x->_ = _; /* closure, sort of */
x->num = 0;
return x;
}
 
func call(func f, func n) {
return f->fn(f, n);
}
 
func Y(func(*f)(func, func)) {
func g = new(f, 0);
g->_ = g;
return g;
}
 
func num(int n) {
func x = new(0, 0);
x->num = n;
return x;
}
 
 
func fac(func self, func n) {
int nn = n->num;
return nn > 1 ? num(nn * call(self->_, num(nn - 1))->num)
: num(1);
}
 
func fib(func self, func n) {
int nn = n->num;
return nn > 1
? num( call(self->_, num(nn - 1))->num +
call(self->_, num(nn - 2))->num )
: num(1);
}
 
void show(func n) { printf(" %d", n->num); }
 
int main() {
int i;
func f = Y(fac);
printf("fac: ");
for (i = 1; i < 10; i++)
show( call(f, num(i)) );
printf("\n");
 
f = Y(fib);
printf("fib: ");
for (i = 1; i < 10; i++)
show( call(f, num(i)) );
printf("\n");
 
return 0;
}
 
Output:
fac:  1 2 6 24 120 720 5040 40320 362880
fib:  1 2 3 5 8 13 21 34 55

C#[edit]

using System;
 
static class YCombinator<T> {
delegate Func<T, T> Recursive(Recursive recursive);
public static Func<Func<Func<T, T>, Func<T, T>>, Func<T, T>> Fix =
f => ((Recursive)(g =>
(f(x => g(g)(x)))))((Recursive)(g => f(x => g(g)(x))));
}
 
class Program {
static Func<int, int> fac =
YCombinator<int>.Fix(f => x => x < 2 ? 1 : x * f(x - 1));
static Func<int, int> fib =
YCombinator<int>.Fix(f => x => x < 2 ? x : f(x - 1) + f(x - 2));
 
static void Main() {
Console.WriteLine(fac(10));
Console.WriteLine(fib(10));
}
}
Output:
3628800
55

C++[edit]

Works with: C++11

Known to work with GCC 4.7.2. Compile with

g++ --std=c++11 ycomb.cc
#include <iostream>
#include <functional>
 
template <typename F>
struct RecursiveFunc {
std::function<F(RecursiveFunc)> o;
};
 
template <typename A, typename B>
std::function<B(A)> Y (std::function<std::function<B(A)>(std::function<B(A)>)> f) {
RecursiveFunc<std::function<B(A)>> r = {
std::function<std::function<B(A)>(RecursiveFunc<std::function<B(A)>>)>([f](RecursiveFunc<std::function<B(A)>> w) {
return f(std::function<B(A)>([w](A x) {
return w.o(w)(x);
}));
})
};
return r.o(r);
}
 
typedef std::function<int(int)> Func;
typedef std::function<Func(Func)> FuncFunc;
FuncFunc almost_fac = [](Func f) {
return Func([f](int n) {
if (n <= 1) return 1;
return n * f(n - 1);
});
};
 
FuncFunc almost_fib = [](Func f) {
return Func([f](int n) {
if (n <= 2) return 1;
return f(n - 1) + f(n - 2);
});
};
 
int main() {
auto fib = Y(almost_fib);
auto fac = Y(almost_fac);
std::cout << "fib(10) = " << fib(10) << std::endl;
std::cout << "fac(10) = " << fac(10) << std::endl;
return 0;
}
Output:
fib(10) = 55
fac(10) = 3628800
Works with: C++14

A shorter version, taking advantage of generic lambdas. Known to work with GCC 5.2.0, but likely some earlier versions as well. Compile with

g++ --std=c++14 ycomb.cc
#include <iostream>
#include <functional>
int main () {
auto y = ([] (auto f) { return
([] (auto x) { return x (x); }
([=] (auto y) -> std:: function <int (int)> { return
f ([=] (auto a) { return
(y (y)) (a) ;});}));});
 
auto almost_fib = [] (auto f) { return
[=] (auto n) { return
n < 2? n: f (n - 1) + f (n - 2) ;};};
auto almost_fac = [] (auto f) { return
[=] (auto n) { return
n <= 1? n: n * f (n - 1); };};
 
auto fib = y (almost_fib);
auto fac = y (almost_fac);
std:: cout << fib (10) << '\n'
<< fac (10) << '\n';
}
Output:
fib(10) = 55
fac(10) = 3628800

The usual version using recursion, disallowed by the task:

template <typename A, typename B>
std::function<B(A)> Y (std::function<std::function<B(A)>(std::function<B(A)>)> f) {
return [f](A x) {
return f(Y(f))(x);
};
}

Another version which is disallowed because a function passes itself, which is also a kind of recursion:

template <typename A, typename B>
struct YFunctor {
const std::function<std::function<B(A)>(std::function<B(A)>)> f;
YFunctor(std::function<std::function<B(A)>(std::function<B(A)>)> _f) : f(_f) {}
B operator()(A x) const {
return f(*this)(x);
}
};
 
template <typename A, typename B>
std::function<B(A)> Y (std::function<std::function<B(A)>(std::function<B(A)>)> f) {
return YFunctor<A,B>(f);
}

Ceylon[edit]

Using a class for the recursive type:

Result(*Args) y1<Result,Args>(
Result(*Args)(Result(*Args)) f)
given Args satisfies Anything[] {
 
class RecursiveFunction(o) {
shared Result(*Args)(RecursiveFunction) o;
}
 
value r = RecursiveFunction((RecursiveFunction w)
=> f(flatten((Args args) => w.o(w)(*args))));
 
return r.o(r);
}
 
value factorialY1 = y1((Integer(Integer) fact)(Integer x)
=> if (x > 1) then x * fact(x - 1) else 1);
 
value fibY1 = y1((Integer(Integer) fib)(Integer x)
=> if (x > 2) then fib(x - 1) + fib(x - 2) else 2);
 
print(factorialY1(10)); // 3628800
print(fibY1(10)); // 110

Using Anything to erase the function type:

Result(*Args) y2<Result,Args>(
Result(*Args)(Result(*Args)) f)
given Args satisfies Anything[] {
 
function r(Anything w) {
assert (is Result(*Args)(Anything) w);
return f(flatten((Args args) => w(w)(*args)));
}
 
return r(r);
}

Using recursion, this does not satisfy the task requirements, but is included here for illustrative purposes:

Result(*Args) y3<Result, Args>(
Result(*Args)(Result(*Args)) f)
given Args satisfies Anything[]
=> flatten((Args args) => f(y3(f))(*args));

Clojure[edit]

(defn Y [f]
((fn [x] (x x))
(fn [x]
(f (fn [& args]
(apply (x x) args))))))
 
(def fac
(fn [f]
(fn [n]
(if (zero? n) 1 (* n (f (dec n)))))))
 
(def fib
(fn [f]
(fn [n]
(condp = n
0 0
1 1
(+ (f (dec n))
(f (dec (dec n))))))))
Output:
user> ((Y fac) 10)
3628800
user> ((Y fib) 10)
55

Y can be written slightly more concisely via syntax sugar:

(defn Y [f]
(#(% %) #(f (fn [& args] (apply (% %) args)))))

Common Lisp[edit]

(defun Y (f)
((lambda (g) (funcall g g))
(lambda (g)
(funcall f (lambda (&rest a)
(apply (funcall g g) a))))))
 
(defun fac (n)
(funcall
(Y (lambda (f)
(lambda (n)
(if (zerop n)
1
(* n (funcall f (1- n)))))))
n))
 
(defun fib (n)
(funcall
(Y (lambda (f)
(lambda (n a b)
(if (< n 1)
a
(funcall f (1- n) b (+ a b))))))
n 0 1))
 
? (mapcar #'fac '(1 2 3 4 5 6 7 8 9 10))
(1 2 6 24 120 720 5040 40320 362880 3628800))
 
? (mapcar #'fib '(1 2 3 4 5 6 7 8 9 10))
(1 1 2 3 5 8 13 21 34 55)

CoffeeScript[edit]

Y = (f) -> g = f( (t...) -> g(t...) )

or

Y = (f) -> ((h)->h(h))((h)->f((t...)->h(h)(t...)))
fac = Y( (f) -> (n) -> if n > 1 then n * f(n-1) else 1 )
fib = Y( (f) -> (n) -> if n > 1 then f(n-1) + f(n-2) else n )
 

D[edit]

A stateless generic Y combinator:

import std.stdio, std.traits, std.algorithm, std.range;
 
auto Y(S, T...)(S delegate(T) delegate(S delegate(T)) f) {
static struct F {
S delegate(T) delegate(F) f;
alias f this;
}
return (x => x(x))(F(x => f((T v) => x(x)(v))));
}
 
void main() { // Demo code:
auto factorial = Y((int delegate(int) self) =>
(int n) => 0 == n ? 1 : n * self(n - 1)
);
 
auto ackermann = Y((ulong delegate(ulong, ulong) self) =>
(ulong m, ulong n) {
if (m == 0) return n + 1;
if (n == 0) return self(m - 1, 1);
return self(m - 1, self(m, n - 1));
});
 
writeln("factorial: ", 10.iota.map!factorial);
writeln("ackermann(3, 5): ", ackermann(3, 5));
}
Output:
factorial: [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
ackermann(3, 5): 253

Déjà Vu[edit]

Translation of: Python
Y f:
labda y:
labda:
call y @y
f
labda x:
x @x
call
 
labda f:
labda n:
if < 1 n:
* n f -- n
else:
1
set :fac Y
 
labda f:
labda n:
if < 1 n:
+ f - n 2 f -- n
else:
1
set :fib Y
 
!. fac 6
!. fib 6
Output:
720
13

Delphi[edit]

May work with Delphi 2009 and 2010 too.

Translation of: C++

(The translation is not literal; e.g. the function argument type is left unspecified to increase generality.)

program Y;
 
{$APPTYPE CONSOLE}
 
uses
SysUtils;
 
type
YCombinator = class sealed
class function Fix<T> (F: TFunc<TFunc<T, T>, TFunc<T, T>>): TFunc<T, T>; static;
end;
 
TRecursiveFuncWrapper<T> = record // workaround required because of QC #101272 (http://qc.embarcadero.com/wc/qcmain.aspx?d=101272)
type
TRecursiveFunc = reference to function (R: TRecursiveFuncWrapper<T>): TFunc<T, T>;
var
O: TRecursiveFunc;
end;
 
class function YCombinator.Fix<T> (F: TFunc<TFunc<T, T>, TFunc<T, T>>): TFunc<T, T>;
var
R: TRecursiveFuncWrapper<T>;
begin
R.O := function (W: TRecursiveFuncWrapper<T>): TFunc<T, T>
begin
Result := F (function (I: T): T
begin
Result := W.O (W) (I);
end);
end;
Result := R.O (R);
end;
 
 
type
IntFunc = TFunc<Integer, Integer>;
 
function AlmostFac (F: IntFunc): IntFunc;
begin
Result := function (N: Integer): Integer
begin
if N <= 1 then
Result := 1
else
Result := N * F (N - 1);
end;
end;
 
function AlmostFib (F: TFunc<Integer, Integer>): TFunc<Integer, Integer>;
begin
Result := function (N: Integer): Integer
begin
if N <= 2 then
Result := 1
else
Result := F (N - 1) + F (N - 2);
end;
end;
 
var
Fib, Fac: IntFunc;
begin
Fib := YCombinator.Fix<Integer> (AlmostFib);
Fac := YCombinator.Fix<Integer> (AlmostFac);
Writeln ('Fib(10) = ', Fib (10));
Writeln ('Fac(10) = ', Fac (10));
end.

E[edit]

Translation of: Python
def y := fn f { fn x { x(x) }(fn y { f(fn a { y(y)(a) }) }) }
def fac := fn f { fn n { if (n<2) {1} else { n*f(n-1) } }}
def fib := fn f { fn n { if (n == 0) {0} else if (n == 1) {1} else { f(n-1) + f(n-2) } }}
? pragma.enable("accumulator")
? accum [] for i in 0..!10 { _.with(y(fac)(i)) }
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
 
? accum [] for i in 0..!10 { _.with(y(fib)(i)) }
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

EchoLisp[edit]

 
;; Ref : http://www.ece.uc.edu/~franco/C511/html/Scheme/ycomb.html
 
(define Y
(lambda (X)
((lambda (procedure)
(X (lambda (arg) ((procedure procedure) arg))))
(lambda (procedure)
(X (lambda (arg) ((procedure procedure) arg)))))))
 
; Fib
(define Fib* (lambda (func-arg)
(lambda (n) (if (< n 2) n (+ (func-arg (- n 1)) (func-arg (- n 2)))))))
(define fib (Y Fib*))
(fib 6)
8
 
; Fact
(define F*
(lambda (func-arg) (lambda (n) (if (zero? n) 1 (* n (func-arg (- n 1)))))))
(define fact (Y F*))
 
(fact 10)
3628800
 

Eero[edit]

Translated from Objective-C example on this page.

#import <Foundation/Foundation.h>
 
typedef int (^Func)(int)
typedef Func (^FuncFunc)(Func)
typedef Func (^RecursiveFunc)(id) // hide recursive typing behind dynamic typing
 
Func fix(FuncFunc f)
Func r(RecursiveFunc g)
int s(int x)
return g(g)(x)
return f(s)
return r(r)
 
int main(int argc, const char *argv[])
autoreleasepool
 
Func almost_fac(Func f)
return (int n | return n <= 1 ? 1 : n * f(n - 1))
 
Func almost_fib(Func f)
return (int n | return n <= 2 ? 1 : f(n - 1) + f(n - 2))
 
fib := fix(almost_fib)
fac := fix(almost_fac)
 
Log('fib(10) = %d', fib(10))
Log('fac(10) = %d', fac(10))
 
return 0

Ela[edit]

fix = \f -> (\x -> & f (x x)) (\x -> & f (x x))
 
fac _ 0 = 1
fac f n = n * f (n - 1)
 
fib _ 0 = 0
fib _ 1 = 1
fib f n = f (n - 1) + f (n - 2)
 
(fix fac 12, fix fib 12)
Output:
(479001600,144)

Elena[edit]

Translation of: Smalltalk

ELENA 3.4 :

import extensions.
 
singleton YCombinator
{
fix : func
= (:f)[(:x)[ x(x) ]((:g)[ f((:x)[ (g(g))(x) ])])](func).
}
 
public program
[
var fib := YCombinator fix(:f)((:i)( (i <= 1) if:[^i] else:[^f(i-1) + f(i-2) ] )).
var fact := YCombinator fix(:f)((:i)((i == 0) if:[^1] else:[^f(i-1) * i] )).
 
console printLine("fib(10)=",fib(10)).
console printLine("fact(10)=",fact(10)).
]
Output:
fib(10)=55
fact(10)=3628800

Elixir[edit]

Translation of: Python
 
iex(1)> yc = fn f -> (fn x -> x.(x) end).(fn y -> f.(fn arg -> y.(y).(arg) end) end) end
#Function<6.90072148/1 in :erl_eval.expr/5>
iex(2)> fac = fn f -> fn n -> if n < 2 do 1 else n * f.(n-1) end end end
#Function<6.90072148/1 in :erl_eval.expr/5>
iex(3)> for i <- 0..9, do: yc.(fac).(i)
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
iex(4)> fib = fn f -> fn n -> if n == 0 do 0 else (if n == 1 do 1 else f.(n-1) + f.(n-2) end) end end end
#Function<6.90072148/1 in :erl_eval.expr/5>
iex(5)> for i <- 0..9, do: yc.(fib).(i)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
 

Elm[edit]

This is similar to the Haskell solution below, but uses a strict fixed-point combinator since Elm lacks lazy evaluation.

 
import Html exposing (text)
 
type Mu a b = Roll (Mu a b -> a -> b)
 
unroll : Mu a b -> (Mu a b -> a -> b)
unroll (Roll x) = x
 
fix : ((a -> b) -> (a -> b)) -> (a -> b)
fix f = let g r = f (\v -> unroll r r v)
in g (Roll g)
 
fac : Int -> Int
fac = fix <|
\f n -> if n <= 0
then 1
else n * f (n - 1)
 
main = text <| toString <| fac 5
 

Erlang[edit]

Y = fun(M) -> (fun(X) -> X(X) end)(fun (F) -> M(fun(A) -> (F(F))(A) end) end) end.
 
Fac = fun (F) ->
fun (0) -> 1;
(N) -> N * F(N-1)
end
end.
Fib = fun(F) ->
fun(0) -> 0;
(1) -> 1;
(N) -> F(N-1) + F(N-2)
end
end.
(Y(Fac))(5). %% 120
(Y(Fib))(8). %% 21

F#[edit]

type 'a mu = Roll of ('a mu -> 'a)  // ' fixes ease syntax colouring confusion with
 
let unroll (Roll x) = x
// val unroll : 'a mu -> ('a mu -> 'a)
 
// As with most of the strict (non-deferred or non-lazy) languages,
// this is the Z-combinator with the additional 'a' parameter...
let fix f = let g = fun x a -> f (unroll x x) a in g (Roll g)
// val fix : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b = <fun>
 
// Although true to the factorial definition, the
// recursive call is not in tail call position, so can't be optimized
// and will overflow the call stack for the recursive calls for large ranges...
//let fac = fix (fun f n -> if n < 2 then 1I else bigint n * f (n - 1))
// val fac : (int -> BigInteger) = <fun>
 
// much better progressive calculation in tail call position...
let fac = fix (fun f n i -> if i < 2 then n else f () (bigint i * n) (i - 1)) <| 1I
// val fac : (int -> BigInteger) = <fun>
 
// Although true to the definition of Fibonacci numbers,
// this can't be tail call optimized and recursively repeats calculations
// for a horrendously inefficient exponential performance fib function...
// let fib = fix (fun fnc n -> if n < 2 then n else fnc (n - 1) + fnc (n - 2))
// val fib : (int -> BigInteger) = <fun>
 
// much better progressive calculation in tail call position...
let fib = fix (fun fnc f s i -> if i < 2 then f else fnc s (f + s) (i - 1)) 1I 1I
// val fib : (int -> BigInteger) = <fun>
 
[<EntryPoint>]
let main argv =
fac 10 |> printfn "%A" // prints 3628800
fib 10 |> printfn "%A" // prints 55
0 // return an integer exit code
Output:
3628800
55

Note that the first `fac` definition isn't really very good as the recursion is not in tail call position and thus will build stack, although for these functions one will likely never use it to stack overflow as the result would be exceedingly large; it is better defined as per the second definition as a steadily increasing function controlled by an `int` indexing argument and thus be in tail call position as is done for the `fib` function.

Also note that the above isn't the true fix point Y-combinator which would race without the beta conversion to the Z-combinator with the included `a` argument; the Z-combinator can't be used in all cases that require a true Y-combinator such as in the formation of deferred execution sequences in the last example, as follows:

// same as previous...
type 'a mu = Roll of ('a mu -> 'a) // ' fixes ease syntax colouring confusion with
 
// same as previous...
let unroll (Roll x) = x
// val unroll : 'a mu -> ('a mu -> 'a)
 
// break race condition with some deferred execution - laziness...
let fix f = let g = fun x -> f <| fun() -> (unroll x x) in g (Roll g)
// val fix : ((unit -> 'a) -> 'a -> 'a) = <fun>
 
// same efficient version of factorial with added deferred execution...
let fac = fix (fun f n i -> if i < 2 then n else f () (bigint i * n) (i - 1)) <| 1I
// val fac : (int -> BigInteger) = <fun>
 
// same efficient version of factorial with added deferred execution...
let fib = fix (fun fnc f s i -> if i < 2 then f else fnc () s (f + s) (i - 1)) 1I 1I
// val fib : (int -> BigInteger) = <fun>
 
// given the following definition for an infinite Co-Inductive Stream (CIS)...
type CIS<'a> = CIS of 'a * (unit -> CIS<'a>) // ' fix formatting
 
// define a continuous stream of Fibonacci numbers; there are other simpler ways,
// this way does not use recursion at all by using the Y-combinator, although it is
// much slower than other ways due to the many additional function calls and memory allocations,
// it demonstrates something that can't be done with the Z-combinator...
let fibs() =
let fbsgen : (CIS<bigint> -> CIS<bigint>) =
fix (fun fnc f (CIS(s, rest)) ->
CIS(f + s, fun() -> fnc () s <| rest())) 1I
Seq.unfold
(fun (CIS(hd, rest)) -> Some(hd, rest()))
<| fix (fun cis -> fbsgen (CIS(0I, cis)))
 
[<EntryPoint>]
let main argv =
fac 10 |> printfn "%A" // prints 3628800
fib 10 |> printfn "%A" // prints 55
fibs() |> Seq.take 20 |> Seq.iter (printf "%A ") // prints 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765
printfn ""
0 // return an integer exit code
Output:
3628800
55
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 

The above would be useful if F# did not have recursive functions (functions that can call themselves in their own definition), but as for most modern languages, F# does have function recursion by the use of the `rec` keyword before the function name, thus the above `fac` and `fib` functions can be written much more simply (and to run faster using tail recursion) with a recursion definition for the `fix` Y-combinator as follows, with a simple injected deferred execution to prevent race:

let rec fix f = f <| fun() -> fix f
// val fix : f:((unit -> 'a) -> 'a) -> 'a
 
// the application of this true Y-combinator is the same as for the above non function recursive version.

Using the Y-combinator (or Z-combinator) as expressed here is pointless as in unnecessary and makes the code slower due to the extra function calls through the call stack, with the first non-function recursive implementation even slower than the second function recursion one; a non Y-combinator version can use function recursion with tail call optimization to simplify looping for about 100 times the speed in the actual loop overhead; thus, this is primarily an intellectual exercise.

Also note that these Y-combinators/Z-combinator are the non sharing kind; for certain types of algorithms that require that the input and output recursive values be the same (such as the same sequence or lazy list but made reference at difference stages), these will work but may be many times slower as in over 10 times slower than using binding recursion if the language allows it; F# allows binding recursion with a warning.

Factor[edit]

In rosettacode/Y.factor

USING: fry kernel math ;
IN: rosettacode.Y
: Y ( quot -- quot )
'[ [ dup call call ] curry @ ] dup call ; inline
 
: almost-fac ( quot -- quot )
'[ dup zero? [ drop 1 ] [ dup 1 - @ * ] if ] ;
 
: almost-fib ( quot -- quot )
'[ dup 2 >= [ 1 2 [ - @ ] [email protected] bi + ] when ] ;

In rosettacode/Y-tests.factor

USING: kernel tools.test rosettacode.Y ;
IN: rosettacode.Y.tests
 
[ 120 ] [ 5 [ almost-fac ] Y call ] unit-test
[ 8 ] [ 6 [ almost-fib ] Y call ] unit-test

running the tests :

 ( scratchpad - auto ) "rosettacode.Y" test
Loading resource:work/rosettacode/Y/Y-tests.factor
Unit Test: { [ 120 ] [ 5 [ almost-fac ] Y call ] }
Unit Test: { [ 8 ]   [ 6 [ almost-fib ] Y call ] }

Forth[edit]

\ Address of an xt.
variable 'xt
\ Make room for an xt.
: xt, ( -- ) here 'xt ! 1 cells allot ;
\ Store xt.
: !xt ( xt -- ) 'xt @ ! ;
\ Compile fetching the xt.
: @xt, ( -- ) 'xt @ postpone literal postpone @ ;
\ Compile the Y combinator.
: y, ( xt1 -- xt2 ) >r :noname @xt, r> compile, postpone ; ;
\ Make a new instance of the Y combinator.
: y ( xt1 -- xt2 ) xt, y, dup !xt ;

Samples:

\ Factorial
10 :noname ( u1 xt -- u2 ) over ?dup if 1- swap execute * else 2drop 1 then ;
y execute . 3628800 ok
 
\ Fibonacci
10 :noname ( u1 xt -- u2 ) over 2 < if drop else >r 1- dup [email protected] execute swap 1- r> execute + then ;
y execute . 55 ok
 

Falcon[edit]

 
Y = { f => {x=> {n => f(x(x))(n)}} ({x=> {n => f(x(x))(n)}}) }
facStep = { f => {x => x < 1 ? 1 : x*f(x-1) }}
fibStep = { f => {x => x == 0 ? 0 : (x == 1 ? 1 : f(x-1) + f(x-2))}}
 
YFac = Y(facStep)
YFib = Y(fibStep)
 
> "Factorial 10: ", YFac(10)
> "Fibonacci 10: ", YFib(10)
 

GAP[edit]

Y := function(f)
local u;
u := x -> x(x);
return u(y -> f(a -> y(y)(a)));
end;
 
fib := function(f)
local u;
u := function(n)
if n < 2 then
return n;
else
return f(n-1) + f(n-2);
fi;
end;
return u;
end;
 
Y(fib)(10);
# 55
 
fac := function(f)
local u;
u := function(n)
if n < 2 then
return 1;
else
return n*f(n-1);
fi;
end;
return u;
end;
 
Y(fac)(8);
# 40320

Genyris[edit]

Translation of: Scheme
def fac (f)
function (n)
if (equal? n 0) 1
* n (f (- n 1))
def fib (f)
function (n)
cond
(equal? n 0) 0
(equal? n 1) 1
else (+ (f (- n 1)) (f (- n 2)))
 
def Y (f)
(function (x) (x x))
function (y)
f
function (&rest args) (apply (y y) args)
 
assertEqual ((Y fac) 5) 120
assertEqual ((Y fib) 8) 21

Go[edit]

package main
 
import "fmt"
 
type Func func(int) int
type FuncFunc func(Func) Func
type RecursiveFunc func (RecursiveFunc) Func
 
func main() {
fac := Y(almost_fac)
fib := Y(almost_fib)
fmt.Println("fac(10) = ", fac(10))
fmt.Println("fib(10) = ", fib(10))
}
 
func Y(f FuncFunc) Func {
g := func(r RecursiveFunc) Func {
return f(func(x int) int {
return r(r)(x)
})
}
return g(g)
}
 
func almost_fac(f Func) Func {
return func(x int) int {
if x <= 1 {
return 1
}
return x * f(x-1)
}
}
 
func almost_fib(f Func) Func {
return func(x int) int {
if x <= 2 {
return 1
}
return f(x-1)+f(x-2)
}
}
Output:
fac(10) =  3628800
fib(10) =  55

The usual version using recursion, disallowed by the task:

func Y(f FuncFunc) Func {
return func(x int) int {
return f(Y(f))(x)
}
}

Groovy[edit]

Here is the simplest (unary) form of applicative order Y:

def Y = { le -> ({ f -> f(f) })({ f -> le { x -> f(f)(x) } }) }
 
def factorial = Y { fac ->
{ n -> n <= 2 ? n : n * fac(n - 1) }
}
 
assert 2432902008176640000 == factorial(20G)
 
def fib = Y { fibStar ->
{ n -> n <= 1 ? n : fibStar(n - 1) + fibStar(n - 2) }
}
 
assert fib(10) == 55

This version was translated from the one in The Little Schemer by Friedman and Felleisen. The use of the variable name le is due to the fact that the authors derive Y from an ordinary recursive length function.

A variadic version of Y in Groovy looks like this:

def Y = { le -> ({ f -> f(f) })({ f -> le { Object[] args -> f(f)(*args) } }) }
 
def mul = Y { mulStar -> { a, b -> a ? b + mulStar(a - 1, b) : 0 } }
 
1.upto(10) {
assert mul(it, 10) == it * 10
}

Haskell[edit]

The obvious definition of the Y combinator (\f-> (\x -> f (x x)) (\x-> f (x x))) cannot be used in Haskell because it contains an infinite recursive type (a = a -> b). Defining a data type (Mu) allows this recursion to be broken.

newtype Mu a = Roll
{ unroll :: Mu a -> a }
 
fix :: (a -> a) -> a
fix = g <*> (Roll . g)
where
g = (. (>>= id) unroll)
 
- this version is not in tail call position...
-- fac :: Integer -> Integer
-- fac =
-- fix $ \f n -> if n <= 0 then 1 else n * f (n - 1)
 
-- this version builds a progression from tail call position and is more efficient...
fac :: Integer -> Integer
fac =
(fix $ \f n i -> if i <= 0 then n else f (i * n) (i - 1)) 1
 
-- make fibs a function, else memory leak as
-- head of the list can never be released as per:
-- https://wiki.haskell.org/Memory_leak, type 1.1
-- overly complex version...
{--
fibs :: () -> [Integer]
fibs() =
fix $
(0 :) . (1 :) .
(fix
(\f (x:xs) (y:ys) ->
case x + y of n -> n `seq` n : f xs ys) <*> tail)
--}

 
-- easier to read, simpler (faster) version...
fibs :: () -> [Integer]
fibs() = 0 : 1 : fix fibs_ 0 1
where
fibs_ fnc f s =
case f + s of n -> n `seq` n : fnc s n
 
main :: IO ()
main =
mapM_
print
[ map fac [1 .. 20]
, take 20 $ fibs()
]

The usual version uses recursion on a binding, disallowed by the task, to define the fix itself; but the definitions produced by this fix does not use recursion on value bindings although it does use recursion when defining a function (not possible in all languages), so it can be viewed as a true Y-combinator too:

-- note that this version of fix uses function recursion in its own definition;
-- thus its use just means that the recursion has been "pulled" into the "fix" function,
-- instead of the function that uses it...
fix :: (a -> a) -> a
fix f = f (fix f) -- _not_ the {fix f = x where x = f x}
 
fac :: Integer -> Integer
fac =
(fix $
\f n i ->
if i <= 0 then n
else f (i * n) (i - 1)) 1
 
fib :: Integer -> Integer
fib =
(fix $
\fnc f s i ->
if i <= 1 then f
else case f + s of n -> n `seq` fnc s n (i - 1)) 0 1
 
{--
-- compute a lazy infinite list. This is
-- a Y-combinator version of: fibs() = 0:1:zipWith (+) fibs (tail fibs)
-- which is the same as the above version but easier to read...
fibs :: () -> [Integer]
fibs() = fix fibs_
where
zipP f (x:xs) (y:ys) =
case x + y of n -> n `seq` n : f xs ys
fibs_ a = 0 : 1 : fix zipP a (tail a)
--}

 
-- easier to read, simpler (faster) version...
fibs :: () -> [Integer]
fibs() = 0 : 1 : fix fibs_ 0 1
where
fibs_ fnc f s =
case f + s of n -> n `seq` n : fnc s n
 
-- This code shows how the functions can be used:
main :: IO ()
main =
mapM_
print
[ map fac [1 .. 20]
, map fib [1 .. 20]
, take 20 fibs()
]

Now just because something is possible using the Y-combinator doesn't mean that it is practical: the above implementations can't compute much past the 1000th number in the Fibonacci list sequence and is quite slow at doing so; using direct function recursive routines compute about 100 times faster and don't hang for large ranges, nor give problems compiling as the first version does (GHC version 8.4.3 at -O1 optimization level).

If one has recursive functions as Haskell does and as used by the second `fix`, there is no need to use `fix`/the Y-combinator at all since one may as well just write the recursion directly. The Y-combinator may be interesting mathematically, but it isn't very practical when one has any other choice.

J[edit]

Non-tacit version[edit]

Unfortunately, in principle, J functions cannot take functions of the same type as arguments. In other words, verbs (functions) cannot take verbs, and adverbs or conjunctions (higher-order functions) cannot take adverbs or conjunctions. This implementation uses the body, a literal (string), of an explicit adverb (a higher-order function with a left argument) as the argument for Y, to represent the adverb for which the product of Y is a fixed-point verb; Y itself is also an adverb.

   Y=. (1 :'('':''<@;(1;~":0)<@;<@((":0)&;))u')(1 :'('':''<@;(1;~":0)<@;<@((":0)&;))''u u`:6('',(5!:5<''u''),'')`:6 y''')(1 :'u u`:6')

This Y combinator follows the standard method: it produces a fixed-point which reproduces and transforms itself anonymously according to the adverb represented by Y's argument. All names (variables) refer to arguments of the enclosing adverbs and there are no assignments. The anonymous verb (':'<@;(1;~":0)<@;<@((":0)&;)), applied to the body of an explicit adverb, produces J's atomic representation (AR) of the adverb. This verb is embedded twice in the Y combinator making it longer that one would otherwise expect.

The factorial and Fibonacci examples follow:

   'if. * y do. y * u <: y else. 1 end.' Y 10 NB. Factorial
3628800
'([email protected]:<:@:<: + [email protected]:<:)^:(1 < ])' Y 10 NB. Fibonacci
55

The names u, x, and y are J's standard names for arguments; the name y represents the argument of u and the name u represents the verb argument of the adverb for which Y produces a fixed-point. Any verb can also be expressed tacitly, without any reference to its argument(s), as in the Fibonacci example.

A structured derivation of a Y with states follows (the stateless version is produced by replacing all the names by its referents):

   arb=. ':'<@;(1;~":0)<@;<@((":0)&;)                     NB. AR of an explicit adverb from its body 
 
ara=. 1 :'arb u' NB. The verb arb as an adverb
srt=. 1 :'arb ''u u`:6('' , (5!:5<''u'') , '')`:6 y''' NB. AR of the self-replication and transformation adverb
gab=. 1 :'u u`:6' NB. The AR of the adverb and the adverb itself as a train
 
Y=. ara srt gab NB. Train of adverbs

The adverb Y, apart from using a representation as Y's argument, satisfies the task's requirements. However, it only works for monadic verbs (functions with a right argument). J's verbs can also be dyadic (functions with a left and right arguments) and ambivalent (almost all J's primitive verbs are ambivalent; for example - can be used as in - 1 and 2 - 1). The following adverb (XY) implements anonymous recursion of monadic, dyadic, and ambivalent verbs (the name x represents the left argument of u),

   XY=. (1 :'('':''<@;(1;~":0)<@;<@((":0)&;))u')(1 :'('':''<@;(1;~":0)<@;<@((":0)&;))((''u u`:6('',(5!:5<''u''),'')`:6 y''),(10{a.),'':'',(10{a.),''x(u u`:6('',(5!:5<''u''),'')`:6)y'')')(1 :'u u`:6')

The following are examples of anonymous dyadic and ambivalent recursions,

   1 2 3 '([:`(>:@:])`(<:@:[ u 1:)`(<:@[ u [ u <:@:])@.(#[email protected],&*))'XY"0/  1 2 3 4 5 NB. Ackermann function...
3 4 5 6 7
5 7 9 11 13
13 29 61 125 253
'1:`(<: u <:)@.* : (+ + 2 * [email protected]:])'XY"0/~ i.7 NB. Ambivalent recursion...
2 5 14 35 80 173 362
3 6 15 36 81 174 363
4 7 16 37 82 175 364
5 8 17 38 83 176 365
6 9 18 39 84 177 366
7 10 19 40 85 178 367
8 11 20 41 86 179 368
NB. OEIS A097813 - main diagonal
NB. OEIS A050488 = A097813 - 1 - adyacent upper off-diagonal

J supports directly anonymous tacit recursion via the verb $: and for tacit recursions, XY is equivalent to the adverb,

   YX=. (1 :'('':''<@;(1;~":0)<@;<@((":0)&;))u')($:`)(`:6)

Tacit version[edit]

The Y combinator can be implemented indirectly using, for example, the linear representations of verbs (Y becomes a wrapper which takes an ad hoc verb as an argument and serializes it; the underlying self-referring system interprets the serialized representation of a verb as the corresponding verb):

Y=. ((((&>)/)((((^:_1)b.)(`(<'0';_1)))(`:6)))(&([ 128!:2 ,&<)))

The factorial and Fibonacci examples:

   u=. [ NB. Function (left)
n=. ] NB. Argument (right)
sr=. [ apply f. ,&< NB. Self referring
 
fac=. (1:`(n * u sr n - 1:)) @. (0 < n)
fac f. Y 10
3628800
 
Fib=. ((u sr n - 2:) + u sr n - 1:) ^: (1 < n)
Fib f. Y 10
55

The stateless functions are shown next (the f. adverb replaces all embedded names by its referents):

   fac f. Y NB. Factorial...
'1:`(] * [ ([ 128!:2 ,&<) ] - 1:)@.(0 < ])&>/'&([ 128!:2 ,&<)
 
fac f. NB. Factorial step...
1:`(] * [ ([ 128!:2 ,&<) ] - 1:)@.(0 < ])
 
 
Fib f. Y NB. Fibonacci...
'(([ ([ 128!:2 ,&<) ] - 2:) + [ ([ 128!:2 ,&<) ] - 1:)^:(1 < ])&>/'&([ 128!:2 ,&<)
 
Fib f. NB. Fibonacci step...
(([ ([ 128!:2 ,&<) ] - 2:) + [ ([ 128!:2 ,&<) ] - 1:)^:(1 < ])

A structured derivation of Y follows:

   sr=. [ apply f.,&<                 NB. Self referring
lv=. (((^:_1)b.)(`(<'0';_1)))(`:6) NB. Linear representation of a verb argument
Y=. (&>)/lv(&sr) NB. Y with embedded states
Y=. 'Y'f. NB. Fixing it...
Y NB. ... To make it stateless (i.e., a combinator)
((((&>)/)((((^:_1)b.)(`_1))(`:6)))(&([ 128!:2 ,&<)))

Explicit alternate implementation[edit]

Another approach:

Y=:1 :0
f=. u Defer
(5!:1<'f') f y
)
 
Defer=: 1 :0
:
g=. x&(x`:6)
(5!:1<'g') u y
)
 
almost_factorial=: 4 :0
if. 0 >: y do. 1
else. y * x`:6 y-1 end.
)
 
almost_fibonacci=: 4 :0
if. 2 > y do. y
else. (x`:6 y-1) + x`:6 y-2 end.
)

Example use:

   almost_factorial Y 9
362880
almost_fibonacci Y 9
34
almost_fibonacci Y"0 i. 10
0 1 1 2 3 5 8 13 21 34

Or, if you would prefer to not have a dependency on the definition of Defer, an equivalent expression would be:

Y=:2 :0(0 :0)
NB. this block will be n in the second part
:
g=. x&(x`:6)
(5!:1<'g') u y
)
f=. u (1 :n)
(5!:1<'f') f y
)

That said, if you think of association with a name as state (because in different contexts the association may not exist, or may be different) you might also want to remove that association in the context of the Y combinator.

For example:

   almost_factorial f. Y 10
3628800

Java[edit]

Works with: Java version 8+
import java.util.function.Function;
 
public interface YCombinator {
interface RecursiveFunction<F> extends Function<RecursiveFunction<F>, F> { }
public static <A,B> Function<A,B> Y(Function<Function<A,B>, Function<A,B>> f) {
RecursiveFunction<Function<A,B>> r = w -> f.apply(x -> w.apply(w).apply(x));
return r.apply(r);
}
 
public static void main(String... arguments) {
Function<Integer,Integer> fib = Y(f -> n ->
(n <= 2)
? 1
 : (f.apply(n - 1) + f.apply(n - 2))
);
Function<Integer,Integer> fac = Y(f -> n ->
(n <= 1)
? 1
 : (n * f.apply(n - 1));
);
 
System.out.println("fib(10) = " + fib.apply(10));
System.out.println("fac(10) = " + fac.apply(10));
}
}
Output:
fib(10) = 55
fac(10) = 3628800

The usual version using recursion, disallowed by the task:

    public static <A,B> Function<A,B> Y(Function<Function<A,B>, Function<A,B>> f) {
return x -> f.apply(Y(f)).apply(x);
}

Another version which is disallowed because a function passes itself, which is also a kind of recursion:

    public static <A,B> Function<A,B> Y(Function<Function<A,B>, Function<A,B>> f) {
return new Function<A,B>() {
public B apply(A x) {
return f.apply(this).apply(x);
}
};
}
Works with: Java version pre-8

We define a generic function interface like Java 8's Function.

interface Function<A, B> {
public B call(A x);
}
 
public class YCombinator {
interface RecursiveFunc<F> extends Function<RecursiveFunc<F>, F> { }
 
public static <A,B> Function<A,B> fix(final Function<Function<A,B>, Function<A,B>> f) {
RecursiveFunc<Function<A,B>> r =
new RecursiveFunc<Function<A,B>>() {
public Function<A,B> call(final RecursiveFunc<Function<A,B>> w) {
return f.call(new Function<A,B>() {
public B call(A x) {
return w.call(w).call(x);
}
});
}
};
return r.call(r);
}
 
public static void main(String[] args) {
Function<Function<Integer,Integer>, Function<Integer,Integer>> almost_fib =
new Function<Function<Integer,Integer>, Function<Integer,Integer>>() {
public Function<Integer,Integer> call(final Function<Integer,Integer> f) {
return new Function<Integer,Integer>() {
public Integer call(Integer n) {
if (n <= 2) return 1;
return f.call(n - 1) + f.call(n - 2);
}
};
}
};
 
Function<Function<Integer,Integer>, Function<Integer,Integer>> almost_fac =
new Function<Function<Integer,Integer>, Function<Integer,Integer>>() {
public Function<Integer,Integer> call(final Function<Integer,Integer> f) {
return new Function<Integer,Integer>() {
public Integer call(Integer n) {
if (n <= 1) return 1;
return n * f.call(n - 1);
}
};
}
};
 
Function<Integer,Integer> fib = fix(almost_fib);
Function<Integer,Integer> fac = fix(almost_fac);
 
System.out.println("fib(10) = " + fib.call(10));
System.out.println("fac(10) = " + fac.call(10));
}
}

The following code modifies the Function interface such that multiple parameters (via varargs) are supported, simplifies the y function considerably, and the Ackermann function has been included in this implementation (mostly because both D and PicoLisp include it in their own implementations).

import java.util.function.Function;
 
@FunctionalInterface
public interface SelfApplicable<OUTPUT> extends Function<SelfApplicable<OUTPUT>, OUTPUT> {
public default OUTPUT selfApply() {
return apply(this);
}
}
import java.util.function.Function;
import java.util.function.UnaryOperator;
 
@FunctionalInterface
public interface FixedPoint<FUNCTION> extends Function<UnaryOperator<FUNCTION>, FUNCTION> {}
import java.util.Arrays;
import java.util.Optional;
import java.util.function.Function;
import java.util.function.BiFunction;
 
@FunctionalInterface
public interface VarargsFunction<INPUTS, OUTPUT> extends Function<INPUTS[], OUTPUT> {
@SuppressWarnings("unchecked")
public OUTPUT apply(INPUTS... inputs);
 
public static <INPUTS, OUTPUT> VarargsFunction<INPUTS, OUTPUT> from(Function<INPUTS[], OUTPUT> function) {
return function::apply;
}
 
public static <INPUTS, OUTPUT> VarargsFunction<INPUTS, OUTPUT> upgrade(Function<INPUTS, OUTPUT> function) {
return inputs -> function.apply(inputs[0]);
}
 
public static <INPUTS, OUTPUT> VarargsFunction<INPUTS, OUTPUT> upgrade(BiFunction<INPUTS, INPUTS, OUTPUT> function) {
return inputs -> function.apply(inputs[0], inputs[1]);
}
 
@SuppressWarnings("unchecked")
public default <POST_OUTPUT> VarargsFunction<INPUTS, POST_OUTPUT> andThen(
VarargsFunction<OUTPUT, POST_OUTPUT> after) {
return inputs -> after.apply(apply(inputs));
}
 
@SuppressWarnings("unchecked")
public default Function<INPUTS, OUTPUT> toFunction() {
return input -> apply(input);
}
 
@SuppressWarnings("unchecked")
public default BiFunction<INPUTS, INPUTS, OUTPUT> toBiFunction() {
return (input, input2) -> apply(input, input2);
}
 
@SuppressWarnings("unchecked")
public default <PRE_INPUTS> VarargsFunction<PRE_INPUTS, OUTPUT> transformArguments(Function<PRE_INPUTS, INPUTS> transformer) {
return inputs -> apply((INPUTS[]) Arrays.stream(inputs).parallel().map(transformer).toArray());
}
}
import java.math.BigDecimal;
import java.math.BigInteger;
import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;
import java.util.function.Function;
import java.util.function.UnaryOperator;
import java.util.stream.Collectors;
import java.util.stream.LongStream;
 
@FunctionalInterface
public interface Y<FUNCTION> extends SelfApplicable<FixedPoint<FUNCTION>> {
public static void main(String... arguments) {
BigInteger TWO = BigInteger.ONE.add(BigInteger.ONE);
 
Function<Number, Long> toLong = Number::longValue;
Function<Number, BigInteger> toBigInteger = toLong.andThen(BigInteger::valueOf);
 
/* Based on https://gist.github.com/aruld/3965968/#comment-604392 */
Y<VarargsFunction<Number, Number>> combinator = y -> f -> x -> f.apply(y.selfApply().apply(f)).apply(x);
FixedPoint<VarargsFunction<Number, Number>> fixedPoint = combinator.selfApply();
 
VarargsFunction<Number, Number> fibonacci = fixedPoint.apply(
f -> VarargsFunction.upgrade(
toBigInteger.andThen(
n -> (n.compareTo(TWO) <= 0)
? 1
 : new BigInteger(f.apply(n.subtract(BigInteger.ONE)).toString())
.add(new BigInteger(f.apply(n.subtract(TWO)).toString()))
)
)
);
 
VarargsFunction<Number, Number> factorial = fixedPoint.apply(
f -> VarargsFunction.upgrade(
toBigInteger.andThen(
n -> (n.compareTo(BigInteger.ONE) <= 0)
? 1
 : n.multiply(new BigInteger(f.apply(n.subtract(BigInteger.ONE)).toString()))
)
)
);
 
VarargsFunction<Number, Number> ackermann = fixedPoint.apply(
f -> VarargsFunction.upgrade(
(BigInteger m, BigInteger n) -> m.equals(BigInteger.ZERO)
? n.add(BigInteger.ONE)
 : f.apply(
m.subtract(BigInteger.ONE),
n.equals(BigInteger.ZERO)
? BigInteger.ONE
 : f.apply(m, n.subtract(BigInteger.ONE))
)
).transformArguments(toBigInteger)
);
 
Map<String, VarargsFunction<Number, Number>> functions = new HashMap<>();
functions.put("fibonacci", fibonacci);
functions.put("factorial", factorial);
functions.put("ackermann", ackermann);
 
Map<VarargsFunction<Number, Number>, Number[]> parameters = new HashMap<>();
parameters.put(functions.get("fibonacci"), new Number[]{20});
parameters.put(functions.get("factorial"), new Number[]{10});
parameters.put(functions.get("ackermann"), new Number[]{3, 2});
 
functions.entrySet().stream().parallel().map(
entry -> entry.getKey()
+ Arrays.toString(parameters.get(entry.getValue()))
+ " = "
+ entry.getValue().apply(parameters.get(entry.getValue()))
).forEach(System.out::println);
}
}
Output:
(may depend on which function gets processed first):
factorial[10] = 3628800
ackermann[3, 2] = 29
fibonacci[20] = 6765

JavaScript[edit]

The standard version of the Y combinator does not use lexically bound local variables (or any local variables at all), which necessitates adding a wrapper function and some code duplication - the remaining locale variables are only there to make the relationship to the previous implementation more explicit:

function Y(f) {
var g = f((function(h) {
return function() {
var g = f(h(h));
return g.apply(this, arguments);
}
})(function(h) {
return function() {
var g = f(h(h));
return g.apply(this, arguments);
}
}));
return g;
}
 
var fac = Y(function(f) {
return function (n) {
return n > 1 ? n * f(n - 1) : 1;
};
});
 
var fib = Y(function(f) {
return function(n) {
return n > 1 ? f(n - 1) + f(n - 2) : n;
};
});

Changing the order of function application (i.e. the place where f gets called) and making use of the fact that we're generating a fixed-point, this can be reduced to

function Y(f) {
return (function(h) {
return h(h);
})(function(h) {
return f(function() {
return h(h).apply(this, arguments);
});
});
}

A functionally equivalent version using the implicit this parameter is also possible:

function pseudoY(f) {
return (function(h) {
return h(h);
})(function(h) {
return f.bind(function() {
return h(h).apply(null, arguments);
});
});
}
 
var fac = pseudoY(function(n) {
return n > 1 ? n * this(n - 1) : 1;
});
 
var fib = pseudoY(function(n) {
return n > 1 ? this(n - 1) + this(n - 2) : n;
});

However, pseudoY() is not a fixed-point combinator.

The usual version using recursion, disallowed by the task:

function Y(f) {
return function() {
return f(Y(f)).apply(this, arguments);
};
}

Another version which is disallowed because it uses arguments.callee for a function to get itself recursively:

function Y(f) {
return function() {
return f(arguments.callee).apply(this, arguments);
};
}

ECMAScript 2015 (ES6) variants[edit]

Since ECMAScript 2015 (ES6) just reached final draft, there are new ways to encode the applicative order Y combinator. These use the new fat arrow function expression syntax, and are made to allow functions of more than one argument through the use of new rest parameters syntax and the corresponding new spread operator syntax. Also showcases new default parameter value syntax:

let
Y= // Except for the η-abstraction necessary for applicative order languages, this is the formal Y combinator.
f=>((g=>(f((...x)=>g(g)(...x))))
(g=>(f((...x)=>g(g)(...x))))),
Y2= // Using β-abstraction to eliminate code repetition.
f=>((f=>f(f))
(g=>(f((...x)=>g(g)(...x))))),
Y3= // Using β-abstraction to separate out the self application combinator δ.
((δ=>f=>δ(g=>(f((...x)=>g(g)(...x)))))
((f=>f(f)))),
fix= // β/η-equivalent fix point combinator. Easier to convert to memoise than the Y combinator.
(((f)=>(g)=>(h)=>(f(h)(g(h)))) // The Substitute combinator out of SKI calculus
((f)=>(g)=>(...x)=>(f(g(g)))(...x)) // S((S(KS)K)S(S(KS)K))(KI)
((f)=>(g)=>(...x)=>(f(g(g)))(...x))),
fix2= // β/η-converted form of fix above into a more compact form
f=>(f=>f(f))(g=>(...x)=>f(g(g))(...x)),
opentailfact= // Open version of the tail call variant of the factorial function
fact=>(n,m=1)=>n<2?m:fact(n-1,n*m);
tailfact= // Tail call version of factorial function
Y(opentailfact);

ECMAScript 2015 (ES6) also permits a really compact polyvariadic variant for mutually recursive functions:

let
polyfix= // A version that takes an array instead of multiple arguments would simply use l instead of (...l) for parameter
(...l)=>(
(f=>f(f))
(g=>l.map(f=>(...x)=>f(...g(g))(...x)))),
[even,odd]= // The new destructive assignment syntax for arrays
polyfix(
(even,odd)=>n=>(n===0)||odd(n-1),
(even,odd)=>n=>(n!==0)&&even(n-1));

A minimalist version:

var Y = f => (x => x(x))(y => f(x => y(y)(x)));
var fac = Y(f => n => n > 1 ? n * f(n-1) : 1);

Joy[edit]

DEFINE y == [dup cons] swap concat dup cons i;
 
fac == [ [pop null] [pop succ] [[dup pred] dip i *] ifte ] y.

Julia[edit]

 
julia> """
# Y combinator
 
* `λf. (λx. f (x x)) (λx. f (x x))`
"""
Y = f -> (x -> x(x))(y -> f((t...) -> y(y)(t...)))
 

Usage:

 
julia> "# Factorial"
fac = f -> (n -> n < 2 ? 1 : n * f(n - 1))
 
julia> "# Fibonacci"
fib = f -> (n -> n == 0 ? 0 : (n == 1 ? 1 : f(n - 1) + f(n - 2)))
 
julia> [Y(fac)(i) for i = 1:10]
10-element Array{Any,1}:
1
2
6
24
120
720
5040
40320
362880
3628800
 
julia> [Y(fib)(i) for i = 1:10]
10-element Array{Any,1}:
1
1
2
3
5
8
13
21
34
55
 

Kitten[edit]

define y<S..., T...> (S..., (S..., (S... -> T...) -> T...) -> T...):
-> f; { f y } f call
 
define fac (Int32, (Int32 -> Int32) -> Int32):
-> x, rec;
if (x <= 1) { 1 } else { (x - 1) rec call * x }
 
define fib (Int32, (Int32 -> Int32) -> Int32):
-> x, rec;
if (x <= 2):
1
else:
(x - 1) rec call -> a;
(x - 2) rec call -> b;
a + b
 
5 \fac y say // 120
10 \fib y say // 55
 

Kotlin[edit]

// version 1.1.2
 
typealias Func<T, R> = (T) -> R
 
class RecursiveFunc<T, R>(val p: (RecursiveFunc<T, R>) -> Func<T, R>)
 
fun <T, R> y(f: (Func<T, R>) -> Func<T, R>): Func<T, R> {
val rec = RecursiveFunc<T, R> { r -> f { r.p(r)(it) } }
return rec.p(rec)
}
 
fun fac(f: Func<Int, Int>) = { x: Int -> if (x <= 1) 1 else x * f(x - 1) }
 
fun fib(f: Func<Int, Int>) = { x: Int -> if (x <= 2) 1 else f(x - 1) + f(x - 2) }
 
fun main(args: Array<String>) {
print("Factorial(1..10)  : ")
for (i in 1..10) print("${y(::fac)(i)} ")
print("\nFibonacci(1..10)  : ")
for (i in 1..10) print("${y(::fib)(i)} ")
println()
}
Output:
Factorial(1..10)   : 1  2  6  24  120  720  5040  40320  362880  3628800  
Fibonacci(1..10)   : 1  1  2  3  5  8  13  21  34  55  

Lambdatalk[edit]

Tested in http://epsilonwiki.free.fr/lambdaway/?view=Ycombinator

 
1) defining the Ycombinator
{def Y
{lambda {:f :n}
{:f :f :n}}}
 
2) defining non recursive functions
2.1) factorial
{def almost-fac
{lambda {:f :n}
{if {= :n 1}
then 1
else {* :n {:f :f {- :n 1}}}}}}
 
2.2) fibonacci
{def almost-fibo
{lambda {:f :n}
{if {<  :n 2}
then 1
else {+ {:f :f {- :n 1}} {:f :f {- :n 2}}}}}}
 
3) testing
{Y almost-fac 6}
-> 720
{Y almost-fibo 8}
-> 34
 
We could also forget the Ycombinator and names:
 
1) fac:
{{lambda {:f :n} {:f :f :n}}
{lambda {:f :n}
{if {= :n 1}
then 1
else {* :n {:f :f {- :n 1}}}}} 6}
-> 720
 
2) fibo:
{{lambda {:f :n} {:f :f :n}}
{{lambda {:f :n}
{if {<  :n 2} then 1
else {+ {:f :f {- :n 1}} {:f :f {- :n 2}}}}}} 8}
-> 34
 
 

Lua[edit]

Y = function (f)
return function(...)
return (function(x) return x(x) end)(function(x) return f(function(y) return x(x)(y) end) end)(...)
end
end
 

Usage:

almostfactorial = function(f) return function(n) return n > 0 and n * f(n-1) or 1 end end
almostfibs = function(f) return function(n) return n < 2 and n or f(n-1) + f(n-2) end end
factorial, fibs = Y(almostfactorial), Y(almostfibs)
print(factorial(7))


M2000 Interpreter[edit]

Lambda functions in M2000 are value types. They have a list of closures, but closures are copies, except for those closures which are reference types. Lambdas can keep state in closures (they are mutable). But here we didn't do that. Y combinator is a lambda which return a lambda with a closure as f function. This function called passing as first argument itself by value.

 
Module Ycombinator {
\\ y() return value. no use of closure
y=lambda (g, x)->g(g, x)
Print y(lambda (g, n)->if(n=0->1, n*g(g, n-1)), 10)
Print y(lambda (g, n)->if(n<=1->n,g(g, n-1)+g(g, n-2)), 10)
 
\\ Using closure in y, y() return function
y=lambda (g)->lambda g (x) -> g(g, x)
fact=y((lambda (g, n)-> if(n=0->1, n*g(g, n-1))))
Print fact(6), fact(24)
fib=y(lambda (g, n)->if(n<=1->n,g(g, n-1)+g(g, n-2)))
Print fib(10)
}
Ycombinator
 
 
Module Checkit {
\\ all lambda arguments passed by value in this example
\\ There is no recursion in these lambdas
\\ Y combinator make argument f as closure, as a copy of f
\\ m(m, argument) pass as first argument a copy of m
\\ so never a function, here, call itself, only call a copy who get it as argument before the call.
Y=lambda (f)-> {
=lambda f (x)->f(f,x)
}
fac_step=lambda (m, n)-> {
if n<2 then {
=1
} else {
=n*m(m, n-1)
}
}
fac=Y(fac_step)
fib_step=lambda (m, n)-> {
if n<=1 then {
=n
} else {
=m(m, n-1)+m(m, n-2)
}
}
fib=Y(fib_step)
For i=1 to 10
Print fib(i), fac(i)
Next i
}
Checkit
Module CheckRecursion {
fac=lambda (n) -> {
if n<2 then {
=1
} else {
=n*Lambda(n-1)
}
}
fib=lambda (n) -> {
if n<=1 then {
=n
} else {
=lambda(n-1)+lambda(n-2)
}
}
For i=1 to 10
Print fib(i), fac(i)
Next i
}
CheckRecursion
 

Maple[edit]

 
> Y:=f->(x->x(x))(g->f((()->g(g)(args)))):
> Yfac:=Y(f->(x->`if`(x<2,1,x*f(x-1)))):
> seq( Yfac( i ), i = 1 .. 10 );
1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800
> Yfib:=Y(f->(x->`if`(x<2,x,f(x-1)+f(x-2)))):
> seq( Yfib( i ), i = 1 .. 10 );
1, 1, 2, 3, 5, 8, 13, 21, 34, 55
 

Mathematica / Wolfram Language[edit]

Y = Function[f, #[#] &[Function[g, f[g[g][##] &]]]];
factorial = Y[Function[f, If[# < 1, 1, # f[# - 1]] &]];
fibonacci = Y[Function[f, If[# < 2, #, f[# - 1] + f[# - 2]] &]];

Moonscript[edit]

Z = (f using nil) -> ((x) -> x x) (x) -> f (...) -> (x x) ...
factorial = Z (f using nil) -> (n) -> if n == 0 then 1 else n * f n - 1

Objective-C[edit]

Works with: Mac OS X version 10.6+
Works with: iOS version 4.0+
#import <Foundation/Foundation.h>
 
typedef int (^Func)(int);
typedef Func (^FuncFunc)(Func);
typedef Func (^RecursiveFunc)(id); // hide recursive typing behind dynamic typing
 
Func Y(FuncFunc f) {
RecursiveFunc r =
^(id y) {
RecursiveFunc w = y; // cast value back into desired type
return f(^(int x) {
return w(w)(x);
});
};
return r(r);
}
 
int main (int argc, const char *argv[]) {
@autoreleasepool {
 
Func fib = Y(^Func(Func f) {
return ^(int n) {
if (n <= 2) return 1;
return f(n - 1) + f(n - 2);
};
});
Func fac = Y(^Func(Func f) {
return ^(int n) {
if (n <= 1) return 1;
return n * f(n - 1);
};
});
 
Func fib = fix(almost_fib);
Func fac = fix(almost_fac);
NSLog(@"fib(10) = %d", fib(10));
NSLog(@"fac(10) = %d", fac(10));
 
}
return 0;
}

The usual version using recursion, disallowed by the task:

Func Y(FuncFunc f) {
return ^(int x) {
return f(Y(f))(x);
};
}

OCaml[edit]

The Y-combinator over functions may be written directly in OCaml provided rectypes are enabled:

let fix f g = (fun x a -> f (x x) a) (fun x a -> f (x x) a) g

Polymorphic variants are the simplest workaround in the absence of rectypes:

let fix f = (fun (`X x) -> f(x (`X x))) (`X(fun (`X x) y -> f(x (`X x)) y));;

Otherwise, an ordinary variant can be defined and used:

type 'a mu = Roll of ('a mu -> 'a);;
 
let unroll (Roll x) = x;;
 
let fix f = (fun x a -> f (unroll x x) a) (Roll (fun x a -> f (unroll x x) a));;
 
let fac f = function
0 -> 1
| n -> n * f (n-1)
;;
 
let fib f = function
0 -> 0
| 1 -> 1
| n -> f (n-1) + f (n-2)
;;
 
(* val unroll : 'a mu -> 'a mu -> 'a = <fun>
val fix : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b = <fun>
val fac : (int -> int) -> int -> int = <fun>
val fib : (int -> int) -> int -> int = <fun> *)

 
fix fac 5;;
(* - : int = 120 *)
 
fix fib 8;;
(* - : int = 21 *)

The usual version using recursion, disallowed by the task:

let rec fix f x = f (fix f) x;;

Oforth[edit]

These combinators work for any number of parameters (see Ackermann usage)

With recursion into Y definition (so non stateless Y) :

: Y(f)   #[ f Y f perform ] ;

Without recursion into Y definition (stateless Y).

: X(me, f)   #[ me f me perform f perform ] ;
: Y(f) #X f X ;

Usage :

: almost-fact(n, f)   n ifZero: [ 1 ] else: [ n n 1 - f perform * ] ;
#almost-fact Y => fact
 
: almost-fib(n, f) n 1 <= ifTrue: [ n ] else: [ n 1 - f perform n 2 - f perform + ] ;
#almost-fib Y => fib
 
: almost-Ackermann(m, n, f)
m 0 == ifTrue: [ n 1 + return ]
n 0 == ifTrue: [ 1 m 1 - f perform return ]
n 1 - m f perform m 1 - f perform ;
#almost-Ackermann Y => Ackermann

Order[edit]

#include <order/interpreter.h>
 
#define ORDER_PP_DEF_8y \
ORDER_PP_FN(8fn(8F, \
8let((8R, 8fn(8G, \
8ap(8F, 8fn(8A, 8ap(8ap(8G, 8G), 8A))))), \
8ap(8R, 8R))))

 
#define ORDER_PP_DEF_8fac \
ORDER_PP_FN(8fn(8F, 8X, \
8if(8less_eq(8X, 0), 1, 8times(8X, 8ap(8F, 8minus(8X, 1))))))

 
#define ORDER_PP_DEF_8fib \
ORDER_PP_FN(8fn(8F, 8X, \
8if(8less(8X, 2), 8X, 8plus(8ap(8F, 8minus(8X, 1)), \
8ap(8F, 8minus(8X, 2))))))

 
ORDER_PP(8to_lit(8ap(8y(8fac), 10))) // 3628800
ORDER_PP(8ap(8y(8fib), 10)) // 55

Oz[edit]

declare
Y = fun {$ F}
{fun {$ X} {X X} end
fun {$ X} {F fun {$ Z} {{X X} Z} end} end}
end
 
Fac = {Y fun {$ F}
fun {$ N}
if N == 0 then 1 else N*{F N-1} end
end
end}
 
Fib = {Y fun {$ F}
fun {$ N}
case N of 0 then 0
[] 1 then 1
else {F N-1} + {F N-2}
end
end
end}
in
{Show {Fac 5}}
{Show {Fib 8}}

PARI/GP[edit]

As of 2.8.0, GP cannot make general self-references in closures declared inline, so the Y combinator is required to implement these functions recursively in that environment, e.g., for use in parallel processing.

Y(f)=x->f(f,x);
fact=Y((f,n)->if(n,n*f(f,n-1),1));
fib=Y((f,n)->if(n>1,f(f,n-1)+f(f,n-2),n));
apply(fact, [1..10])
apply(fib, [1..10])
Output:
%1 = [1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
%2 = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

Perl[edit]

sub Y { my $f = shift;                                # λf.
sub { my $x = shift; $x->($x) }->( # (λx.x x)
sub {my $y = shift; $f->(sub {$y->($y)(@_)})} # λy.f λz.y y z
)
}
my $fac = sub {my $f = shift;
sub {my $n = shift; $n < 2 ? 1 : $n * $f->($n-1)}
};
my $fib = sub {my $f = shift;
sub {my $n = shift; $n == 0 ? 0 : $n == 1 ? 1 : $f->($n-1) + $f->($n-2)}
};
for my $f ($fac, $fib) {
print join(' ', map Y($f)->($_), 0..9), "\n";
}
Output:
1 1 2 6 24 120 720 5040 40320 362880
0 1 1 2 3 5 8 13 21 34

The usual version using recursion, disallowed by the task:

sub Y { my $f = shift;
sub {$f->(Y($f))->(@_)}
}

Perl 6[edit]

sub Y (&f) { sub (&x) { x(&x) }( sub (&y) { f(sub ($x) { y(&y)($x) }) } ) }
sub fac (&f) { sub ($n) { $n < 2 ?? 1 !! $n * f($n - 1) } }
sub fib (&f) { sub ($n) { $n < 2 ?? $n !! f($n - 1) + f($n - 2) } }
say map Y($_), ^10 for &fac, &fib;
Output:
(1 1 2 6 24 120 720 5040 40320 362880)
(0 1 1 2 3 5 8 13 21 34)

Note that Perl 6 doesn't actually need a Y combinator because you can name anonymous functions from the inside:

say .(10) given sub (Int $x) { $x < 2 ?? 1 !! $x * &?ROUTINE($x - 1); }

Phix[edit]

Translation of: C

After (over) simplifying things, the Y function has become a bit of a joke, but at least the recursion has been shifted out of fib/fac

Before saying anything too derogatory about Y(f)=f, it is clearly a fixed-point combinator, and I feel compelled to quote from the Mike Vanier link above:
"It doesn't matter whether you use cos or (lambda (x) (cos x)) as your cosine function; they will both do the same thing."
Anyone thinking they can do better may find some inspiration at Currying, Closures/Value_capture, Partial_function_application, and/or Function_composition

function call_fn(integer f, n)
return call_func(f,{f,n})
end function
 
function Y(integer f)
return f
end function
 
function fac(integer self, integer n)
return iff(n>1?n*call_fn(self,n-1):1)
end function
 
function fib(integer self, integer n)
return iff(n>1?call_fn(self,n-1)+call_fn(self,n-2):n)
end function
 
procedure test(string name, integer rid=routine_id(name))
integer f = Y(rid)
printf(1,"%s: ",{name})
for i=1 to 10 do
printf(1," %d",call_fn(f,i))
end for
printf(1,"\n");
end procedure
test("fac")
test("fib")
Output:
fac:  1 2 6 24 120 720 5040 40320 362880 3628800
fib:  1 1 2 3 5 8 13 21 34 55

PHP[edit]

Works with: PHP version 5.3+
<?php
function Y($f) {
$g = function($w) use($f) {
return $f(function() use($w) {
return call_user_func_array($w($w), func_get_args());
});
};
return $g($g);
}
 
$fibonacci = Y(function($f) {
return function($i) use($f) { return ($i <= 1) ? $i : ($f($i-1) + $f($i-2)); };
});
 
echo $fibonacci(10), "\n";
 
$factorial = Y(function($f) {
return function($i) use($f) { return ($i <= 1) ? 1 : ($f($i - 1) * $i); };
});
 
echo $factorial(10), "\n";
?>

The usual version using recursion, disallowed by the task:

function Y($f) {
return function() use($f) {
return call_user_func_array($f(Y($f)), func_get_args());
};
}
Works with: PHP version pre-5.3 and 5.3+

with create_function instead of real closures. A little far-fetched, but...

<?php
function Y($f) {
$g = create_function('$w', '$f = '.var_export($f,true).';
return $f(create_function(\'\', \'$w = \'.var_export($w,true).\';
return call_user_func_array($w($w), func_get_args());
\'));
'
);
return $g($g);
}
 
function almost_fib($f) {
return create_function('$i', '$f = '.var_export($f,true).';
return ($i <= 1) ? $i : ($f($i-1) + $f($i-2));
'
);
};
$fibonacci = Y('almost_fib');
echo $fibonacci(10), "\n";
 
function almost_fac($f) {
return create_function('$i', '$f = '.var_export($f,true).';
return ($i <= 1) ? 1 : ($f($i - 1) * $i);
'
);
};
$factorial = Y('almost_fac');
echo $factorial(10), "\n";
?>

A functionally equivalent version using the $this parameter in closures is also possible:

Works with: PHP version 5.4+
<?php
function pseudoY($f) {
$g = function($w) use ($f) {
return $f->bindTo(function() use ($w) {
return call_user_func_array($w($w), func_get_args());
});
};
return $g($g);
}
 
$factorial = pseudoY(function($n) {
return $n > 1 ? $n * $this($n - 1) : 1;
});
echo $factorial(10), "\n";
 
$fibonacci = pseudoY(function($n) {
return $n > 1 ? $this($n - 1) + $this($n - 2) : $n;
});
echo $fibonacci(10), "\n";
?>

However, pseudoY() is not a fixed-point combinator.

PicoLisp[edit]

Translation of: Common Lisp
(de Y (F)
(let X (curry (F) (Y) (F (curry (Y) @ (pass (Y Y)))))
(X X) ) )

Factorial[edit]

# Factorial
(de fact (F)
(curry (F) (N)
(if (=0 N)
1
(* N (F (dec N))) ) ) )
 
: ((Y fact) 6)
-> 720

Fibonacci sequence[edit]

# Fibonacci
(de fibo (F)
(curry (F) (N)
(if (> 2 N)
1
(+ (F (dec N)) (F (- N 2))) ) ) )
 
: ((Y fibo) 22)
-> 28657

Ackermann function[edit]

# Ackermann
(de ack (F)
(curry (F) (X Y)
(cond
((=0 X) (inc Y))
((=0 Y) (F (dec X) 1))
(T (F (dec X) (F X (dec Y)))) ) ) )
 
: ((Y ack) 3 4)
-> 125

Pop11[edit]

define Y(f);
procedure (x); x(x) endprocedure(
procedure (y);
f(procedure(z); (y(y))(z) endprocedure)
endprocedure
)
enddefine;
 
define fac(h);
procedure (n);
if n = 0 then 1 else n * h(n - 1) endif
endprocedure
enddefine;
 
define fib(h);
procedure (n);
if n < 2 then 1 else h(n - 1) + h(n - 2) endif
endprocedure
enddefine;
 
Y(fac)(5) =>
Y(fib)(5) =>
Output:
** 120
** 8

PostScript[edit]

Translation of: Joy
Library: initlib
y {
{dup cons} exch concat dup cons i
}.
 
/fac {
{ {pop zero?} {pop succ} {{dup pred} dip i *} ifte }
y
}.

PowerShell[edit]

Translation of: Python

PowerShell Doesn't have true closure, in order to fake it, the script-block is converted to text and inserted whole into the next function using variable expansion in double-quoted strings. For simple translation of lambda calculus, translates as param inside of a ScriptBlock, translates as Invoke-Expression "{}", invocation (written as a space) translates to InvokeReturnAsIs.

$fac = {
param([ScriptBlock] $f)
invoke-expression @"
{
param([int] `$n)
if (`$n -le 0) {1}
else {`$n * {$f}.InvokeReturnAsIs(`$n - 1)}
}
"
@
}
 
$fib = {
param([ScriptBlock] $f)
invoke-expression @"
{
param([int] `$n)
switch (`$n)
{
0 {1}
1 {1}
default {{$f}.InvokeReturnAsIs(`$n-1)+{$f}.InvokeReturnAsIs(`$n-2)}
}
}
"
@
}
 
$Z = {
param([ScriptBlock] $f)
invoke-expression @"
{
param([ScriptBlock] `$x)
{$f}.InvokeReturnAsIs(`$(invoke-expression @`"
{
param(```$y)
{`$x}.InvokeReturnAsIs({`$x}).InvokeReturnAsIs(```$y)
}
`"@))
}.InvokeReturnAsIs({
param([ScriptBlock] `$x)
{$f}.InvokeReturnAsIs(`$(invoke-expression @`"
{
param(```$y)
{`$x}.InvokeReturnAsIs({`$x}).InvokeReturnAsIs(```$y)
}
`"@))
})
"
@
}
 
$Z.InvokeReturnAsIs($fac).InvokeReturnAsIs(5)
$Z.InvokeReturnAsIs($fib).InvokeReturnAsIs(5)


GetNewClosure() was added in Powershell 2, allowing for an implementation without metaprogramming. The following was tested with Powershell 4.

$Y = {
param ($f)
 
{
param ($x)
 
$f.InvokeReturnAsIs({
param ($y)
 
$x.InvokeReturnAsIs($x).InvokeReturnAsIs($y)
}.GetNewClosure())
 
}.InvokeReturnAsIs({
param ($x)
 
$f.InvokeReturnAsIs({
param ($y)
 
$x.InvokeReturnAsIs($x).InvokeReturnAsIs($y)
}.GetNewClosure())
 
}.GetNewClosure())
}
 
$fact = {
param ($f)
 
{
param ($n)
 
if ($n -eq 0) { 1 } else { $n * $f.InvokeReturnAsIs($n - 1) }
 
}.GetNewClosure()
}
 
$fib = {
param ($f)
 
{
param ($n)
 
if ($n -lt 2) { 1 } else { $f.InvokeReturnAsIs($n - 1) + $f.InvokeReturnAsIs($n - 2) }
 
}.GetNewClosure()
}
 
$Y.invoke($fact).invoke(5)
$Y.invoke($fib).invoke(5)

Prolog[edit]

Works with SWI-Prolog and module lambda, written by Ulrich Neumerkel found there http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl.

The code is inspired from this page : http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/ISO-Hiord#Hiord (p 106).
Original code is from Hermenegildo and al : Hiord: A Type-Free Higher-Order Logic Programming Language with Predicate Abstraction, pdf accessible here http://www.stups.uni-duesseldorf.de/asap/?id=129.

:- use_module(lambda).
 
% The Y combinator
y(P, Arg, R) :-
Pred = P +\Nb2^F2^call(P,Nb2,F2,P),
call(Pred, Arg, R).
 
 
test_y_combinator :-
% code for Fibonacci function
Fib = \NFib^RFib^RFibr1^(NFib < 2 ->
RFib = NFib
;
NFib1 is NFib - 1,
NFib2 is NFib - 2,
call(RFibr1,NFib1,RFib1,RFibr1),
call(RFibr1,NFib2,RFib2,RFibr1),
RFib is RFib1 + RFib2
),
 
y(Fib, 10, FR), format('Fib(~w) = ~w~n', [10, FR]),
 
% code for Factorial function
Fact = \NFact^RFact^RFactr1^(NFact = 1 ->
RFact = NFact
;
NFact1 is NFact - 1,
call(RFactr1,NFact1,RFact1,RFactr1),
RFact is NFact * RFact1
),
 
y(Fact, 10, FF), format('Fact(~w) = ~w~n', [10, FF]).
Output:
 ?- test_y_combinator.
Fib(10) = 55
Fact(10) = 3628800
true.

Python[edit]

>>> Y = lambda f: (lambda x: x(x))(lambda y: f(lambda *args: y(y)(*args)))
>>> fac = lambda f: lambda n: (1 if n<2 else n*f(n-1))
>>> [ Y(fac)(i) for i in range(10) ]
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
>>> fib = lambda f: lambda n: 0 if n == 0 else (1 if n == 1 else f(n-1) + f(n-2))
>>> [ Y(fib)(i) for i in range(10) ]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

The usual version using recursion, disallowed by the task:

Y = lambda f: lambda *args: f(Y(f))(*args)
Y = lambda b: ((lambda f: b(lambda *x: f(f)(*x)))((lambda f: b(lambda *x: f(f)(*x)))))

R[edit]

Y <- function(f) {
(function(x) { (x)(x) })( function(y) { f( (function(a) {y(y)})(a) ) } )
}
fac <- function(f) {
function(n) {
if (n<2)
1
else
n*f(n-1)
}
}
 
fib <- function(f) {
function(n) {
if (n <= 1)
n
else
f(n-1) + f(n-2)
}
}
for(i in 1:9) print(Y(fac)(i))
for(i in 1:9) print(Y(fib)(i))

Racket[edit]

The lazy implementation

 
#lang lazy
 
(define Y (λ(f)((λ(x)(f (x x)))(λ(x)(f (x x))))))
 
(define Fact
(Y (λ(fact) (λ(n) (if (zero? n) 1 (* n (fact (- n 1))))))))
(define Fib
(Y (λ(fib) (λ(n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2))))))))
 
Output:
> (!! (map Fact '(1 2 4 8 16)))
'(1 2 24 40320 20922789888000)
> (!! (map Fib '(1 2 4 8 16)))
'(0 1 2 13 610)

Strict realization:

 
#lang racket
(define Y (λ(b)((λ(f)(b(λ(x)((f f) x))))
(λ(f)(b(λ(x)((f f) x)))))))
 

Definitions of Fact and Fib functions will be the same as in Lazy Racket.

Finally, a definition in Typed Racket is a little difficult as in other statically typed languages:

 
#lang typed/racket
 
(: make-recursive : (All (S T) ((S -> T) -> (S -> T)) -> (S -> T)))
(define-type Tau (All (S T) (Rec this (this -> (S -> T)))))
(define (make-recursive f)
((lambda: ([x : (Tau S T)]) (f (lambda (z) ((x x) z))))
(lambda: ([x : (Tau S T)]) (f (lambda (z) ((x x) z))))))
 
(: fact : Number -> Number)
(define fact (make-recursive
(lambda: ([fact : (Number -> Number)])
(lambda: ([n : Number])
(if (zero? n)
1
(* n (fact (- n 1))))))))
 
(fact 5)
 

REBOL[edit]

Y: closure [g] [do func [f] [f :f] closure [f] [g func [x] [do f :f :x]]]
usage example
fact*: closure [h] [func [n] [either n <= 1 [1] [n * h n - 1]]]
fact: Y :fact*

REXX[edit]

Programming note:   length,   reverse,   and   trunc   are REXX BIFs   (Built In Functions).

/*REXX program implements and displays  a  stateless   Y   combinator.        */
numeric digits 1000 /*allow big numbers. */
say ' fib' Y(fib (50)) /*Fibonacci series. */
say ' fib' Y(fib (12 11 10 9 8 7 6 5 4 3 2 1 0)) /*Fibonacci series. */
say ' fact' Y(fact (60)) /*single factorial*/
say ' fact' Y(fact (0 1 2 3 4 5 6 7 8 9 10 11)) /*single factorial*/
say ' Dfact' Y(dfact (4 5 6 7 8 9 10 11 12 13)) /*double factorial*/
say ' Tfact' Y(tfact (4 5 6 7 8 9 10 11 12 13)) /*triple factorial*/
say ' Qfact' Y(qfact (4 5 6 7 8 40)) /*quadruple factorial*/
say ' length' Y(length (when for to where whenceforth)) /*lengths of words.*/
say 'reverse' Y(reverse (23 678 1007 45 MAS I MA)) /*reverses strings. */
say ' trunc' Y(trunc (-7.0005 12 3.14159 6.4 78.999)) /*truncates numbers. */
exit /*stick a fork in it, we're all done. */
/*────────────────────────────────────────────────────────────────────────────*/
Y: parse arg Y _; $= /*the Y combinator.*/
do j=1 for words(_); interpret '$=$' Y"("word(_,j)')'; end; return $
fib: procedure; parse arg x; if x<2 then return x; s=0; a=0; b=1
s=0; a=0; b=1; do j=2 to x; s=a+b; a=b; b=s; end; return s
dfact: procedure; parse arg x; !=1; do j=x to 2 by -2; !=!*j; end; return !
tfact: procedure; parse arg x; !=1; do j=x to 2 by -3; !=!*j; end; return !
qfact: procedure; parse arg x; !=1; do j=x to 2 by -4; !=!*j; end; return !
fact: procedure; parse arg x; !=1; do j=2 to x  ; !=!*j; end; return !

output

    fib  12586269025
    fib  144 89 55 34 21 13 8 5 3 2 1 1 0
   fact  8320987112741390144276341183223364380754172606361245952449277696409600000000000000
   fact  1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800
  Dfact  8 15 48 105 384 945 3840 10395 46080 135135
  Tfact  4 10 18 28 80 162 280 880 1944 3640
  Qfact  4 5 12 21 32 3805072588800
 length  4 3 2 5 11
reverse  32 876 7001 54 SAM I AM
  trunc  -7 12 3 6 78

Ruby[edit]

Using a lambda:

y = lambda do |f|
lambda {|g| g[g]}[lambda do |g|
f[lambda {|*args| g[g][*args]}]
end]
end
 
fac = lambda{|f| lambda{|n| n < 2 ? 1 : n * f[n-1]}}
p Array.new(10) {|i| y[fac][i]} #=> [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
 
fib = lambda{|f| lambda{|n| n < 2 ? n : f[n-1] + f[n-2]}}
p Array.new(10) {|i| y[fib][i]} #=> [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

Same as the above, using the new short lambda syntax:

Works with: Ruby version 1.9
y = ->(f) {->(g) {g.(g)}.(->(g) { f.(->(*args) {g.(g).(*args)})})}
 
fac = ->(f) { ->(n) { n < 2 ? 1 : n * f.(n-1) } }
 
p 10.times.map {|i| y.(fac).(i)}
 
fib = ->(f) { ->(n) { n < 2 ? n : f.(n-2) + f.(n-1) } }
 
p 10.times.map {|i| y.(fib).(i)}

Using a method:

Works with: Ruby version 1.9
def y(&f)
lambda do |g|
f.call {|*args| g[g][*args]}
end.tap {|g| break g[g]}
end
 
fac = y {|&f| lambda {|n| n < 2 ? 1 : n * f[n - 1]}}
fib = y {|&f| lambda {|n| n < 2 ? n : f[n - 1] + f[n - 2]}}
 
p Array.new(10) {|i| fac[i]}
# => [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
p Array.new(10) {|i| fib[i]}
# => [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

The usual version using recursion, disallowed by the task:

y = lambda do |f|
lambda {|*args| f[y[f]][*args]}
end

Rust[edit]

Works with: Rust version 1.26.0 stable
 
//! A simple implementation of the Y Combinator
// λf.(λx.xx)(λx.f(xx))
// <=> λf.(λx.f(xx))(λx.f(xx))
 
// CREDITS: A better version of the previous code that was posted here, with detailed explanation.
// See <y> and also <y_apply>.
 
// A function type that takes its own type as an input is an infinite recursive type.
// We introduce a trait that will allow us to have an input with the same type as self, and break the recursion.
// The input is going to be a trait object that implements the desired function in the interface.
// NOTE: We will be coercing a reference to a closure into this trait object.
trait Apply<T, R> {
fn apply( &self, &Apply<T, R >, T ) -> R;
}
 
// In Rust, closures fall into three kinds: FnOnce, FnMut and Fn.
// FnOnce assumed to be able to be called just once if it is not Clone. It is impossible to
// write recursive FnOnce that is not Clone.
// All FnMut are also FnOnce, although you can call them multiple times, they are not allow to
// have a reference to themselves. So it is also not possible to write recursive FnMut closures
// that is not Clone.
// All Fn are also FnMut, and all closures of Fn are also Clone. However, programmers can create
// Fn objects that are not Clone
// The following address all closures that is Clone, and those are Fn.
impl<T, R, F> Apply<T, R> for F where F: FnOnce( &Apply<T, R>, T ) -> R + Clone {
fn apply( &self, f: &Apply<T, R>, t: T ) -> R {
(self.clone())( f, t )
 
// If we were to pass in self as f, we get -
// NOTE: Each letter is an individual symbol.
// λf.λt.sft
// => λs.λt.sst [s/f]
// => λs.ss
}
}
//This will work for all Fn objects, not just closures
//And it is a little bit more efficient for Fn closures as it do not clone itself.
//However under 1.26 it is not possible to define both. We will
//need to wait for specialization.
//impl<T, R, F> Apply<T, R> for F where F: Fn( &Apply<T, R>, T ) -> R {
// fn apply( &self, f: &Apply<T, R>, t: T ) -> R {
// self( f, t )
//}
//Before 1.26 we have some limitations and so we need some workarounds. But now impl Trait is stable and we can
// write the following:
fn y<T,R>(f:impl FnOnce(&Fn(T) -> R, T) -> R + Clone) -> impl FnOnce(T) -> R {
|t| (|x: &Apply<T,R>,y| x.apply(x,y))
(&move |x:&Apply<T,R>,y| f(&|z| x.apply(x,z), y), t)
 
// NOTE: Each letter is an individual symbol.
// (λx.(λy.xxy))(λx.(λy.f(λz.xxz)y))t
// => (λx.xx)(λx.f(xx))t
// => (Yf)t
}
 
//Previous version removed as they are just hacks when impl Trait is not available.
 
fn fac( n: usize ) -> usize {
let almost_fac = |f: &Fn( usize ) -> usize, x| if x == 0 { 1 } else { x * f( x - 1 ) };
let fac = y( almost_fac );
fac( n )
}
 
fn fib( n: usize ) -> usize {
let almost_fib = |f: &Fn( usize ) -> usize, x| if x < 2 { 1 } else { f( x - 2 ) + f( x - 1 ) };
let fib = y( almost_fib );
fib( n )
}
 
fn optimal_fib( n: usize ) -> usize {
let almost_fib = |f: &Fn( (usize,usize,usize) ) -> usize, (i0,i1,x)|
{
match x {
0 => i0,
1 => i1,
x => f((i1,i0+i1, x-1))
}
};
let fib = |x|y( almost_fib )((1,1,x));
fib( n )
}
 
fn main() {
println!( "{}", fac( 10 ) );
println!( "{}", fib( 10 ) );
println!( "{}", optimal_fib( 10 ) );
}
 
Output:
3628800
89
89

Scala[edit]

Credit goes to the thread in scala blog

def Y[A,B](f: (A=>B)=>(A=>B)) = {
case class W(wf: W=>A=>B) {
def apply(w: W) = wf(w)
}
val g: W=>A=>B = w => f(w(w))(_)
g(W(g))
}

Example

val fac = Y[Int, Int](f => i => if (i <= 0) 1 else f(i - 1) * i)
fac(6) //> res0: Int = 720
 
val fib = Y[Int, Int](f => i => if (i < 2) i else f(i - 1) + f(i - 2))
fib(6) //> res1: Int = 8

Scheme[edit]

(define Y                 ; (Y f) = (g g) where
(lambda (f) ; (g g) = (f (lambda a (apply (g g) a)))
((lambda (g) (g g)) ; (Y f) == (f (lambda a (apply (Y f) a)))
(lambda (g)
(f (lambda a (apply (g g) a)))))))
 
;; head-recursive factorial
(define fac ; fac = (Y f) = (f (lambda a (apply (Y f) a)))
(Y (lambda (r) ; = (lambda (x) ... (r (- x 1)) ... )
(lambda (x) ; where r = (lambda a (apply (Y f) a))
(if (< x 2) ; (r ... ) == ((Y f) ... )
1 ; == (lambda (x) ... (fac (- x 1)) ... )
(* x (r (- x 1))))))))
 
;; tail-recursive factorial
(define fac2
(lambda (x)
((Y (lambda (r) ; (Y f) == (f (lambda a (apply (Y f) a)))
(lambda (x acc) ; r == (lambda a (apply (Y f) a))
(if (< x 2) ; (r ... ) == ((Y f) ... )
acc
(r (- x 1) (* x acc))))))
x 1)))
 
; double-recursive Fibonacci
(define fib
(Y (lambda (f)
(lambda (x)
(if (< x 2)
x
(+ (f (- x 1)) (f (- x 2))))))))
 
; tail-recursive Fibonacci
(define fib2
(lambda (x)
((Y (lambda (f)
(lambda (x a b)
(if (< x 1)
a
(f (- x 1) b (+ a b))))))
x 0 1)))
 
(display (fac 6))
(newline)
 
(display (fib2 134))
(newline)
Output:
720
4517090495650391871408712937

If we were allowed to use recursion (with Y referring to itself by name in its body) we could define the equivalent to the above as

(define Yr        ; (Y f) == (f  (lambda a (apply (Y f) a)))
(lambda (f)
(f (lambda a (apply (Yr f) a)))))

And another way is:

(define Y2r
(lambda (f)
(lambda a (apply (f (Y2r f)) a))))

Which, non-recursively, is

(define Y2                ; (Y2 f) = (g g) where
(lambda (f) ; (g g) = (lambda a (apply (f (g g)) a))
((lambda (g) (g g)) ; (Y2 f) == (lambda a (apply (f (Y2 f)) a))
(lambda (g)
(lambda a (apply (f (g g)) a))))))

Shen[edit]

(define y
F -> ((/. X (X X))
(/. X (F (/. Z ((X X) Z))))))
 
(let Fac (y (/. F N (if (= 0 N)
1
(* N (F (- N 1))))))
(output "~A~%~A~%~A~%"
(Fac 0)
(Fac 5)
(Fac 10)))
Output:
1
120
3628800

Sidef[edit]

var y = ->(f) {->(g) {g(g)}(->(g) { f(->(*args) {g(g)(args...)})})}
 
var fac = ->(f) { ->(n) { n < 2 ? 1 : (n * f(n-1)) } }
say 10.of { |i| y(fac)(i) }
 
var fib = ->(f) { ->(n) { n < 2 ? n : (f(n-2) + f(n-1)) } }
say 10.of { |i| y(fib)(i) }
Output:
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

Slate[edit]

The Y combinator is already defined in slate as:

Method traits define: #Y &builder:
[[| :f | [| :x | f applyWith: (x applyWith: x)]
applyWith: [| :x | f applyWith: (x applyWith: x)]]].

Smalltalk[edit]

Works with: GNU Smalltalk
Y := [:f| [:x| x value: x] value: [:g| f value: [:x| (g value: g) value: x] ] ].
 
fib := Y value: [:f| [:i| i <= 1 ifTrue: [i] ifFalse: [(f value: i-1) + (f value: i-2)] ] ].
 
(fib value: 10) displayNl.
 
fact := Y value: [:f| [:i| i = 0 ifTrue: [1] ifFalse: [(f value: i-1) * i] ] ].
 
(fact value: 10) displayNl.
Output:
55
3628800

The usual version using recursion, disallowed by the task:

Y := [:f| [:x| (f value: (Y value: f)) value: x] ].

Standard ML[edit]

- datatype 'a mu = Roll of ('a mu -> 'a)
fun unroll (Roll x) = x
 
fun fix f = (fn x => fn a => f (unroll x x) a) (Roll (fn x => fn a => f (unroll x x) a))
 
fun fac f 0 = 1
| fac f n = n * f (n-1)
 
fun fib f 0 = 0
| fib f 1 = 1
| fib f n = f (n-1) + f (n-2)
;
datatype 'a mu = Roll of 'a mu -> 'a
val unroll = fn : 'a mu -> 'a mu -> 'a
val fix = fn : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b
val fac = fn : (int -> int) -> int -> int
val fib = fn : (int -> int) -> int -> int
- List.tabulate (10, fix fac);
val it = [1,1,2,6,24,120,720,5040,40320,362880] : int list
- List.tabulate (10, fix fib);
val it = [0,1,1,2,3,5,8,13,21,34] : int list

The usual version using recursion, disallowed by the task:

fun fix f x = f (fix f) x


SuperCollider[edit]

Like Ruby, SuperCollider needs an extra level of lambda-abstraction to implement the y-combinator. The z-combinator is straightforward:

// z-combinator
(
z = { |f|
{ |x| x.(x) }.(
{ |y|
f.({ |args| y.(y).(args) })
}
)
};
)
 
// the same in a shorter form
 
(
r = { |x| x.(x) };
z = { |f| r.({ |y| f.(r.(y).(_)) }) };
)
 
 
// factorial
k = { |f| { |x| if(x < 2, 1, { x * f.(x - 1) }) } };
 
g = z.(k);
 
g.(5) // 120
 
(1..10).collect(g) // [ 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ]
 
 
 
// fibonacci
 
k = { |f| { |x| if(x <= 2, 1, { f.(x - 1) + f.(x - 2) }) } };
 
g = z.(k);
 
g.(3)
 
(1..10).collect(g) // [ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ]
 
 
 

Swift[edit]

Using a recursive type:

struct RecursiveFunc<F> {
let o : RecursiveFunc<F> -> F
}
 
func Y<A, B>(f: (A -> B) -> A -> B) -> A -> B {
let r = RecursiveFunc<A -> B> { w in f { w.o(w)($0) } }
return r.o(r)
}
 
let fac = Y { (f: Int -> Int) in
{ $0 <= 1 ? 1 : $0 * f($0-1) }
}
let fib = Y { (f: Int -> Int) in
{ $0 <= 2 ? 1 : f($0-1)+f($0-2) }
}
println("fac(5) = \(fac(5))")
println("fib(9) = \(fib(9))")
Output:
fac(5) = 120
fib(9) = 34

Without a recursive type, and instead using Any to erase the type:

Works with: Swift version 1.2+
(for Swift 1.1 replace as! with as)
func Y<A, B>(f: (A -> B) -> A -> B) -> A -> B {
typealias RecursiveFunc = Any -> A -> B
let r : RecursiveFunc = { (z: Any) in let w = z as! RecursiveFunc; return f { w(w)($0) } }
return r(r)
}

The usual version using recursion, disallowed by the task:

func Y<In, Out>( f: (In->Out) -> (In->Out) ) -> (In->Out) {
return { x in f(Y(f))(x) }
}

Tcl[edit]

Y combinator is derived in great detail here.

TXR[edit]

This prints out 24, the factorial of 4:

;; The Y combinator:
(defun y (f)
[(op @1 @1)
(op f (op [@@1 @@1]))])
 
;; The Y-combinator-based factorial:
(defun fac (f)
(do if (zerop @1)
1
(* @1 [f (- @1 1)])))
 
;; Test:
(format t "~s\n" [[y fac] 4])

Both the op and do operators are a syntactic sugar for currying, in two different flavors. The forms within do that are symbols are evaluated in the normal Lisp-2 style and the first symbol can be an operator. Under op, any forms that are symbols are evaluated in the Lisp-2 style, and the first form is expected to evaluate to a function. The name do stems from the fact that the operator is used for currying over special forms like if in the above example, where there is evaluation control. Operators can have side effects: they can "do" something. Consider (do set a @1) which yields a function of one argument which assigns that argument to a.

The compounded @@... notation allows for inner functions to refer to outer parameters, when the notation is nested. Consider
(op foo @1 (op bar @2 @@2))
. Here the @2 refers to the second argument of the anonymous function denoted by the inner op. The @@2 refers to the second argument of the outer op.

Ursala[edit]

The standard y combinator doesn't work in Ursala due to eager evaluation, but an alternative is easily defined as shown

(r "f") "x" = "f"("f","x")
my_fix "h" = r ("f","x"). ("h" r "f") "x"

or by this shorter expression for the same thing in point free form.

my_fix = //~&R+ ^|H\~&+ ; //~&R

Normally you'd like to define a function recursively by writing , where is just the body of the function with recursive calls to in it. With a fixed point combinator such as my_fix as defined above, you do almost the same thing, except it's my_fix "f". ("f"), where the dot represents lambda abstraction and the quotes signify a dummy variable. Using this method, the definition of the factorial function becomes

#import nat
 
fact = my_fix "f". ~&?\1! product^/~& "f"+ predecessor

To make it easier, the compiler has a directive to let you install your own fixed point combinator for it to use, which looks like this,

#fix my_fix

with your choice of function to be used in place of my_fix. Having done that, you may express recursive functions per convention by circular definitions, as in this example of a Fibonacci function.

fib = {0,1}?</1! sum+ fib~~+ predecessor^~/~& predecessor

Note that this way is only syntactic sugar for the for explicit way shown above. Without a fixed point combinator given in the #fix directive, this definition of fib would not have compiled. (Ursala allows user defined fixed point combinators because they're good for other things besides functions.) To confirm that all this works, here is a test program applying both of the functions defined above to the numbers from 1 to 8.

#cast %nLW
 
examples = (fact* <1,2,3,4,5,6,7,8>,fib* <1,2,3,4,5,6,7,8>)
Output:
(
   <1,2,6,24,120,720,5040,40320>,
   <1,2,3,5,8,13,21,34>)

The fixed point combinator defined above is theoretically correct but inefficient and limited to first order functions, whereas the standard distribution includes a library (sol) providing a hierarchy of fixed point combinators suitable for production use and with higher order functions. A more efficient alternative implementation of my_fix would be general_function_fixer 0 (with 0 signifying the lowest order of fixed point combinators), or if that's too easy, then by this definition.

#import sol
 
#fix general_function_fixer 1
 
my_fix "h" = "h" my_fix "h"

Note that this equation is solved using the next fixed point combinator in the hierarchy.

Verbexx[edit]

/////// Y-combinator function (for single-argument lambdas) ///////
 
y @FN [f]
{ @( x -> { @f (z -> {@(@x x) z}) } ) // output of this expression is treated as a verb, due to outer @( )
( x -> { @f (z -> {@(@x x) z}) } ) // this is the argument supplied to the above verb expression
};
 
 
/////// Function to generate an anonymous factorial function as the return value -- (not tail-recursive) ///////
 
fact_gen @FN [f]
{ n -> { (n<=0) ? {1} {n * (@f n-1)}
}
};
 
 
/////// Function to generate an anonymous fibonacci function as the return value -- (not tail-recursive) ///////
 
fib_gen @FN [f]
{ n -> { (n<=0) ? { 0 }
{ (n<=2) ? {1} { (@f n-1) + (@f n-2) } }
}
};
 
 
/////// loops to test the above functions ///////
 
@VAR factorial = @y fact_gen;
@VAR fibonacci = @y fib_gen;
 
@LOOP init:{@VAR i = -1} while:(i <= 20) next:{i++}
{ @SAY i "factorial =" (@factorial i) };
 
@LOOP init:{ i = -1} while:(i <= 16) next:{i++}
{ @SAY "fibonacci<" i "> =" (@fibonacci i) };

Vim Script[edit]

There is no lambda in Vim (yet?), so here is a way to fake it using a Dictionary. This also provides garbage collection.

" Translated from Python.  Works with: Vim 7.0
 
func! Lambx(sig, expr, dict)
let fanon = {'d': a:dict}
exec printf("
\func fanon.f(%s) dict\n
\ return %s\n
\endfunc",
\ a:sig, a:expr)
return fanon
endfunc
 
func! Callx(fanon, arglist)
return call(a:fanon.f, a:arglist, a:fanon.d)
endfunc
 
let g:Y = Lambx('f', 'Callx(Lambx("x", "Callx(a:x, [a:x])", {}), [Lambx("y", ''Callx(self.f, [Lambx("...", "Callx(Callx(self.y, [self.y]), a:000)", {"y": a:y})])'', {"f": a:f})])', {})
 
let g:fac = Lambx('f', 'Lambx("n", "a:n<2 ? 1 : a:n * Callx(self.f, [a:n-1])", {"f": a:f})', {})
 
echo Callx(Callx(g:Y, [g:fac]), [5])
echo map(range(10), 'Callx(Callx(Y, [fac]), [v:val])')
 

Update: since Vim 7.4.2044 (or so...), the following can be used (the feature check was added with 7.4.2121):

 
if !has("lambda")
echoerr 'Lambda feature required'
finish
endif
let Y = {f -> {x -> x(x)}({y -> f({... -> call(y(y), a:000)})})}
let Fac = {f -> {n -> n<2 ? 1 : n * f(n-1)}}
 
echo Y(Fac)(5)
echo map(range(10), 'Y(Fac)(v:val)')
 

Output:

120
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]

Wart[edit]

# Better names due to Jim Weirich: http://vimeo.com/45140590
def (Y improver)
((fn(gen) gen.gen)
(fn(gen)
(fn(n)
((improver gen.gen) n))))
 
factorial <- (Y (fn(f)
(fn(n)
(if zero?.n
1
(n * (f n-1))))))
 
prn factorial.5

XQuery[edit]

Version 3.0 of the XPath and XQuery specifications added support for function items.

let $Y := function($f) {  
(function($x) { ($x)($x) })( function($g) { $f( (function($a) { $g($g) ($a)}) ) } )
}
let $fac := $Y(function($f) { function($n) { if($n < 2) then 1 else $n * $f($n - 1) } })
let $fib := $Y(function($f) { function($n) { if($n <= 1) then $n else $f($n - 1) + $f($n - 2) } })
return (
$fac(6),
$fib(6)
)
 
Output:
720 8

zkl[edit]

fcn Y(f){ fcn(g){ g(g) }( 'wrap(h){ f( 'wrap(a){ h(h)(a) }) }) }

Functions don't get to look outside of their scope so data in enclosing scopes needs to be bound to a function, the fp (function application/cheap currying) method does this. 'wrap is syntactic sugar for fp.

fcn almost_factorial(f){ fcn(n,f){ if(n<=1) 1 else n*f(n-1) }.fp1(f) }
Y(almost_factorial)(6).println();
[0..10].apply(Y(almost_factorial)).println();
Output:
720
L(1,1,2,6,24,120,720,5040,40320,362880,3628800)
fcn almost_fib(f){ fcn(n,f){ if(n<2) 1 else f(n-1)+f(n-2) }.fp1(f) }
Y(almost_fib)(9).println();
[0..10].apply(Y(almost_fib)).println();
Output:
55
L(1,1,2,3,5,8,13,21,34,55,89)