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{{task|Classic CS problems and programs}} {{requires|First class functions}} [[Category:Recursion]]

In strict [[wp:Functional programming|functional programming]] and the [[wp:lambda calculus|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 [http://mvanier.livejournal.com/2897.html 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 [[wp:Fixed-point combinator|fixed-point combinators]].

;Task: Define the stateless Y combinator and use it to compute [[wp:Factorial|factorials]] and [[wp:Fibonacci number|Fibonacci numbers]] from other stateless functions or lambda expressions.

;Cf:

• [http://vimeo.com/45140590 Jim Weirich: Adventures in Functional Programming]

ALGOL 68

{{trans|Python}} Note: This specimen retains the original [[#Python|Python]] coding style. {{wont work with|ALGOL 68|Revision 1 - scoping extensions to language used.}} {{works with|ALGOL 68S|from Amsterdam Compiler Kit ( [[wp:Guido van Rossum|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

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


{{Out}}

{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}}


ARM Assembly

{{works with|as|Raspberry Pi}}



/* ARM assembly Raspberry PI  */
/*  program Ycombi.s   */

/* REMARK 1 : this program use routines in a include file
see task Include a file language arm assembly
for the routine affichageMess conversion10
see at end of this program the instruction include */

/* Constantes    */
.equ STDOUT, 1                           @ Linux output console
.equ EXIT,   1                           @ Linux syscall
.equ WRITE,  4                           @ Linux syscall

/*******************************************/
/* Structures                               */
/********************************************/
/* structure function*/
.struct  0
func_fn:                    @ next element
.struct  func_fn + 4
func_f_:                    @ next element
.struct  func_f_ + 4
func_num:
.struct  func_num + 4
func_fin:

/* Initialized data */
.data
szMessStartPgm:            .asciz "Program start \n"
szMessEndPgm:              .asciz "Program normal end.\n"
szMessError:               .asciz "\033[31mError Allocation !!!\n"

szFactorielle:             .asciz "Function factorielle : \n"
szFibonacci:               .asciz "Function Fibonacci : \n"
szCarriageReturn:          .asciz "\n"

/* datas message display */
szMessResult:            .ascii "Result value :"
sValue:                  .space 12,' '
.asciz "\n"

/* UnInitialized data */
.bss

/*  code section */
.text
.global main
main:                                           @ program start
ldr r0,iAdrszMessStartPgm                   @ display start message
bl affichageMess
bl YFunc                                    @ create Ycombinator
mov r5,r0                                   @ save Ycombinator
bl affichageMess
mov r4,#1                                   @ loop counter
1:  @ start loop
mov r0,r4
bl numFunc                                  @ create number structure
cmp r0,#-1                                  @ allocation error ?
beq 99f
mov r1,r0                                   @ structure number address
bl callFunc                                 @ call
ldr r1,iAdrsValue                           @ and convert ascii string
bl conversion10
ldr r0,iAdrszMessResult                     @ display result message
bl affichageMess
add r4,#1                                   @ increment loop counter
cmp r4,#10                                  @ end ?
ble 1b                                      @ no -> loop
/*********Fibonacci  *************/
bl YFunc                                    @ create Ycombinator
mov r5,r0                                   @ save Ycombinator
bl affichageMess
mov r4,#1                                   @ loop counter
2:  @ start loop
mov r0,r4
bl numFunc                                  @ create number structure
cmp r0,#-1                                  @ allocation error ?
beq 99f
mov r1,r0                                   @ structure number address
bl callFunc                                 @ call
ldr r1,iAdrsValue                           @ and convert ascii string
bl conversion10
ldr r0,iAdrszMessResult                     @ display result message
bl affichageMess
add r4,#1                                   @ increment loop counter
cmp r4,#10                                  @ end ?
ble 2b                                      @ no -> loop
ldr r0,iAdrszMessEndPgm                     @ display end message
bl affichageMess
b 100f
99:                                             @ display error message
bl affichageMess
100:                                            @ standard end of the program
mov r0, #0                                  @ return code
mov r7, #EXIT                               @ request to exit program
svc 0                                       @ perform system call
/******************************************************************/
/*     factorielle function                         */
/******************************************************************/
/* r0 contains the Y combinator address  */
/* r1 contains the number structure  */
facFunc:
push {r1-r3,lr}             @ save  registers
mov r2,r0                   @ save Y combinator address
cmp r0,#1                   @ > 1 ?
bgt 1f                      @ yes
mov r0,#1                   @ create structure number value 1
bl numFunc
b 100f
1:
mov r3,r0                   @ save number
sub r0,#1                   @ decrement number
bl numFunc                  @ and create new structure number
cmp r0,#-1                  @ allocation error ?
beq 100f
mov r1,r0                   @ new structure number -> param 1
bl callFunc                 @ call
ldr r1,[r0,#func_num]       @ load new result
mul r0,r1,r3                @ and multiply by precedent
bl numFunc                  @ and create new structure number
@ and return her address in r0
100:
pop {r1-r3,lr}              @ restaur registers
bx lr                       @ return
/******************************************************************/
/*     fibonacci function                         */
/******************************************************************/
/* r0 contains the Y combinator address  */
/* r1 contains the number structure  */
fibFunc:
push {r1-r4,lr}             @ save  registers
mov r2,r0                   @ save Y combinator address
cmp r0,#1                   @ > 1 ?
bgt 1f                      @ yes
mov r0,#1                   @ create structure number value 1
bl numFunc
b 100f
1:
mov r3,r0                   @ save number
sub r0,#1                   @ decrement number
bl numFunc                  @ and create new structure number
cmp r0,#-1                  @ allocation error ?
beq 100f
mov r1,r0                   @ new structure number -> param 1
bl callFunc                 @ call
ldr r4,[r0,#func_num]       @ load new result
sub r0,r3,#2                @ new number - 2
bl numFunc                  @ and create new structure number
cmp r0,#-1                  @ allocation error ?
beq 100f
mov r1,r0                   @ new structure number -> param 1
bl callFunc                 @ call
ldr r1,[r0,#func_num]       @ load new result
bl numFunc                  @ and create new structure number
@ and return her address in r0
100:
pop {r1-r4,lr}              @ restaur registers
bx lr                       @ return
/******************************************************************/
/*     call function                         */
/******************************************************************/
/* r0 contains the address of the function  */
/* r1 contains the address of the function 1 */
callFunc:
push {r2,lr}                                @ save  registers
blx r2                                      @ and call it
pop {r2,lr}                                 @ restaur registers
bx lr                                       @ return
/******************************************************************/
/*     create Y combinator function                         */
/******************************************************************/
/* r0 contains the address of the function  */
YFunc:
push {r1,lr}                                @ save  registers
mov r1,#0
bl newFunc
cmp r0,#-1                                  @ allocation error ?
strne r0,[r0,#func_f_]                      @ store function and return in r0
pop {r1,lr}                                 @ restaur registers
bx lr                                       @ return
/******************************************************************/
/*     create structure number function                         */
/******************************************************************/
/* r0 contains the number  */
numFunc:
push {r1,r2,lr}                             @ save  registers
mov r2,r0                                   @ save number
mov r0,#0                                   @ function null
mov r1,#0                                   @ function null
bl newFunc
cmp r0,#-1                                  @ allocation error ?
strne r2,[r0,#func_num]                     @ store number in new structure
pop {r1,r2,lr}                              @ restaur registers
bx lr                                       @ return
/******************************************************************/
/*     new function                                               */
/******************************************************************/
/* r0 contains the function address   */
/* r1 contains the function address 1   */
newFunc:
push {r2-r7,lr}                             @ save  registers
mov r5,r1                                   @ save adresse 1
@ allocation place on the heap
mov r0,#0                                   @ allocation place heap
mov r7,#0x2D                                @ call system 'brk'
svc #0
mov r3,r0                                   @ save address heap for output string
add r0,#func_fin                            @ reservation place one element
mov r7,#0x2D                                @ call system 'brk'
svc #0
cmp r0,#-1                                  @ allocation error
beq 100f
mov r0,r3
str r5,[r0,#func_f_]
mov r2,#0
str r2,[r0,#func_num]                       @ store zero to number
100:
pop {r2-r7,lr}                              @ restaur registers
bx lr                                       @ return
/***************************************************/
/*      ROUTINES INCLUDE                 */
/***************************************************/
.include "../affichage.inc"



{{output}}


Program start
Function factorielle :
Result value :1
Result value :2
Result value :6
Result value :24
Result value :120
Result value :720
Result value :5040
Result value :40320
Result value :362880
Result value :3628800
Function Fibonacci :
Result value :1
Result value :2
Result value :3
Result value :5
Result value :8
Result value :13
Result value :21
Result value :34
Result value :55
Result value :89
Program normal end.



ATS


(* ****** ****** *)
//
//
(* ****** ****** *)
//
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 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 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))
)
&
)


{{out}}

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

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;
}



{{out}}

fac:  1 2 6 24 120 720 5040 40320 362880
fib:  1 2 3 5 8 13 21 34 55


C#

Like many other statically typed languages, this involves a recursive type, and like other strict languages, it is the Z-combinator instead.

The combinator here is expressed entirely as a lambda expression and is a static property of the generic YCombinator class. Both it and the RecursiveFunc type thus "inherit" the type parameters of the containing class—there effectively exists a separate specialized copy of both for each generic instantiation of YCombinator.

''Note: in the code, Func<T, TResult> is a delegate type (the CLR equivalent of a function pointer) that has a parameter of type T and return type of TResult. See [[Higher-order functions#C#]] or [https://docs.microsoft.com/en-us/dotnet/standard/delegates-lambdas the documentation] for more information.''

using System;

static class YCombinator<T, TResult>
{
// RecursiveFunc is not needed to call Fix() and so can be private.
private delegate Func<T, TResult> RecursiveFunc(RecursiveFunc r);

public static Func<Func<Func<T, TResult>, Func<T, TResult>>, Func<T, TResult>> Fix { get; } =
f => ((RecursiveFunc)(g => f(x => g(g)(x))))(g => f(x => g(g)(x)));
}

static class Program
{
static void Main()
{
var fac = YCombinator<int, int>.Fix(f => x => x < 2 ? 1 : x * f(x - 1));
var fib = YCombinator<int, int>.Fix(f => x => x < 2 ? x : f(x - 1) + f(x - 2));

Console.WriteLine(fac(10));
Console.WriteLine(fib(10));
}
}



{{out}}

3628800
55


Alternatively, with a non-generic holder class (note that Fix is now a method, as properties cannot be generic):

static class YCombinator
{
private delegate Func<T, TResult> RecursiveFunc<T, TResult>(RecursiveFunc<T, TResult> r);

public static Func<T, TResult> Fix<T, TResult>(Func<Func<T, TResult>, Func<T, TResult>> f)
=> ((RecursiveFunc<T, TResult>)(g => f(x => g(g)(x))))(g => f(x => g(g)(x)));
}


Using the late-binding offered by dynamic to eliminate the recursive type:

static class YCombinator<T, TResult>
{
public static Func<Func<Func<T, TResult>, Func<T, TResult>>, Func<T, TResult>> Fix { get; } =
f => ((Func<dynamic, Func<T, TResult>>)(g => f(x => g(g)(x))))((Func<dynamic, Func<T, TResult>>)(g => f(x => g(g)(x))));
}


The usual version using recursion, disallowed by the task (implemented as a generic method):

static class YCombinator
{
static Func<T, TResult> Fix<T, TResult>(Func<Func<T, TResult>, Func<T, TResult>> f) => x => f(Fix(f))(x);
}


Translations

To compare differences in language and runtime instead of in approaches to the task, the following are translations of solutions from other languages. Two versions of each translation are provided, one seeking to resemble the original as closely as possible, and another that is identical in program control flow but syntactically closer to idiomatic C#.

====[http://rosettacode.org/mw/index.php?oldid=287744#C++ C++]==== std::function<TResult(T)> in C++ corresponds to Func<T, TResult> in C#.

'''Verbatim'''

using Func = System.Func<int, int>;
using FuncFunc = System.Func<System.Func<int, int>, System.Func<int, int>>;

static class Program {
struct RecursiveFunc<F> {
public System.Func<RecursiveFunc<F>, F> o;
}

static System.Func<A, B> Y<A, B>(System.Func<System.Func<A, B>, System.Func<A, B>> f) {
var r = new RecursiveFunc<System.Func<A, B>>() {
o = new System.Func<RecursiveFunc<System.Func<A, B>>, System.Func<A, B>>((RecursiveFunc<System.Func<A, B>> w) => {
return f(new System.Func<A, B>((A x) => {
return w.o(w)(x);
}));
})
};
return r.o(r);
}

static FuncFunc almost_fac = (Func f) => {
return new Func((int n) => {
if (n <= 1) return 1;
return n * f(n - 1);
});
};

static FuncFunc almost_fib = (Func f) => {
return new Func((int n) => {
if (n <= 2) return 1;
return f(n - 1) + f(n - 2);
});
};

static int Main() {
var fib = Y(almost_fib);
var fac = Y(almost_fac);
System.Console.WriteLine("fib(10) = " + fib(10));
System.Console.WriteLine("fac(10) = " + fac(10));
return 0;
}
}


'''Semi-idiomatic'''

using System;

using FuncFunc = System.Func<System.Func<int, int>, System.Func<int, int>>;

static class Program {
struct RecursiveFunc<F> {
public Func<RecursiveFunc<F>, F> o;
}

static Func<A, B> Y<A, B>(Func<Func<A, B>, Func<A, B>> f) {
var r = new RecursiveFunc<Func<A, B>> {
o = w => f(x => w.o(w)(x))
};
return r.o(r);
}

static FuncFunc almost_fac = f => n => n <= 1 ? 1 : n * f(n - 1);

static FuncFunc almost_fib = f => n => n <= 2 ? 1 : f(n - 1) + f(n - 2);

static void Main() {
var fib = Y(almost_fib);
var fac = Y(almost_fac);
Console.WriteLine("fib(10) = " + fib(10));
Console.WriteLine("fac(10) = " + fac(10));
}
}


====[http://rosettacode.org/mw/index.php?oldid=287744#Ceylon Ceylon]==== TResult(T) in Ceylon corresponds to Func<T, TResult> in C#.

Since C# does not have local classes, RecursiveFunc and y1 are declared in a class of their own. Moving the type parameters to the class also prevents type parameter inference.

'''Verbatim'''

using System;
using System.Diagnostics;

class Program {
public delegate TResult ParamsFunc<T, TResult>(params T[] args);

static class Y<Result, Args> {
class RecursiveFunction {
public Func<RecursiveFunction, ParamsFunc<Args, Result>> o;
public RecursiveFunction(Func<RecursiveFunction, ParamsFunc<Args, Result>> o) => this.o = o;
}

public static ParamsFunc<Args, Result> y1(
Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f) {

var r = new RecursiveFunction((RecursiveFunction w)
=> f((Args[] args) => w.o(w)(args)));

return r.o(r);
}
}

static ParamsFunc<Args, Result> y2<Args, Result>(
Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f) {

Func<dynamic, ParamsFunc<Args, Result>> r = w => {
Debug.Assert(w is Func<dynamic, ParamsFunc<Args, Result>>);
return f((Args[] args) => w(w)(args));
};

return r(r);
}

static ParamsFunc<Args, Result> y3<Args, Result>(
Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f)
=> (Args[] args) => f(y3(f))(args);

static void Main() {
var factorialY1 = Y<int, int>.y1((ParamsFunc<int, int> fact) => (int[] x)
=> (x[0] > 1) ? x[0] * fact(x[0] - 1) : 1);

var fibY1 = Y<int, int>.y1((ParamsFunc<int, int> fib) => (int[] x)
=> (x[0] > 2) ? fib(x[0] - 1) + fib(x[0] - 2) : 2);

Console.WriteLine(factorialY1(10)); // 362880
Console.WriteLine(fibY1(10));       // 110
}
}


'''Semi-idiomatic'''

using System;
using System.Diagnostics;

static class Program {
delegate TResult ParamsFunc<T, TResult>(params T[] args);

static class Y<Result, Args> {
class RecursiveFunction {
public Func<RecursiveFunction, ParamsFunc<Args, Result>> o;
public RecursiveFunction(Func<RecursiveFunction, ParamsFunc<Args, Result>> o) => this.o = o;
}

public static ParamsFunc<Args, Result> y1(
Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f) {

var r = new RecursiveFunction(w => f(args => w.o(w)(args)));

return r.o(r);
}
}

static ParamsFunc<Args, Result> y2<Args, Result>(
Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f) {

Func<dynamic, ParamsFunc<Args, Result>> r = w => {
Debug.Assert(w is Func<dynamic, ParamsFunc<Args, Result>>);
return f(args => w(w)(args));
};

return r(r);
}

static ParamsFunc<Args, Result> y3<Args, Result>(
Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f)
=> args => f(y3(f))(args);

static void Main() {
var factorialY1 = Y<int, int>.y1(fact => x => (x[0] > 1) ? x[0] * fact(x[0] - 1) : 1);
var fibY1 = Y<int, int>.y1(fib => x => (x[0] > 2) ? fib(x[0] - 1) + fib(x[0] - 2) : 2);

Console.WriteLine(factorialY1(10));
Console.WriteLine(fibY1(10));
}
}


====[http://rosettacode.org/mw/index.php?oldid=287744#Go Go]==== func(T) TResult in Go corresponds to Func<T, TResult> in C#.

'''Verbatim'''

using System;

// Func and FuncFunc can be defined using using aliases and the System.Func<T, TReult> type, but RecursiveFunc must be a delegate type of its own.
using Func = System.Func<int, int>;
using FuncFunc = System.Func<System.Func<int, int>, System.Func<int, int>>;

delegate Func RecursiveFunc(RecursiveFunc f);

static class Program {
static void Main() {
var fac = Y(almost_fac);
var fib = Y(almost_fib);
Console.WriteLine("fac(10) = " + fac(10));
Console.WriteLine("fib(10) = " + fib(10));
}

static Func Y(FuncFunc f) {
RecursiveFunc g = delegate (RecursiveFunc r) {
return f(delegate (int x) {
return r(r)(x);
});
};
return g(g);
}

static Func almost_fac(Func f) {
return delegate (int x) {
if (x <= 1) {
return 1;
}
return x * f(x-1);
};
}

static Func almost_fib(Func f) {
return delegate (int x) {
if (x <= 2) {
return 1;
}
return f(x-1)+f(x-2);
};
}
}


Recursive:

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


'''Semi-idiomatic'''

using System;

delegate int Func(int i);
delegate Func FuncFunc(Func f);
delegate Func RecursiveFunc(RecursiveFunc f);

static class Program {
static void Main() {
var fac = Y(almost_fac);
var fib = Y(almost_fib);
Console.WriteLine("fac(10) = " + fac(10));
Console.WriteLine("fib(10) = " + fib(10));
}

static Func Y(FuncFunc f) {
RecursiveFunc g = r => f(x => r(r)(x));
return g(g);
}

static Func almost_fac(Func f) => x => x <= 1 ? 1 : x * f(x - 1);

static Func almost_fib(Func f) => x => x <= 2 ? 1 : f(x - 1) + f(x - 2);
}


Recursive:

    static Func Y(FuncFunc f) => x => f(Y(f))(x);


====[http://rosettacode.org/mw/index.php?oldid=287744#Java Java]====

'''Verbatim'''

Since Java uses interfaces and C# uses delegates, which are the only type that the C# compiler will coerce lambda expressions to, this code declares a Functions class for providing a means of converting CLR delegates to objects that implement the Function and RecursiveFunction interfaces.

using System;

static class Program {
interface Function<T, R> {
R apply(T t);
}

interface RecursiveFunction<F> : Function<RecursiveFunction<F>, F> {
}

static class Functions {
class Function<T, R> : Program.Function<T, R> {

public Function(Func<T, R> inner) => this._inner = inner;

public R apply(T t) => this._inner(t);
}

class RecursiveFunction<F> : Function<Program.RecursiveFunction<F>, F>, Program.RecursiveFunction<F> {
public RecursiveFunction(Func<Program.RecursiveFunction<F>, F> inner) : base(inner) {
}
}

public static Program.Function<T, R> Create<T, R>(Func<T, R> inner) => new Function<T, R>(inner);
public static Program.RecursiveFunction<F> Create<F>(Func<Program.RecursiveFunction<F>, F> inner) => new RecursiveFunction<F>(inner);
}

static Function<A, B> Y<A, B>(Function<Function<A, B>, Function<A, B>> f) {
var r = Functions.Create<Function<A, B>>(w => f.apply(Functions.Create<A, B>(x => w.apply(w).apply(x))));
return r.apply(r);
}

static void Main(params String[] arguments) {
Function<int, int> fib = Y(Functions.Create<Function<int, int>, Function<int, int>>(f => Functions.Create<int, int>(n =>
(n <= 2)
? 1
: (f.apply(n - 1) + f.apply(n - 2))))
);
Function<int, int> fac = Y(Functions.Create<Function<int, int>, Function<int, int>>(f => Functions.Create<int, int>(n =>
(n <= 1)
? 1
: (n * f.apply(n - 1))))
);

Console.WriteLine("fib(10) = " + fib.apply(10));
Console.WriteLine("fac(10) = " + fac.apply(10));
}
}


'''"Idiomatic"'''

For demonstrative purposes, to completely avoid using CLR delegates, lambda expressions can be replaced with explicit types that implement the functional interfaces. Closures are thus implemented by replacing all usages of the original local variable with a field of the type that represents the lambda expression; this process, called "hoisting" is actually how variable capturing is implemented by the C# compiler (for more information, see [https://blogs.msdn.microsoft.com/abhinaba/2005/10/18/c-anonymous-methods-are-not-closures/ this Microsoft blog post].

using System;

static class YCombinator {
interface Function<T, R> {
R apply(T t);
}

interface RecursiveFunction<F> : Function<RecursiveFunction<F>, F> {
}

static class Y<A, B> {
class __1 : RecursiveFunction<Function<A, B>> {
class __2 : Function<A, B> {

public __2(RecursiveFunction<Function<A, B>> w) {
this.w = w;
}

public B apply(A x) {
return w.apply(w).apply(x);
}
}

Function<Function<A, B>, Function<A, B>> f;

public __1(Function<Function<A, B>, Function<A, B>> f) {
this.f = f;
}

public Function<A, B> apply(RecursiveFunction<Function<A, B>> w) {
return f.apply(new __2(w));
}
}

public static Function<A, B> _(Function<Function<A, B>, Function<A, B>> f) {
var r = new __1(f);
return r.apply(r);
}
}

class __1 : Function<Function<int, int>, Function<int, int>> {
class __2 : Function<int, int> {

public __2(Function<int, int> f) {
this.f = f;
}

public int apply(int n) {
return
(n <= 2)
? 1
: (f.apply(n - 1) + f.apply(n - 2));
}
}

public Function<int, int> apply(Function<int, int> f) {
return new __2(f);
}
}

class __2 : Function<Function<int, int>, Function<int, int>> {
class __3 : Function<int, int> {

public __3(Function<int, int> f) {
this.f = f;
}

public int apply(int n) {
return
(n <= 1)
? 1
: (n * f.apply(n - 1));
}
}

public Function<int, int> apply(Function<int, int> f) {
return new __3(f);
}
}

static void Main(params String[] arguments) {
Function<int, int> fib = Y<int, int>._(new __1());
Function<int, int> fac = Y<int, int>._(new __2());

Console.WriteLine("fib(10) = " + fib.apply(10));
Console.WriteLine("fac(10) = " + fac.apply(10));
}
}


'''C# 1.0'''

To conclude this chain of decreasing reliance on language features, here is above code translated to C# 1.0. The largest change is the replacement of the generic interfaces with the results of manually substituting their type parameters.

using System;

class Program {
interface Func {
int apply(int i);
}

interface FuncFunc {
Func apply(Func f);
}

interface RecursiveFunc {
Func apply(RecursiveFunc f);
}

class Y {
class __1 : RecursiveFunc {
class __2 : Func {

public __2(RecursiveFunc w) {
this.w = w;
}

public int apply(int x) {
return w.apply(w).apply(x);
}
}

public __1(FuncFunc f) {
this.f = f;
}

public Func apply(RecursiveFunc w) {
return f.apply(new __2(w));
}
}

public static Func _(FuncFunc f) {
__1 r = new __1(f);
return r.apply(r);
}
}

class __fib : FuncFunc {
class __1 : Func {

public __1(Func f) {
this.f = f;
}

public int apply(int n) {
return
(n <= 2)
? 1
: (f.apply(n - 1) + f.apply(n - 2));
}

}

public Func apply(Func f) {
return new __1(f);
}
}

class __fac : FuncFunc {
class __1 : Func {

public __1(Func f) {
this.f = f;
}

public int apply(int n) {
return
(n <= 1)
? 1
: (n * f.apply(n - 1));
}
}

public Func apply(Func f) {
return new __1(f);
}
}

static void Main(params String[] arguments) {
Func fib = Y._(new __fib());
Func fac = Y._(new __fac());

Console.WriteLine("fib(10) = " + fib.apply(10));
Console.WriteLine("fac(10) = " + fac.apply(10));
}
}


'''Modified/varargs (the last implementation in the Java section)'''

Since C# delegates cannot declare members, extension methods are used to simulate doing so.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;

static class Func {
public static Func<T, TResult2> andThen<T, TResult, TResult2>(
this Func<T, TResult> @this,
Func<TResult, TResult2> after)
=> _ => after(@this(_));
}

delegate OUTPUT SelfApplicable<OUTPUT>(SelfApplicable<OUTPUT> s);
static class SelfApplicable {
public static OUTPUT selfApply<OUTPUT>(this SelfApplicable<OUTPUT> @this) => @this(@this);
}

delegate FUNCTION FixedPoint<FUNCTION>(Func<FUNCTION, FUNCTION> f);

delegate OUTPUT VarargsFunction<INPUTS, OUTPUT>(params INPUTS[] inputs);
static class VarargsFunction {
public static VarargsFunction<INPUTS, OUTPUT> from<INPUTS, OUTPUT>(
Func<INPUTS[], OUTPUT> function)
=> function.Invoke;

public static VarargsFunction<INPUTS, OUTPUT> upgrade<INPUTS, OUTPUT>(
Func<INPUTS, OUTPUT> function) {
return inputs => function(inputs[0]);
}

public static VarargsFunction<INPUTS, OUTPUT> upgrade<INPUTS, OUTPUT>(
Func<INPUTS, INPUTS, OUTPUT> function) {
return inputs => function(inputs[0], inputs[1]);
}

public static VarargsFunction<INPUTS, POST_OUTPUT> andThen<INPUTS, OUTPUT, POST_OUTPUT>(
this VarargsFunction<INPUTS, OUTPUT> @this,
VarargsFunction<OUTPUT, POST_OUTPUT> after) {
return inputs => after(@this(inputs));
}

public static Func<INPUTS, OUTPUT> toFunction<INPUTS, OUTPUT>(
this VarargsFunction<INPUTS, OUTPUT> @this) {
return input => @this(input);
}

public static Func<INPUTS, INPUTS, OUTPUT> toBiFunction<INPUTS, OUTPUT>(
this VarargsFunction<INPUTS, OUTPUT> @this) {
return (input, input2) => @this(input, input2);
}

public static VarargsFunction<PRE_INPUTS, OUTPUT> transformArguments<PRE_INPUTS, INPUTS, OUTPUT>(
this VarargsFunction<INPUTS, OUTPUT> @this,
Func<PRE_INPUTS, INPUTS> transformer) {
return inputs => @this(inputs.AsParallel().AsOrdered().Select(transformer).ToArray());
}
}

delegate FixedPoint<FUNCTION> Y<FUNCTION>(SelfApplicable<FixedPoint<FUNCTION>> y);

static class Program {
static TResult Cast<TResult>(this Delegate @this) where TResult : Delegate {
return (TResult)Delegate.CreateDelegate(typeof(TResult), @this.Target, @this.Method);
}

static void Main(params String[] arguments) {
BigInteger TWO = BigInteger.One + BigInteger.One;

Func<IFormattable, long> toLong = x => long.Parse(x.ToString());
Func<IFormattable, BigInteger> toBigInteger = x => new BigInteger(toLong(x));

/* Based on https://gist.github.com/aruld/3965968/#comment-604392 */
Y<VarargsFunction<IFormattable, IFormattable>> combinator = y => f => x => f(y.selfApply()(f))(x);
FixedPoint<VarargsFunction<IFormattable, IFormattable>> fixedPoint =
combinator.Cast<SelfApplicable<FixedPoint<VarargsFunction<IFormattable, IFormattable>>>>().selfApply();

VarargsFunction<IFormattable, IFormattable> fibonacci = fixedPoint(
toBigInteger.andThen(
n => (IFormattable)(
(n.CompareTo(TWO) <= 0)
? 1
: BigInteger.Parse(f(n - BigInteger.One).ToString())
+ BigInteger.Parse(f(n - TWO).ToString()))
)
)
);

VarargsFunction<IFormattable, IFormattable> factorial = fixedPoint(
toBigInteger.andThen(
n => (IFormattable)((n.CompareTo(BigInteger.One) <= 0)
? 1
: n * BigInteger.Parse(f(n - BigInteger.One).ToString()))
)
)
);

VarargsFunction<IFormattable, IFormattable> ackermann = fixedPoint(
(BigInteger m, BigInteger n) => m.Equals(BigInteger.Zero)
? n + BigInteger.One
: f(
m - BigInteger.One,
n.Equals(BigInteger.Zero)
? BigInteger.One
: f(m, n - BigInteger.One)
)
).transformArguments(toBigInteger)
);

var functions = new Dictionary<String, VarargsFunction<IFormattable, IFormattable>>();

var parameters = new Dictionary<VarargsFunction<IFormattable, IFormattable>, IFormattable[]>();
parameters.Add(functions["fibonacci"], new IFormattable[] { 20 });
parameters.Add(functions["factorial"], new IFormattable[] { 10 });
parameters.Add(functions["ackermann"], new IFormattable[] { 3, 2 });

functions.AsParallel().Select(
entry => entry.Key
+ "[" + String.Join(", ", parameters[entry.Value].Select(x => x.ToString())) + "]"
+ " = "
+ entry.Value(parameters[entry.Value])
).ForAll(Console.WriteLine);
}
}


====[http://rosettacode.org/mw/index.php?oldid=287744#Swift Swift]==== T -> TResult in Swift corresponds to Func<T, TResult> in C#.

'''Verbatim'''

The more idiomatic version doesn't look much different.

using System;

static class Program {
struct RecursiveFunc<F> {
public Func<RecursiveFunc<F>, F> o;
}

static Func<A, B> Y<A, B>(Func<Func<A, B>, Func<A, B>> f) {
var r = new RecursiveFunc<Func<A, B>> { o = w => f(_0 => w.o(w)(_0)) };
return r.o(r);
}

static void Main() {
// C# can't infer the type arguments to Y either; either it or f must be explicitly typed.
var fac = Y((Func<int, int> f) => _0 => _0 <= 1 ? 1 : _0 * f(_0 - 1));
var fib = Y((Func<int, int> f) => _0 => _0 <= 2 ? 1 : f(_0 - 1) + f(_0 - 2));

Console.WriteLine($"fac(5) = {fac(5)}"); Console.WriteLine($"fib(9) = {fib(9)}");
}
}


Without recursive type:

    public static Func<A, B> Y<A, B>(Func<Func<A, B>, Func<A, B>> f) {
Func<dynamic, Func<A, B>> r = z => { var w = (Func<dynamic, Func<A, B>>)z; return f(_0 => w(w)(_0)); };
return r(r);
}


Recursive:

    public static Func<In, Out> Y<In, Out>(Func<Func<In, Out>, Func<In, Out>> f) {
return x => f(Y(f))(x);
}


C++

{{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;
}


{{out}}


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';
}


{{out}}


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

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

(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))))))))


{{out}}

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

(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

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

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));
}


{{out}}

factorial: [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
ackermann(3, 5): 253


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


{{out}}

720
13


Delphi

{{works with|Delphi XE and higher}} May work with Delphi 2009 and 2010 too. {{trans|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 {{trans|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  ;; 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 Translated from Objective-C example on this page.  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 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)  {{out}} (479001600,144)  Elena {{trans|Smalltalk}} ELENA 4.x : 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) ? i : (f(i-1) + f(i-2)) )); var fact := YCombinator.fix:(f => (i => (i == 0) ? 1 : (f(i-1) * i) )); console.printLine("fib(10)=",fib(10)); console.printLine("fact(10)=",fact(10)); }  {{out}}  fib(10)=55 fact(10)=3628800  Elixir {{trans|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 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 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  =={{header|F Sharp|F#}}== 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 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 [ - @ ] bi-curry@ 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 \ 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 r@ execute swap 1- r> execute + then ; y execute . 55 ok  Falcon  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 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 {{trans|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 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) } }  {{out}}  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 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 '''''le'''ngth'' 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 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 ===Non-tacit version=== 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;~":0)<@;<@((":0)&;))'(2 : 0 '') (1 : (m,'u'))(1 : (m,'''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 factorial and Fibonacci examples follow:  'if. * y do. y * u <: y else. 1 end.' Y 10 NB. Factorial 3628800 '(u@:<:@:<: + u@:<:)^:(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 can be 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 <:@:])@.(#.@,&*))'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 * u@:])'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 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 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 {{works with|Java|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)); } }  {{out}}  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|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#Java|Ackermann function]] has been included in this implementation (mostly because both [[Y combinator#D|D]] and [[Y combinator#PicoLisp|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); } }  {{out}} (may depend on which function gets processed first): factorial[10] = 3628800 ackermann[3, 2] = 29 fibonacci[20] = 6765  ## JavaScript 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: javascript 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=== 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:  (x => x(x))(y => f(x => y(y)(x))); var fac = Y(f => n => n > 1 ? n * f(n-1) : 1);  Joy DEFINE y == [dup cons] swap concat dup cons i; fac == [ [pop null] [pop succ] [[dup pred] dip i *] ifte ] y.  Julia  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 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 // 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() }  {{out}}  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 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 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 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  MANOOL Here one additional technique is demonstrated: the Y combinator is applied to a function ''during compilation'' due to the $ operator, which is optional:


{ {extern "manool.org.18/std/0.3/all"} in
: let { Y = {proc {F} as {proc {X} as X[X]}[{proc {X} with {F} as F[{proc {Y} with {X} as X[X][Y]}]}]} } in
{ for { N = Range[10] } do
: (WriteLine) Out; N "! = "
{Y: proc {Rec} as {proc {N} with {Rec} as: if N == 0 then 1 else N * Rec[N - 1]}}$[N] } { for { N = Range[10] } do : (WriteLine) Out; "Fib " N " = " {Y: proc {Rec} as {proc {N} with {Rec} as: if N == 0 then 0 else: if N == 1 then 1 else Rec[N - 2] + Rec[N - 1]}}$[N]
}
}



Using less syntactic sugar:


{ {extern "manool.org.18/std/0.3/all"} in
: let { Y = {proc {F} as {proc {X} as X[X]}[{proc {F; X} as F[{proc {X; Y} as X[X][Y]}.Bind[X]]}.Bind[F]]} } in
{ for { N = Range[10] } do
: (WriteLine) Out; N "! = "
{Y: proc {Rec} as {proc {Rec; N} as: if N == 0 then 1 else N * Rec[N - 1]}.Bind[Rec]}$[N] } { for { N = Range[10] } do : (WriteLine) Out; "Fib " N " = " {Y: proc {Rec} as {proc {Rec; N} as: if N == 0 then 0 else: if N == 1 then 1 else Rec[N - 2] + Rec[N - 1]}.Bind[Rec]}$[N]
}
}



{{output}}


0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
Fib 0 = 0
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



Maple


> 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



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

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


=={{header|Objective-C}}== {{works with|Mac OS X|10.6+}}{{works with|iOS|4.0+}}



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

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

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

#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

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

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])


{{out}}

%1 = [1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
%2 = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]


Perl

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)->($_), 0..9), "\n"; }  {{out}} 1 1 2 6 24 120 720 5040 40320 362880 0 1 1 2 3 5 8 13 21 34  The usual version using recursion, disallowed by the task: sub Y { my$f = shift;
sub {$f->(Y($f))->(@_)}
}


Perl 6

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;


{{out}}

(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

{{trans|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#Phix|Currying]], [[Closures/Value_capture#Phix|Closures/Value_capture]], [[Partial_function_application#Phix|Partial_function_application]], and/or [[Function_composition#Phix|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")


{{out}}


fac:  1 2 6 24 120 720 5040 40320 362880 3628800
fib:  1 1 2 3 5 8 13 21 34 55



PHP

{{works with|PHP|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|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 $this parameter in closures is also possible: {{works with|PHP|5.4+}} <?php function pseudoY($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 {{trans|Common Lisp}} (de Y (F) (let X (curry (F) (Y) (F (curry (Y) @ (pass (Y Y))))) (X X) ) )  Factorial # Factorial (de fact (F) (curry (F) (N) (if (=0 N) 1 (* N (F (dec N))) ) ) ) : ((Y fact) 6) -> 720  Fibonacci sequence # Fibonacci (de fibo (F) (curry (F) (N) (if (> 2 N) 1 (+ (F (dec N)) (F (- N 2))) ) ) ) : ((Y fibo) 22) -> 28657  Ackermann function # 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 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) =>  {{out}}  ** 120 ** 8  PostScript {{trans|Joy}} {{libheader|initlib}} y { {dup cons} exch concat dup cons i }. /fac { { {pop zero?} {pop succ} {{dup pred} dip i *} ifte } y }.  PowerShell {{trans|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, $lambda$ translates as param inside of a ScriptBlock, $\left(\ldots\right)$ translates as Invoke-Expression "{}", invocation (written as a space) translates to InvokeReturnAsIs. $\begin\left\{array\right\}\left\{lcl\right\} fac & := & \lambda f.\left(\lambda n.\mbox\left\{if \right\}n\leq0\mbox\left\{ then \right\}1\mbox\left\{ else \right\}n*\left(f\ n-1\right)\right) \ fib & := & \lambda f.\left(\lambda n. \mbox\left\{if \right\}n=0\mbox\left\{ or \right\}n=1\mbox\left\{ then \right\}1\mbox\left\{ else \right\}\left(f\ n-1\right)+\left(f\ n-2\right)\right) \ Z & := & \lambda f.\left(\lambda x.f\ \left(\lambda y.x\ x\ y\right)\right)\ \left(\lambda x.f\ \left(\lambda y.x\ x\ y\right)\right) \ \end\left\{array\right\}$ $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)

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 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|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|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 {{works with|Rust|1.35.0 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 // 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. 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) // 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 } } // This works for 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 // } // } // 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 Fn(&Fn(T) -> R, T) -> R) -> impl Fn(T) -> R { move |t| ( |x: &Apply<T,R>, y| x.apply(x, y) ) ( &|x: &Apply<T,R>, y| f( &|z| x.apply(x,z), y ), t ) } // 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 Credit goes to the thread in [http://scala-blogs.org/2008/09/y-combinator-in-scala.html 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 (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)  {{out}} 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 (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)))  {{out}} 1 120 3628800  Sidef 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) }  {{out}}  [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]  Slate 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 {{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.  {{out}} 55 3628800  The usual version using recursion, disallowed by the task: Y := [:f| [:x| (f value: (Y value: f)) value: x] ].  Standard ML - 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 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 Using a recursive type:  { 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))")


{{out}}


fac(5) = 120
fib(9) = 34



Without a recursive type, and instead using Any to erase the type: {{works with|Swift|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)($0) } } return r(r) }  The usual version using recursion, disallowed by the task: func Y<In, Out>( f: (In->Out) -> (In->Out) ) -> (In->Out) { return { x in f(Y(f))(x) } }  Tailspin  // YCombinator is not needed since tailspin supports recursion readily, but this demonstrates passing functions as parameters templates combinator@{stepper:} templates makeStep@{rec:}$ -> stepper@{next: rec@{rec: rec}} !
end makeStep
$-> makeStep@{rec: makeStep} ! end combinator templates factorial templates seed@{next:} <0> 1 ! <>$ * ($- 1 -> next) ! end seed$ -> combinator@{stepper: seed} !
end factorial

5 -> factorial -> 'factorial 5: $; ' -> !OUT::write templates fibonacci templates seed@{next:} <..1>$ !
<>
($-2 -> next) + ($ - 1 -> next) !
end seed
$-> combinator@{stepper: seed} ! end fibonacci 5 -> fibonacci -> 'fibonacci 5:$;
' -> !OUT::write



{{out}}


factorial 5: 120
fibonacci 5: 5



Tcl

Y combinator is derived in great detail [http://wiki.tcl.tk/4833 here].

TXR

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

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 $f = h\left(f\right)$, where $h\left(f\right)$ is just the body of the function with recursive calls to $f$ in it. With a fixed point combinator such as my_fix as defined above, you do almost the same thing, except it's $f =$my_fix "f". $h$("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>)


{{out}}


(
<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.

VBA

{{trans|Phix}} The IIf as translation of Iff can not be used as IIf executes both true and false parts and will cause a stack overflow.

Private Function call_fn(f As String, n As Long) As Long
call_fn = Application.Run(f, f, n)
End Function

Private Function Y(f As String) As String
Y = f
End Function

Private Function fac(self As String, n As Long) As Long
If n > 1 Then
fac = n * call_fn(self, n - 1)
Else
fac = 1
End If
End Function

Private Function fib(self As String, n As Long) As Long
If n > 1 Then
fib = call_fn(self, n - 1) + call_fn(self, n - 2)
Else
fib = n
End If
End Function

Private Sub test(name As String)
Dim f As String: f = Y(name)
Dim i As Long
Debug.Print name
For i = 1 To 10
Debug.Print call_fn(f, i);
Next i
Debug.Print
End Sub

Public Sub main()
test "fac"
test "fib"
End Sub


{{out}}

fac
1  2  6  24  120  720  5040  40320  362880  3628800
fib
1  1  2  3  5  8  13  21  34  55


Verbexx

/////// 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

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

# 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


{{omit from|ACL2}} {{omit from|Ada}} {{omit from|PureBasic}} {{omit from|TI-89 BASIC}}

XQuery

Version 3.0 of the [http://www.w3.org/TR/xpath-30/ XPath] and [http://www.w3.org/TR/xquery-30/ 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),
$fib(6) )  {{out}}  ## Yabasic Yabasic sub fac(self$, n)
if n > 1 then
return n * execute(self$, self$, n - 1)
else
return 1
end if
end sub

sub fib(self$, n) if n > 1 then return execute(self$, self$, n - 1) + execute(self$, self$, n - 2) else return n end if end sub sub test(name$)
local i

print name$, ": "; for i = 1 to 10 print execute(name$, name\$, i);
next
print
end sub

test("fac")
test("fib")


zkl

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();


{{out}}


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();


{{out}}


55
L(1,1,2,3,5,8,13,21,34,55,89)