Y combinator: Difference between revisions
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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]
<br><br>
=={{header|AArch64 Assembly}}==
{{works with|as|Raspberry Pi 3B version Buster 64 bits}}
<lang AArch64 Assembly>
/* ARM assembly AARCH64 Raspberry PI 3B */
/* program Ycombi64.s */
/*******************************************/
/* Constantes file */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
/*******************************************/
/* Structures */
/********************************************/
/* structure function*/
.struct 0
func_fn: // next element
.struct func_fn + 8
func_f_: // next element
.struct func_f_ + 8
func_num:
.struct func_num + 8
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 : @ \n"
/* UnInitialized data */
.bss
sZoneConv: .skip 100
/* code section */
.text
.global main
main: // program start
ldr x0,qAdrszMessStartPgm // display start message
bl affichageMess
adr x0,facFunc // function factorielle address
bl YFunc // create Ycombinator
mov x19,x0 // save Ycombinator
ldr x0,qAdrszFactorielle // display message
bl affichageMess
mov x20,#1 // loop counter
1: // start loop
mov x0,x20
bl numFunc // create number structure
cmp x0,#-1 // allocation error ?
beq 99f
mov x1,x0 // structure number address
mov x0,x19 // Ycombinator address
bl callFunc // call
ldr x0,[x0,#func_num] // load result
ldr x1,qAdrsZoneConv // and convert ascii string
bl conversion10S // decimal conversion
ldr x0,qAdrszMessResult
ldr x1,qAdrsZoneConv
bl strInsertAtCharInc // insert result at @ character
bl affichageMess // display message final
add x20,x20,#1 // increment loop counter
cmp x20,#10 // end ?
ble 1b // no -> loop
/*********Fibonacci *************/
adr x0,fibFunc // function fibonacci address
bl YFunc // create Ycombinator
mov x19,x0 // save Ycombinator
ldr x0,qAdrszFibonacci // display message
bl affichageMess
mov x20,#1 // loop counter
2: // start loop
mov x0,x20
bl numFunc // create number structure
cmp x0,#-1 // allocation error ?
beq 99f
mov x1,x0 // structure number address
mov x0,x19 // Ycombinator address
bl callFunc // call
ldr x0,[x0,#func_num] // load result
ldr x1,qAdrsZoneConv // and convert ascii string
bl conversion10S
ldr x0,qAdrszMessResult
ldr x1,qAdrsZoneConv
bl strInsertAtCharInc // insert result at @ character
bl affichageMess
add x20,x20,#1 // increment loop counter
cmp x20,#10 // end ?
ble 2b // no -> loop
ldr x0,qAdrszMessEndPgm // display end message
bl affichageMess
b 100f
99: // display error message
ldr x0,qAdrszMessError
bl affichageMess
100: // standard end of the program
mov x0,0 // return code
mov x8,EXIT // request to exit program
svc 0 // perform system call
qAdrszMessStartPgm: .quad szMessStartPgm
qAdrszMessEndPgm: .quad szMessEndPgm
qAdrszFactorielle: .quad szFactorielle
qAdrszFibonacci: .quad szFibonacci
qAdrszMessError: .quad szMessError
qAdrszCarriageReturn: .quad szCarriageReturn
qAdrszMessResult: .quad szMessResult
qAdrsZoneConv: .quad sZoneConv
/******************************************************************/
/* factorielle function */
/******************************************************************/
/* x0 contains the Y combinator address */
/* x1 contains the number structure */
facFunc:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
mov x2,x0 // save Y combinator address
ldr x0,[x1,#func_num] // load number
cmp x0,#1 // > 1 ?
bgt 1f // yes
mov x0,#1 // create structure number value 1
bl numFunc
b 100f
1:
mov x3,x0 // save number
sub x0,x0,#1 // decrement number
bl numFunc // and create new structure number
cmp x0,#-1 // allocation error ?
beq 100f
mov x1,x0 // new structure number -> param 1
ldr x0,[x2,#func_f_] // load function address to execute
bl callFunc // call
ldr x1,[x0,#func_num] // load new result
mul x0,x1,x3 // and multiply by precedent
bl numFunc // and create new structure number
// and return her address in x0
100:
ldp x2,x3,[sp],16 // restaur 2 registers
ldp x1,lr,[sp],16 // restaur 2 registers
ret // return to address lr x30
/******************************************************************/
/* fibonacci function */
/******************************************************************/
/* x0 contains the Y combinator address */
/* x1 contains the number structure */
fibFunc:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
stp x4,x5,[sp,-16]! // save registers
mov x2,x0 // save Y combinator address
ldr x0,[x1,#func_num] // load number
cmp x0,#1 // > 1 ?
bgt 1f // yes
mov x0,#1 // create structure number value 1
bl numFunc
b 100f
1:
mov x3,x0 // save number
sub x0,x0,#1 // decrement number
bl numFunc // and create new structure number
cmp x0,#-1 // allocation error ?
beq 100f
mov x1,x0 // new structure number -> param 1
ldr x0,[x2,#func_f_] // load function address to execute
bl callFunc // call
ldr x4,[x0,#func_num] // load new result
sub x0,x3,#2 // new number - 2
bl numFunc // and create new structure number
cmp x0,#-1 // allocation error ?
beq 100f
mov x1,x0 // new structure number -> param 1
ldr x0,[x2,#func_f_] // load function address to execute
bl callFunc // call
ldr x1,[x0,#func_num] // load new result
add x0,x1,x4 // add two results
bl numFunc // and create new structure number
// and return her address in x0
100:
ldp x4,x5,[sp],16 // restaur 2 registers
ldp x2,x3,[sp],16 // restaur 2 registers
ldp x1,lr,[sp],16 // restaur 2 registers
ret // return to address lr x30
/******************************************************************/
/* call function */
/******************************************************************/
/* x0 contains the address of the function */
/* x1 contains the address of the function 1 */
callFunc:
stp x2,lr,[sp,-16]! // save registers
ldr x2,[x0,#func_fn] // load function address to execute
blr x2 // and call it
ldp x2,lr,[sp],16 // restaur 2 registers
ret // return to address lr x30
/******************************************************************/
/* create Y combinator function */
/******************************************************************/
/* x0 contains the address of the function */
YFunc:
stp x1,lr,[sp,-16]! // save registers
mov x1,#0
bl newFunc
cmp x0,#-1 // allocation error ?
beq 100f
str x0,[x0,#func_f_] // store function and return in x0
100:
ldp x1,lr,[sp],16 // restaur 2 registers
ret // return to address lr x30
/******************************************************************/
/* create structure number function */
/******************************************************************/
/* x0 contains the number */
numFunc:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
mov x2,x0 // save number
mov x0,#0 // function null
mov x1,#0 // function null
bl newFunc
cmp x0,#-1 // allocation error ?
beq 100f
str x2,[x0,#func_num] // store number in new structure
100:
ldp x2,x3,[sp],16 // restaur 2 registers
ldp x1,lr,[sp],16 // restaur 2 registers
ret // return to address lr x30
/******************************************************************/
/* new function */
/******************************************************************/
/* x0 contains the function address */
/* x1 contains the function address 1 */
newFunc:
stp x1,lr,[sp,-16]! // save registers
stp x3,x4,[sp,-16]! // save registers
stp x5,x8,[sp,-16]! // save registers
mov x4,x0 // save address
mov x5,x1 // save adresse 1
// allocation place on the heap
mov x0,#0 // allocation place heap
mov x8,BRK // call system 'brk'
svc #0
mov x6,x0 // save address heap for output string
add x0,x0,#func_fin // reservation place one element
mov x8,BRK // call system 'brk'
svc #0
cmp x0,#-1 // allocation error
beq 100f
mov x0,x6
str x4,[x0,#func_fn] // store address
str x5,[x0,#func_f_]
str xzr,[x0,#func_num] // store zero to number
100:
ldp x5,x8,[sp],16 // restaur 2 registers
ldp x3,x4,[sp],16 // restaur 2 registers
ldp x1,lr,[sp],16 // restaur 2 registers
ret // return to address lr x30
/********************************************************/
/* File Include fonctions */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"
</lang>
=={{header|ALGOL 68}}==
Line 39 ⟶ 317:
=={{header|AppleScript}}==
AppleScript is not
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 '
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.
<lang AppleScript>
on |Y|(f)
script
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|(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</lang>
{{Out}}
<lang AppleScript>{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}}</lang>
=={{header|ARM Assembly}}==
{{works with|as|Raspberry Pi}}
<lang ARM Assembly>
/* 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
adr r0,facFunc @ function factorielle address
bl YFunc @ create Ycombinator
mov r5,r0 @ save Ycombinator
ldr r0,iAdrszFactorielle @ display message
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
mov r0,r5 @ Ycombinator address
bl callFunc @ call
ldr r0,[r0,#func_num] @ load result
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 *************/
adr r0,fibFunc @ function factorielle address
bl YFunc @ create Ycombinator
mov r5,r0 @ save Ycombinator
ldr r0,iAdrszFibonacci @ display message
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
mov r0,r5 @ Ycombinator address
bl callFunc @ call
ldr r0,[r0,#func_num] @ load result
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
ldr r0,iAdrszMessError
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
iAdrszMessStartPgm: .int szMessStartPgm
iAdrszMessEndPgm: .int szMessEndPgm
iAdrszFactorielle: .int szFactorielle
iAdrszFibonacci: .int szFibonacci
iAdrszMessError: .int szMessError
iAdrszCarriageReturn: .int szCarriageReturn
iAdrszMessResult: .int szMessResult
iAdrsValue: .int sValue
/******************************************************************/
/* 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
ldr r0,[r1,#func_num] @ load number
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
ldr r0,[r2,#func_f_] @ load function address to execute
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
ldr r0,[r1,#func_num] @ load number
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
ldr r0,[r2,#func_f_] @ load function address to execute
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
ldr r0,[r2,#func_f_] @ load function address to execute
bl callFunc @ call
ldr r1,[r0,#func_num] @ load new result
add r0,r1,r4 @ add two results
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
ldr r2,[r0,#func_fn] @ load function address to execute
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 r4,r0 @ save address
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 r4,[r0,#func_fn] @ store address
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"
</lang>
{{output}}
<pre>
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.
</pre>
=={{header|ATS}}==
<lang ATS>
(* ****** ****** *)
//
#include "share/atspre_staload.hats"
//
(* ****** ****** *)
//
fun
myfix
{a:type}
(
f: lazy(a) -<cloref1> a
) : lazy(a) = $delay(f(myfix(f)))
//
val
fact =
myfix{int-<cloref1>int}
(
lam(ff) => lam(x) => if x > 0 then x * !ff(x-1) else 1
)
(* ****** ****** *)
//
implement main0 () = println! ("fact(10) = ", !fact(10))
//
(* ****** ****** *)
</lang>
=={{header|BlitzMax}}==
Line 351 ⟶ 987:
typedef struct func_t *func;
typedef struct func_t {
func (*
func _;
int num;
} func_t;
Line 357 ⟶ 994:
func new(func(*f)(func, func), func _) {
func x = malloc(sizeof(func_t));
x->
x->_ = _; /* closure, sort of */
x->num = 0;
Line 363 ⟶ 1,000:
}
func call(func f, func
return f->
}
func Y(func(*f)(func, func)) {
func
g->_ = g;
return g;
Line 382 ⟶ 1,016:
}
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
}
Line 421 ⟶ 1,048:
return 0;
}
</lang>
{{out}}
Line 427 ⟶ 1,055:
fib: 1 2 3 5 8 13 21 34 55</pre>
=={{header|C sharp
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 <code>YCombinator</code> class. Both it and the <code>RecursiveFunc</code> type thus "inherit" the type parameters of the containing class—there effectively exists a separate specialized copy of both for each generic instantiation of <code>YCombinator</code>.
''Note: in the code, <code>Func<T, TResult></code> is a delegate type (the CLR equivalent of a function pointer) that has a parameter of type <code>T</code> and return type of <code>TResult</code>. See [[Higher-order functions#C#]] or [https://docs.microsoft.com/en-us/dotnet/standard/delegates-lambdas the documentation] for more information.''
<lang csharp>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 fib = YCombinator<int, int>.Fix(f =>
Console.WriteLine(fib(10));
}
}
</lang>
{{out}}
<pre>3628800
55</pre>
Alternatively, with a non-generic holder class (note that <code>Fix</code> is now a method, as properties cannot be generic):
<lang csharp>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)));
}</lang>
Using the late-binding offered by <code>dynamic</code> to eliminate the recursive type:
<lang csharp>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))));
}</lang>
The usual version using recursion, disallowed by the task (implemented as a generic method):
<lang csharp>static class YCombinator
{
static Func<T, TResult> Fix<T, TResult>(Func<Func<T, TResult>, Func<T, TResult>> f) => x => f(Fix(f))(x);
}</lang>
===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++]====
<code>std::function<TResult(T)></code> in C++ corresponds to <code>Func<T, TResult></code> in C#.
'''Verbatim'''
<lang csharp>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;
}
}</lang>
'''Semi-idiomatic'''
<lang csharp>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));
}
}</lang>
====[http://rosettacode.org/mw/index.php?oldid=287744#Ceylon Ceylon]====
<code>TResult(T)</code> in Ceylon corresponds to <code>Func<T, TResult></code> in C#.
Since C# does not have local classes, <code>RecursiveFunc</code> and <code>y1</code> are declared in a class of their own. Moving the type parameters to the class also prevents type parameter inference.
'''Verbatim'''
<lang csharp>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
}
}</lang>
'''Semi-idiomatic'''
<lang csharp>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));
}
}</lang>
====[http://rosettacode.org/mw/index.php?oldid=287744#Go Go]====
<code>func(T) TResult</code> in Go corresponds to <code>Func<T, TResult></code> in C#.
'''Verbatim'''
<lang csharp>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);
};
}
}</lang>
Recursive:
<lang csharp> static Func Y(FuncFunc f) {
return delegate (int x) {
return f(Y(f))(x);
};
}</lang>
'''Semi-idiomatic'''
<lang csharp>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);
}</lang>
Recursive:
<lang csharp> static Func Y(FuncFunc f) => x => f(Y(f))(x);</lang>
====[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 <code>Functions</code> class for providing a means of converting CLR delegates to objects that implement the <code>Function</code> and <code>RecursiveFunction</code> interfaces.
<lang csharp>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> {
readonly Func<T, R> _inner;
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));
}
}</lang>
'''"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].
<lang csharp>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> {
readonly RecursiveFunction<Function<A, B>> w;
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> {
readonly Function<int, int> f;
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> {
readonly Function<int, int> f;
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));
}
}</lang>
'''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.
<lang csharp>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 {
readonly RecursiveFunc w;
public __2(RecursiveFunc w) {
this.w = w;
}
public int apply(int x) {
return w.apply(w).apply(x);
}
}
readonly FuncFunc f;
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 {
readonly Func f;
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 {
readonly Func f;
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));
}
}</lang>
'''Modified/varargs (the last implementation in the Java section)'''
Since C# delegates cannot declare members, extension methods are used to simulate doing so.
<lang csharp>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(
f => VarargsFunction.upgrade(
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(
f => VarargsFunction.upgrade(
toBigInteger.andThen(
n => (IFormattable)((n.CompareTo(BigInteger.One) <= 0)
? 1
: n * BigInteger.Parse(f(n - BigInteger.One).ToString()))
)
)
);
VarargsFunction<IFormattable, IFormattable> ackermann = fixedPoint(
f => VarargsFunction.upgrade(
(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>>();
functions.Add("fibonacci", fibonacci);
functions.Add("factorial", factorial);
functions.Add("ackermann", ackermann);
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);
}
}</lang>
====[http://rosettacode.org/mw/index.php?oldid=287744#Swift Swift]====
<code>T -> TResult</code> in Swift corresponds to <code>Func<T, TResult></code> in C#.
'''Verbatim'''
The more idiomatic version doesn't look much different.
<lang csharp>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)}");
}
}</lang>
Without recursive type:
<lang csharp> 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);
}</lang>
Recursive:
<lang csharp> public static Func<In, Out> Y<In, Out>(Func<Func<In, Out>, Func<In, Out>> f) {
return x => f(Y(f))(x);
}</lang>
=={{header|C++}}==
Line 500 ⟶ 1,846:
}</lang>
{{out}}
<pre>
fib(10) = 55
fac(10) = 3628800
</pre>
{{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
<lang cpp>#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';
}</lang>
{{out}}
<pre>
fib(10) = 55
Line 571 ⟶ 1,949:
given Args satisfies Anything[]
=> flatten((Args args) => f(y3(f))(*args));</lang>
=={{header|Chapel}}==
Strict (non-lazy = non-deferred execution) languages will race with the usually defined Y combinator (call-by-name) so most implementations are the Z combinator which lack one Beta Reduction from the true Y combinator (they are call-by-value). Although one can inject laziness so as to make the true Y combinator work with strict languages, the following code implements the usual Z call-by-value combinator using records to represent closures as Chapel does not have First Class Functions that can capture bindings from outside their scope other than from global scope:
{{works with|Chapel version 1.24.1}}
<lang chapel>proc fixz(f) {
record InnerFunc {
const xi;
proc this(a) { return xi(xi)(a); }
}
record XFunc {
const fi;
proc this(x) { return fi(new InnerFunc(x)); }
}
const g = new XFunc(f);
return g(g);
}
record Facz {
record FacFunc {
const fi;
proc this(n: int): int {
return if n <= 1 then 1 else n * fi(n - 1); }
}
proc this(f) { return new FacFunc(f); }
}
record Fibz {
record FibFunc {
const fi;
proc this(n: int): int {
return if n <= 1 then n else fi(n - 2) + fi(n - 1); }
}
proc this(f) { return new FibFunc(f); }
}
const facz = fixz(new Facz());
const fibz = fixz(new Fibz());
writeln(facz(10));
writeln(fibz(10));</lang>
{{out}}
<pre>3628800
55</pre>
One can write a true call-by-name Y combinator by injecting one level of laziness or deferred execution at the defining function level as per the following code:
{{works with|Chapel version 1.24.1}}
<lang chapel>// this is the longer version...
/*
proc fixy(f) {
record InnerFunc {
const xi;
proc this() { return xi(xi); }
}
record XFunc {
const fi;
proc this(x) { return fi(new InnerFunc(x)); }
}
const g = new XFunc(f);
return g(g);
}
// */
// short version using direct recursion as Chapel has...
// 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...
proc fixy(f) {
record InnerFunc { const fi; proc this() { return fixy(fi); } }
return f(new InnerFunc(f));
}
record Facy {
record FacFunc {
const fi;
proc this(n: int): int {
return if n <= 1 then 1 else n * fi()(n - 1); }
}
proc this(f) { return new FacFunc(f); }
}
record Fiby {
record FibFunc {
const fi;
proc this(n: int): int {
return if n <= 1 then n else fi()(n - 2) + fi()(n - 1); }
}
proc this(f) { return new FibFunc(f); }
}
const facy = fixy(new Facy());
const fibz = fixy(new Fiby());
writeln(facy(10));
writeln(fibz(10));</lang>
The output is the same as the above.
=={{header|Clojure}}==
Line 603 ⟶ 2,079:
<lang lisp>(defn Y [f]
(#(% %) #(f (fn [& args] (apply (% %) args)))))</lang>
=={{header|CoffeeScript}}==
<lang coffeescript>Y = (f) -> g = f( (t...) -> g(t...) )</lang>
or
<lang coffeescript>Y = (f) -> ((h)->h(h))((h)->f((t...)->h(h)(t...)))</lang>
<lang coffeescript>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 )
</lang>
=={{header|Common Lisp}}==
<lang lisp>(defun Y (f)
((lambda (
(lambda (
(funcall f (lambda (&rest
(apply (funcall
(defun fac (
(
(lambda (n)
1
(* n (funcall f (1- n)))))))
n))
(defun fib (
(
(Y (
(
(
n 0 1))
? (mapcar
(1 2 6 24 120 720 5040 40320 362880 3628800))
? (mapcar
(1 1 2 3 5 8 13 21 34 55)</lang>
=={{header|D}}==
Line 671 ⟶ 2,149:
<pre>factorial: [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
ackermann(3, 5): 253</pre>
=={{header|Delphi}}==
Line 777 ⟶ 2,222:
Writeln ('Fac(10) = ', Fac (10));
end.</lang>
=={{header|Dhall}}==
Dhall is not a turing complete language, so there's no way to implement the real Y combinator. That being said, you can replicate the effects of the Y combinator to any arbitrary but finite recursion depth using the builtin function Natural/Fold, which acts as a bounded fixed-point combinator that takes a natural argument to describe how far to recurse.
Here's an example using Natural/Fold to define recursive definitions of fibonacci and factorial:
<lang Dhall>let const
: ∀(b : Type) → ∀(a : Type) → a → b → a
= λ(r : Type) → λ(a : Type) → λ(x : a) → λ(y : r) → x
let fac
: ∀(n : Natural) → Natural
= λ(n : Natural) →
let factorial =
λ(f : Natural → Natural → Natural) →
λ(n : Natural) →
λ(i : Natural) →
if Natural/isZero i then n else f (i * n) (Natural/subtract 1 i)
in Natural/fold
n
(Natural → Natural → Natural)
factorial
(const Natural Natural)
1
n
let fib
: ∀(n : Natural) → Natural
= λ(n : Natural) →
let fibFunc = Natural → Natural → Natural → Natural
let fibonacci =
λ(f : fibFunc) →
λ(a : Natural) →
λ(b : Natural) →
λ(i : Natural) →
if Natural/isZero i
then a
else f b (a + b) (Natural/subtract 1 i)
in Natural/fold
n
fibFunc
fibonacci
(λ(a : Natural) → λ(_ : Natural) → λ(_ : Natural) → a)
0
1
n
in [fac 50, fib 50]</lang>
The above dhall file gets rendered down to:
<lang Dhall>[ 30414093201713378043612608166064768844377641568960512000000000000
, 12586269025
]</lang>
=={{header|Déjà Vu}}==
{{trans|Python}}
<lang dejavu>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</lang>
{{out}}
<pre>720
13</pre>
=={{header|E}}==
Line 790 ⟶ 2,326:
? accum [] for i in 0..!10 { _.with(y(fib)(i)) }
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]</lang>
=={{header|EchoLisp}}==
<lang scheme>
;; 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
</lang>
=={{header|Eero}}==
Line 837 ⟶ 2,400:
{{out}}
<pre>(479001600,144)</pre>
=={{header|Elena}}==
{{trans|Smalltalk}}
ELENA 4.x :
<lang elena>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));
}</lang>
{{out}}
<pre>
fib(10)=55
fact(10)=3628800
</pre>
=={{header|Elixir}}==
Line 852 ⟶ 2,440:
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
</lang>
=={{header|Elm}}==
This is similar to the Haskell solution below, but the first `fixz` is a strict fixed-point combinator lacking one beta reduction as compared to the Y-combinator; the second `fixy` injects laziness using a "thunk" (a unit argument function whose return value is deferred until the function is called/applied).
Note: the Fibonacci sequence is defined to start with zero or one, with the first exactly the same but with a zero prepended; these Fibonacci calculations use the second definition.
<lang elm>module Main exposing ( main )
import Html exposing ( Html, text )
-- As with most of the strict (non-deferred or non-lazy) languages,
-- this is the Z-combinator with the additional value parameter...
-- wrap type conversion to avoid recursive type definition...
type Mu a b = Roll (Mu a b -> a -> b)
unroll : Mu a b -> (Mu a b -> a -> b) -- unwrap it...
unroll (Roll x) = x
-- note lack of beta reduction using values...
fixz : ((a -> b) -> (a -> b)) -> (a -> b)
fixz f = let g r = f (\ v -> unroll r r v) in g (Roll g)
facz : Int -> Int
-- facz = fixz <| \ f n -> if n < 2 then 1 else n * f (n - 1) -- inefficient recursion
facz = fixz (\ f n i -> if i < 2 then n else f (i * n) (i - 1)) 1 -- efficient tailcall
fibz : Int -> Int
-- fibz = fixz <| \ f n -> if n < 2 then n else f (n - 1) + f (n - 2) -- inefficient recursion
fibz = fixz (\ fn f s i -> if i < 2 then f else fn s (f + s) (i - 1)) 1 1 -- efficient tailcall
-- by injecting laziness, we can get the true Y-combinator...
-- as this includes laziness, there is no need for the type wrapper!
fixy : ((() -> a) -> a) -> a
fixy f = f <| \ () -> fixy f -- direct function recursion
-- the above is not value recursion but function recursion!
-- fixv f = let x = f x in x -- not allowed by task or by Elm!
-- we can make Elm allow it by injecting laziness...
-- fixv f = let x = f () x in x -- but now value recursion not function recursion
facy : Int -> Int
-- facy = fixy <| \ f n -> if n < 2 then 1 else n * f () (n - 1) -- inefficient recursion
facy = fixy (\ f n i -> if i < 2 then n else f () (i * n) (i - 1)) 1 -- efficient tailcall
fiby : Int -> Int
-- fiby = fixy <| \ f n -> if n < 2 then n else f () (n - 1) + f (n - 2) -- inefficient recursion
fiby = fixy (\ fn f s i -> if i < 2 then f else fn () s (f + s) (i - 1)) 1 1 -- efficient tailcall
-- something that can be done with a true Y-Combinator that
-- can't be done with the Z combinator...
-- given an infinite Co-Inductive Stream (CIS) defined as...
type CIS a = CIS a (() -> CIS a) -- infinite lazy stream!
mapCIS : (a -> b) -> CIS a -> CIS b -- uses function to map
mapCIS cf cis =
let mp (CIS head restf) = CIS (cf head) <| \ () -> mp (restf()) in mp cis
-- now we can define a Fibonacci stream as follows...
fibs : () -> CIS Int
fibs() = -- two recursive fix's, second already lazy...
let fibsgen = fixy (\ fn (CIS (f, s) restf) ->
CIS (s, f + s) (\ () -> fn () (restf())))
in fixy (\ cisthnk -> fibsgen (CIS (0, 1) cisthnk))
|> mapCIS (\ (v, _) -> v)
nCISs2String : Int -> CIS a -> String -- convert n CIS's to String
nCISs2String n cis =
let loop i (CIS head restf) rslt =
if i <= 0 then rslt ++ " )" else
loop (i - 1) (restf()) (rslt ++ " " ++ Debug.toString head)
in loop n cis "("
-- unfortunately, if we need CIS memoization so as
-- to make a true lazy list, Elm doesn't support it!!!
main : Html Never
main =
String.fromInt (facz 10) ++ " " ++ String.fromInt (fibz 10)
++ " " ++ String.fromInt (facy 10) ++ " " ++ String.fromInt (fiby 10)
++ " " ++ nCISs2String 20 (fibs())
|> text</lang>
{{out}}
<pre>3628800 55 3628800 55 ( 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 )</pre>
=={{header|Erlang}}==
Line 871 ⟶ 2,543:
=={{header|F Sharp|F#}}==
<lang fsharp>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</lang>
{{output}}
<pre>3628800
55</pre>
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:
<lang fsharp>// 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 functionb with added deferred execution...
// val fac : (int -> BigInteger) = <fun>
// same efficient version of Fibonacci function 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
// given the following definition for an infinite Co-Inductive Stream (CIS)...
type CIS<'a> = CIS of 'a * (unit -> CIS<'a>) // ' fix formatting
// Using a double Y-Combinator recursion...
// defines a continuous stream of Fibonacci numbers; there are other simpler ways,
// this way implements recursion by using the Y-combinator, although it is
// much slower than other ways due to the many additional function calls,
// it demonstrates something that can't be done with the Z-combinator...
let fibs() =
let fbsgen = fix (fun fnc (CIS((f, s), rest)) ->
CIS((s, f + s), fun() -> fnc () <| rest()))
Seq.unfold (fun (CIS((v, _), rest)) -> Some(v, rest()))
<| fix (fun cis -> fbsgen (CIS((1I, 0I), cis))) // cis is a lazy thunk!
[<EntryPoint>]
let main argv =
fac 10 |> printfn "%A" // prints 3628800
fib 10 |> printfn "%A" // prints 55
fibs() |> Seq.take 20 |> Seq.iter (printf "%A ")
printfn ""
0 // return an integer exit code</lang>
{{output}}
<pre>3628800
55
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 </pre>
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:
<lang fsharp>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.</lang>
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.
=={{header|Factor}}==
Line 931 ⟶ 2,676:
> "Factorial 10: ", YFac(10)
> "Fibonacci 10: ", YFib(10)
</lang>
=={{header|Forth}}==
<lang 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 ;</lang>
Samples:
<lang Forth>\ 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
</lang>
Line 1,074 ⟶ 2,843:
=={{header|Haskell}}==
The obvious definition of the Y combinator <code>(\f-> (\x -> f (x x)) (\x-> f (x x)))</code> cannot be used in Haskell because it contains an infinite recursive type (<code>a = a -> b</code>). Defining a data type (Mu) allows this recursion to be broken.
<lang haskell>newtype Mu a = Roll
{ unroll :: Mu a -> a }
fix :: (a -> a) -> a
fix =
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()
]</lang>
The usual version uses recursion on a binding, disallowed by the task, to define the <code>fix</code> itself; but the definitions produced by this <code>fix</code> 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:
<lang haskell>-- 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 $
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
main =
mapM_
print
[ map fac [1 .. 20]
, map fib [1 .. 20]
, take 20 fibs()
]</lang>
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.
=={{header|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.
<lang j>Y=. '('':''<@;(1;~":0)<@;<@((":0)&;))'(2 : 0 '')
(1 : (m,'u'))(1 : (m,'''u u`:6('',(5!:5<''u''),'')`:6 y'''))(1 :'u u`:6')
)
</lang>
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:
<lang j> 'if. * y do. y * u <: y else. 1 end.' Y 10 NB. Factorial
3628800
'(u@:<:@:<: + u@:<:)^:(1 < ])' Y 10 NB. Fibonacci
55</lang>
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):
<lang j> 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</lang>
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),
<lang j>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')</lang>
The following are examples of anonymous dyadic and ambivalent recursions,
<lang j> 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</lang>
J supports directly anonymous tacit recursion via the verb $: and for tacit recursions, XY is equivalent to the adverb,
<lang j>YX=. (1 :'('':''<@;(1;~":0)<@;<@((":0)&;))u')($:`)(`:6)</lang>
===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):
<lang j>Y=. ((((&>)/)((((^:_1)b.)(`(<'0';_1)))(`:6)))(&([ 128!:2 ,&<)))</lang>
The factorial and Fibonacci examples:
<lang j> u=. [ NB. Function (left)
n=. ] NB. Argument (right)
sr=. [
fac=. (1:`(n * u sr n - 1:)) @. (0
fac f. Y 10
3628800
Fib=. ((u sr n - 2:) + u sr n - 1:) ^: (1
Fib f. Y 10
55</lang>
The
<lang j> fac f. Y NB.
'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 < ])</lang>
A structured derivation of Y follows:
<lang j> 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 ,&<)))</lang>
===Explicit alternate implementation===
Another approach:
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.
)</lang>
Example use:
<lang J>
362880
almost_fibonacci Y 9
34
0 1 1 2 3 5 8 13 21 34</lang>
Or, if you would prefer to not have a dependency on the definition of Defer, an equivalent expression would be:
<lang J>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
)</lang>
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:
<lang J> almost_factorial f. Y 10
3628800</lang>
=={{header|Java}}==
Line 1,232 ⟶ 3,102:
(n <= 1)
? 1
: (n * f.apply(n - 1))
);
Line 1,548 ⟶ 3,418:
fact=>(n,m=1)=>n<2?m:fact(n-1,n*m);
tailfact= // Tail call version of factorial function
Y(
ECMAScript 2015 (ES6) also permits a really compact polyvariadic variant for mutually recursive functions:
<lang javascript>let
Line 1,559 ⟶ 3,429:
(even,odd)=>n=>(n===0)||odd(n-1),
(even,odd)=>n=>(n!==0)&&even(n-1));</lang>
A minimalist version:
<lang javascript>var Y = f => (x => x(x))(y => f(x => y(y)(x)));
var fac = Y(f => n => n > 1 ? n * f(n-1) : 1);</lang>
=={{header|Joy}}==
Line 1,567 ⟶ 3,442:
=={{header|Julia}}==
<lang julia>
julia> """
# Y combinator
* `λf. (λx. f (x x)) (λx. f (x x))`
Y = f -> (x -> x(x))(y -> f((t...) -> y(y)(t...)))
</lang>
Line 1,590 ⟶ 3,453:
<lang julia>
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)
10-element
1
2
Line 1,609 ⟶ 3,472:
3628800
julia> [Y(fib)
10-element
1
1
Line 1,621 ⟶ 3,484:
34
55
</lang>
=={{header|Kitten}}==
<lang 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
</lang>
=={{header|Klingphix}}==
<lang Klingphix>:fac
dup 1 great [dup 1 sub fac mult] if
;
:fib
dup 1 great [dup 1 sub fib swap 2 sub fib add] if
;
:test
print ": " print
10 [over exec print " " print] for
nl
;
@fib "fib" test
@fac "fac" test
"End " input</lang>
{{out}}
<pre>fib: 1 1 2 3 5 8 13 21 34 55
fac: 1 2 6 24 120 720 5040 40320 362880 3628800
End</pre>
=={{header|Kotlin}}==
<lang scala>// 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()
}</lang>
{{out}}
<pre>
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
</pre>
=={{header|Lambdatalk}}==
Tested in http://lambdaway.free.fr/lambdawalks/?view=Ycombinator
<lang Scheme>
1) defining the Ycombinator
{def Y {lambda {:f} {:f :f}}}
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
</lang>
Line 1,637 ⟶ 3,609:
factorial, fibs = Y(almostfactorial), Y(almostfibs)
print(factorial(7))</lang>
=={{header|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.
<lang M2000 Interpreter>
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
</lang>
<lang M2000 Interpreter>
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
</lang>
=={{header|MANOOL}}==
Here one additional technique is demonstrated: the Y combinator is applied to a function ''during compilation'' due to the <code>$</code> operator, which is optional:
<lang MANOOL>
{ {extern "manool.org.18/std/0.3/all"} in
: let { Y = {proc {F} as {proc {X} as X[X]}[{proc {X} with {F} as F[{proc {Y} with {X} as X[X][Y]}]}]} } in
{ for { N = Range[10] } do
: (WriteLine) Out; N "! = "
{Y: proc {Rec} as {proc {N} with {Rec} as: if N == 0 then 1 else N * Rec[N - 1]}}$[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]}}$[N]
}
}
</lang>
Using less syntactic sugar:
<lang MANOOL>
{ {extern "manool.org.18/std/0.3/all"} in
: let { Y = {proc {F} as {proc {X} as X[X]}[{proc {F; X} as F[{proc {X; Y} as X[X][Y]}.Bind[X]]}.Bind[F]]} } in
{ for { N = Range[10] } do
: (WriteLine) Out; N "! = "
{Y: proc {Rec} as {proc {Rec; N} as: if N == 0 then 1 else N * Rec[N - 1]}.Bind[Rec]}$[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]
}
}
</lang>
{{output}}
<pre>
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
</pre>
=={{header|Maple}}==
Line 1,652 ⟶ 3,749:
<lang Mathematica>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]] &]];</lang>
=={{header|Moonscript}}==
<lang 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</lang>
=={{header|Nim}}==
<lang nim># The following is implemented for a strict language as a Z-Combinator;
# Z-combinators differ from Y-combinators in lacking one Beta reduction of
# the extra `T` argument to the function to be recursed...
import sugar
proc fixz[T, TResult](f: ((T) -> TResult) -> ((T) -> TResult)): (T) -> TResult =
type RecursiveFunc = object # any entity that wraps the recursion!
recfnc: ((RecursiveFunc) -> ((T) -> TResult))
let g = (x: RecursiveFunc) => f ((a: T) => x.recfnc(x)(a))
g(RecursiveFunc(recfnc: g))
let facz = fixz((f: (int) -> int) =>
((n: int) => (if n <= 1: 1 else: n * f(n - 1))))
let fibz = fixz((f: (int) -> int) =>
((n: int) => (if n < 2: n else: f(n - 2) + f(n - 1))))
echo facz(10)
echo fibz(10)
# by adding some laziness, we can get a true Y-Combinator...
# note that there is no specified parmater(s) - truly fix point!...
#[
proc fixy[T](f: () -> T -> T): T =
type RecursiveFunc = object # any entity that wraps the recursion!
recfnc: ((RecursiveFunc) -> T)
let g = ((x: RecursiveFunc) => f(() => x.recfnc(x)))
g(RecursiveFunc(recfnc: g))
# ]#
# same thing using direct recursion as Nim has...
# 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...
proc fixy[T](f: () -> T -> T): T = f(() => (fixy(f)))
# these are dreadfully inefficient as they becursively build stack!...
let facy = fixy((f: () -> (int -> int)) =>
((n: int) => (if n <= 1: 1 else: n * f()(n - 1))))
let fiby = fixy((f: () -> (int -> int)) =>
((n: int) => (if n < 2: n else: f()(n - 2) + f()(n - 1))))
echo facy 10
echo fiby 10
# something that can be done with the Y-Combinator that con't be done with the Z...
# given the following Co-Inductive Stream (CIS) definition...
type CIS[T] = object
head: T
tail: () -> CIS[T]
# Using a double Y-Combinator recursion...
# defines a continuous stream of Fibonacci numbers; there are other simpler ways,
# this way implements recursion by using the Y-combinator, although it is
# much slower than other ways due to the many additional function calls,
# it demonstrates something that can't be done with the Z-combinator...
iterator fibsy: int {.closure.} = # two recursions...
let fbsfnc: (CIS[(int, int)] -> CIS[(int, int)]) = # first one...
fixy((fnc: () -> (CIS[(int,int)] -> CIS[(int,int)])) =>
((cis: CIS[(int,int)]) => (
let (f,s) = cis.head;
CIS[(int,int)](head: (s, f + s), tail: () => fnc()(cis.tail())))))
var fbsgen: CIS[(int, int)] = # second recursion
fixy((cis: () -> CIS[(int,int)]) => # cis is a lazy thunk used directly below!
fbsfnc(CIS[(int,int)](head: (1,0), tail: cis)))
while true: yield fbsgen.head[0]; fbsgen = fbsgen.tail()
let fibs = fibsy
for _ in 1 .. 20: stdout.write fibs(), " "
echo()</lang>
{{out}}
<pre>3628800
55
3628800
55
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181</pre>
At least this last example version building a sequence of Fibonacci numbers doesn't build stack as it the use of CIS's means that it is a type of continuation passing/trampolining style.
Note that these would likely never be practically used in Nim as the language offers both direct variable binding recursion and recursion on proc's as well as other forms of recursion so it would never normally be necessary. Also note that these implementations not using recursive bindings on variables are "non-sharing" fix point combinators, whereas sharing is sometimes desired/required and thus recursion on variable bindings is required.
=={{header|Objective-C}}==
Line 1,746 ⟶ 3,932:
With recursion into Y definition (so non stateless Y) :
<lang Oforth>: Y(f)
Without recursion into Y definition (stateless Y).
<lang Oforth>: X(me, f)
: Y(f)
Usage :
<lang Oforth>: almost-fact(
: almost-fib(
: almost-Ackermann(
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 </lang>
=={{header|Order}}==
Line 1,848 ⟶ 4,032:
}</lang>
=={{header|
{{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:<br>
"It doesn't matter whether you use cos or (lambda (x) (cos x)) as your cosine function; they will both do the same thing."<br>
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]]
<lang Phix>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")</lang>
{{out}}
<pre>
fib: 1 1 2 3 5 8 13 21 34 55
</pre>
=={{header|Phixmonti}}==
<lang Phixmonti>0 var subr
def fac
dup 1 > if
dup 1 - subr exec *
endif
enddef
def fib
dup 1 > if
dup 1 - subr exec swap 2 - subr exec +
endif
enddef
def test
print ": " print
var subr
10 for
subr exec print " " print
endfor
nl
enddef
getid fac "fac" test
getid fib "fib" test</lang>
=={{header|PHP}}==
Line 2,085 ⟶ 4,326:
$Z.InvokeReturnAsIs($fac).InvokeReturnAsIs(5)
$Z.InvokeReturnAsIs($fib).InvokeReturnAsIs(5)</lang>
GetNewClosure() was added in Powershell 2, allowing for an implementation without metaprogramming. The following was tested with Powershell 4.
<lang PowerShell>$Y = {
param ($f)
{
param ($x)
$f.InvokeReturnAsIs({
param ($y)
$x.InvokeReturnAsIs($x).InvokeReturnAsIs($y)
}.GetNewClosure())
}.InvokeReturnAsIs({
param ($x)
$f.InvokeReturnAsIs({
param ($y)
$x.InvokeReturnAsIs($x).InvokeReturnAsIs($y)
}.GetNewClosure())
}.GetNewClosure())
}
$fact = {
param ($f)
{
param ($n)
if ($n -eq 0) { 1 } else { $n * $f.InvokeReturnAsIs($n - 1) }
}.GetNewClosure()
}
$fib = {
param ($f)
{
param ($n)
if ($n -lt 2) { 1 } else { $f.InvokeReturnAsIs($n - 1) + $f.InvokeReturnAsIs($n - 2) }
}.GetNewClosure()
}
$Y.invoke($fact).invoke(5)
$Y.invoke($fib).invoke(5)</lang>
=={{header|Prolog}}==
Line 2,141 ⟶ 4,434:
The usual version using recursion, disallowed by the task:
<lang python>Y = lambda f: lambda *args: f(Y(f))(*args)</lang>
<lang python>Y = lambda b: ((lambda f: b(lambda *x: f(f)(*x)))((lambda f: b(lambda *x: f(f)(*x)))))</lang>
=={{header|Q}}==
<lang Q>> Y: {{x x} {({y {(x x) y} x} y) x} x}
> fac: {{$[y<2; 1; y*x y-1]} x}
> (Y fac) 6
720j
</lang>
=={{header|R}}==
Line 2,171 ⟶ 4,473:
The lazy implementation
<lang racket>#lang lazy
(define Y (λ (f) ((λ (x) (f (x x))) (λ (x) (f (x x))))))
(define Fact
(Y (λ (fact) (λ (n) (if (zero? n) 1 (* n (fact (- n 1))))))))
(define Fib
(Y (λ (fib) (λ (n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2))))))))</lang>
{{out}}
Line 2,191 ⟶ 4,491:
Strict realization:
<lang racket>#lang racket
(define Y (λ (b) ((λ (f) (b (λ (x) ((f f) x))))
Definitions of <tt>Fact</tt> and <tt>Fib</tt> functions will be the same as in Lazy Racket.
Finally, a definition in Typed Racket is a little difficult as in other statically typed languages:
<lang racket>#lang typed/racket
(: make-recursive : (All (S T) ((S -> T) -> (S -> T)) -> (S -> T)))
Line 2,217 ⟶ 4,514:
(* n (fact (- n 1))))))))
(fact 5)</lang>
=={{header|Raku}}==
(formerly Perl 6)
<lang perl6>sub Y (&f) { sub (&x) { x(&x) }( sub (&y) { f(sub ($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;</lang>
{{out}}
<pre>(1 1 2 6 24 120 720 5040 40320 362880)
(0 1 1 2 3 5 8 13 21 34)</pre>
Note that Raku doesn't actually need a Y combinator because you can name anonymous functions from the inside:
<lang perl6>say .(10) given sub (Int $x) { $x < 2 ?? 1 !! $x * &?ROUTINE($x - 1); }</lang>
=={{header|REBOL}}==
Line 2,227 ⟶ 4,537:
=={{header|REXX}}==
Programming note: '''length''', '''reverse''', '''sign''', '''trunc''', '''b2x''', '''d2x''', and '''x2d''' are REXX BIFs ('''B'''uilt '''I'''n '''F'''unctions).
numeric digits 1000 /*allow big numbers. */
say ' fib' Y(fib (50) )
say ' fib' Y(fib (12 11 10 9 8 7 6 5 4 3 2 1 0
say ' fact' Y(fact (60) ) /*single
say ' fact' Y(fact (0 1 2 3 4 5 6 7 8 9 10 11) ) /*single
say ' Dfact' Y(dfact (4 5 6 7 8 9 10 11 12 13
say ' Tfact' Y(tfact (4 5 6 7 8 9 10 11 12 13
say ' Qfact' Y(qfact (4 5 6 7 8 40) ) /*quadruple
say ' length' Y(length (when for to where whenceforth) ) /*lengths of words. */
say 'reverse' Y(reverse (
say '
say ' b2x' Y(b2x (1 10 11 100 1000 10000 11111 ) ) /*converts BIN──►HEX. */
say ' d2x' Y(d2x (8 9 10 11 12 88 89 90 91 6789) ) /*converts DEC──►HEX. */
say ' x2d' Y
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Y: parse arg Y _;
/*──────────────────────────────────────────────────────────────────────────────────────*/
fib:
/*──────────────────────────────────────────────────────────────────────────────────────*/
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 !</lang>
{{out|output|text= when using the internal default input:}}
<pre>
fib 12586269025
Line 2,261 ⟶ 4,575:
Qfact 4 5 12 21 32 3805072588800
length 4 3 2 5 11
reverse
sign -1 0 1
trunc -7 12 3 6 78
b2x 1 2 3 4 8 10 1F
d2x 8 9 A B C 58 59 5A 5B 1A85
x2d 8 9 16 17 18 136 137 144 145 26505
</pre>
Line 2,315 ⟶ 4,633:
=={{header|Rust}}==
{{works with|Rust|1.44.1 stable}}
<lang rust>
//! A simple implementation of the Y Combinator:
//! λf.(λx.xx)(λx.f(xx))
//! <=> λf.(λx.f(xx))(λx.f(xx))
/// A function type that takes its own type as an input is an infinite recursive type.
/// We introduce the "Apply" trait, which 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.
trait Apply<T, R> {
fn apply(&self, f: &dyn Apply<T, R>, t: T) -> R;
}
/// If we were to pass in self as f, we get:
/// λf.λt.sft
/// => λs.λt.sst [s/f]
/// => λs.ss
impl<T, R, F> Apply<T, R> for F where F: Fn(&dyn Apply<T, R>, T) -> R {
fn apply(&self, f: &dyn Apply<T, R>, t: T) -> R {
self(f, t)
}
}
/// (λt(λx.(λy.xxy))(λx.(λy.f(λz.xxz)y)))t
/// => (λx.xx)(λx.f(xx))
/// => Yf
fn y<T, R>(f: impl Fn(&dyn Fn(T) -> R, T) -> R) -> impl Fn(T) -> R {
move |t| (&|x: &dyn Apply<T, R>, y| x.apply(x, y))
(&|x: &dyn Apply<T, R>, y| f(&|z| x.apply(x, z), y), t)
}
/// Factorial of n.
fn fac(n: usize) -> usize {
let almost_fac = |f: &dyn Fn(usize) -> usize, x| if x == 0 { 1 } else { x * f(x - 1) };
y(almost_fac)(n)
}
/// nth Fibonacci number.
fn fib(n: usize) -> usize {
let almost_fib = |f: &dyn Fn((usize, usize, usize)) -> usize, (a0, a1, x)|
match x {
0 => a0,
1 => a1,
_ => f((a1, a0 + a1, x - 1)),
};
y(almost_fib)((1, 1, n))
}
/// Driver function.
fn main() {
let n = 10;
println!("fac({}) = {}", n, fac(n));
println!("fib({}) = {}", n, fib(n));
}
</lang>
{{output}}
<pre>
fac(10) = 3628800
fib(10) = 89
</pre>
=={{header|Scala}}==
Credit goes to the thread in [https://web.archive.org/web/20160709050901/http://scala-blogs.org/2008/09/y-combinator-in-scala.html scala blog]
<lang scala>
def Y[A,
case class
def apply(w: W): A => B = wf(w)
}
val g: W => (A => B) = w => f(w(w))(_)
g(W(g))
}
</lang>
Example
<lang scala>
val fac: Int => Int = Y[Int, Int](f => i => if (i <= 0) 1 else f(i - 1) * i)
fac(6) //> res0: Int = 720
val fib: Int => Int = Y[Int, Int](f => i => if (i < 2) i else f(i - 1) + f(i - 2))
fib(6) //> res1: Int = 8
</lang>
=={{header|Scheme}}==
<lang scheme>(define Y ; (Y f) = (g g) where
(lambda (f) ; (g g) = (f (lambda a (apply (g g) a)))
((lambda (
(lambda (g)
(
;; 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 (
;; 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 (
(
(+ (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 (
(newline)</lang>
{{out}}
<pre>720
4517090495650391871408712937</pre>
If we were allowed to use recursion (with <code>Y</code> referring to itself by name in its body) we could define the equivalent to the above as
<lang scheme>(define Yr ; (Y f) == (f (lambda a (apply (Y f) a)))
(f (lambda a (apply (Yr f) a)))))</lang>
And another way is:
<lang scheme>(define Y2r
(lambda (f)
(lambda a (apply (f (Y2r f)) a))))</lang>
Which, non-recursively, is
<lang scheme>(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))))))</lang>
=={{header|Shen}}==
<lang 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)))</lang>
{{out}}
<pre>1
120
3628800</pre>
=={{header|Sidef}}==
<lang ruby>var y = ->(f) {->(g) {g(g)}(->(g) { f(->(*args) {g(g)(args...)})})}
var fac = ->(f) { ->(n) { n < 2
say 10.of { |i| y(fac)(i) }
var fib = ->(f) { ->(n) { n < 2
say 10.of { |i| y(fib)(i) }
{{out}}
<pre>
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34
</pre>
Line 2,456 ⟶ 4,867:
The usual version using recursion, disallowed by the task:
<lang sml>fun fix f x = f (fix f) x</lang>
=={{header|SuperCollider}}==
Like Ruby, SuperCollider needs an extra level of lambda-abstraction to implement the y-combinator. The z-combinator is straightforward:
<lang SuperCollider>// 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 ]
</lang>
=={{header|Swift}}==
Line 2,494 ⟶ 4,950:
return { x in f(Y(f))(x) }
}</lang>
=={{header|Tailspin}}==
<lang 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
</lang>
{{out}}
<pre>
factorial 5: 120
fibonacci 5: 5
</pre>
=={{header|Tcl}}==
Line 2,501 ⟶ 4,998:
This prints out 24, the factorial of 4:
<lang
(defun y (f)
[(
(op f
Both the <code>op</code> and <code>do</code> operators are a syntactic sugar for currying, in two different flavors. The forms within <code>do</code> that are symbols are evaluated in the normal Lisp-2 style and the first symbol can be an operator. Under <code>op</code>, 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 <code>do</code> stems from the fact that the operator is used for currying over special forms like <code>if</code> in the above example, where there is evaluation control. Operators can have side effects: they can "do" something. Consider <code>(do set a @1)</code> which yields a function of one argument which assigns that argument to <code>a</code>.
The compounded <code>@@...</code>
=={{header|Ursala}}==
Line 2,576 ⟶ 5,072:
my_fix "h" = "h" my_fix "h"</lang>
Note that this equation is solved using the next fixed point combinator in the hierarchy.
=={{header|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.
<lang vb>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</lang>{{out}}
<pre>fac
1 2 6 24 120 720 5040 40320 362880 3628800
fib
1 1 2 3 5 8 13 21 34 55 </pre>
=={{header|Verbexx}}==
<lang 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) };</lang>
=={{header|Vim Script}}==
Line 2,601 ⟶ 5,180:
echo Callx(Callx(g:Y, [g:fac]), [5])
echo map(range(10), 'Callx(Callx(Y, [fac]), [v:val])')
</lang>
Update: since Vim 7.4.2044 (or so...), the following can be used (the feature check was added with 7.4.2121):
<lang vim>
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)')
</lang>
Output:
Line 2,607 ⟶ 5,198:
=={{header|Wart}}==
<lang python># Better names due to Jim Weirich: http://vimeo.com/45140590
def (Y improver)
((fn(gen) gen.gen)
(fn(gen)
Line 2,625 ⟶ 5,217:
{{omit from|PureBasic}}
{{omit from|TI-89 BASIC}} <!-- no lambdas, no first-class functions except by name string -->
=={{header|Wren}}==
{{trans|Go}}
<lang ecmascript>var y = Fn.new { |f|
var g = Fn.new { |r| f.call { |x| r.call(r).call(x) } }
return g.call(g)
}
var almostFac = Fn.new { |f| Fn.new { |x| x <= 1 ? 1 : x * f.call(x-1) } }
var almostFib = Fn.new { |f| Fn.new { |x| x <= 2 ? 1 : f.call(x-1) + f.call(x-2) } }
var fac = y.call(almostFac)
var fib = y.call(almostFib)
System.print("fac(10) = %(fac.call(10))")
System.print("fib(10) = %(fib.call(10))")</lang>
{{out}}
<pre>
fac(10) = 3628800
fib(10) = 55
</pre>
=={{header|XQuery}}==
Line 2,642 ⟶ 5,257:
{{out}}
<lang XQuery>720 8</lang>
=={{header|Yabasic}}==
<lang 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")</lang>
=={{header|zkl}}==
| |||