Y combinator: Difference between revisions
| Line 3,742: | Line 3,742: | ||
<lang M2000 Interpreter> |
<lang M2000 Interpreter> |
||
Module Ycombinator { |
Module Ycombinator { |
||
\\ y() return value. no use of closure |
|||
y=lambda (g, x)->g(g, x) |
|||
Print y(lambda (g, n as decimal)->if(n=0->1, n*g(g, n-1)), 10)=3628800 ' true |
|||
Print y(lambda (g, n)->if(n<=1->n,g(g, n-1)+g(g, n-2)), 10)=55 ' true |
|||
\\ Using closure in y, y() return function |
|||
y=lambda (g)->lambda g (x) -> g(g, x) |
|||
fact=y((lambda (g, n as decimal)-> if(n=0->1, n*g(g, n-1)))) |
|||
Print fact(6)=720, fact(24)=620448401733239439360000@ |
|||
fib=y(lambda (g, n)->if(n<=1->n, g(g, n-1)+g(g, n-2))) |
|||
Print fib(10)=55 |
|||
} |
} |
||
Ycombinator |
Ycombinator |
||
</lang> |
</lang> |
||
| Line 3,759: | Line 3,759: | ||
<lang M2000 Interpreter> |
<lang M2000 Interpreter> |
||
Module Checkit { |
Module Checkit { |
||
Rem { |
|||
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 |
|||
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. |
|||
| ⚫ | |||
| ⚫ | |||
=lambda f (x)->f(f,x) |
|||
| ⚫ | |||
} |
|||
=lambda f (x)->f(f,x) |
|||
} |
|||
if n<2 then { |
|||
| ⚫ | |||
=1 |
|||
if n<2 then =1 else =n*m(m, n-1) |
|||
} else { |
|||
} |
|||
=n*m(m, n-1) |
|||
| ⚫ | |||
} |
|||
| ⚫ | |||
} |
|||
| ⚫ | |||
| ⚫ | |||
} |
|||
| ⚫ | |||
| ⚫ | |||
if n<=1 then { |
|||
| ⚫ | |||
=n |
|||
| ⚫ | |||
} else { |
|||
} |
|||
| ⚫ | |||
} |
|||
} |
|||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
Next i |
|||
} |
} |
||
Checkit |
Checkit |
||
// same but more compact |
|||
Module Checkit { |
|||
fac=lambda (f)->{=lambda f (x)->f(f,x)}(lambda (m, n)->{=if(n<2->1,n*m(m, n-1))}) |
|||
fib=lambda (f)->{=lambda f (x)->f(f,x)}(lambda (m, n)->{=if(n<=1->n,m(m, n-1)+m(m, n-2))}) |
|||
For i=1 to 10 |
|||
| ⚫ | |||
Next i |
|||
| ⚫ | |||
Checkit |
|||
Module CheckRecursion { |
Module CheckRecursion { |
||
fac=lambda (n) -> { |
|||
if n<2 then =1 else =n*Lambda(n-1) |
|||
} |
|||
=1 |
|||
fib=lambda (n) -> { |
|||
} else { |
|||
| ⚫ | |||
=n*Lambda(n-1) |
|||
} |
|||
} |
|||
| ⚫ | |||
} |
|||
| ⚫ | |||
if n<=1 then { |
|||
=n |
|||
} else { |
|||
| ⚫ | |||
} |
|||
} |
|||
For i=1 to 10 |
|||
Print fib(i), fac(i) |
|||
Next i |
|||
} |
} |
||
CheckRecursion |
CheckRecursion |
||
Revision as of 09:34, 3 March 2022
You are encouraged to solve this task according to the task description, using any language you may know.
In strict functional programming and the lambda calculus, functions (lambda expressions) don't have state and are only allowed to refer to arguments of enclosing functions.
This rules out the usual definition of a recursive function wherein a function is associated with the state of a variable and this variable's state is used in the body of the function.
The Y combinator is itself a stateless function that, when applied to another stateless function, returns a recursive version of the function.
The Y combinator is the simplest of the class of such functions, called fixed-point combinators.
- Task
Define the stateless Y combinator and use it to compute factorials and Fibonacci numbers from other stateless functions or lambda expressions.
- Cf
AArch64 Assembly
<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>
ALGOL 68
Note: This specimen retains the original Python coding style.
<lang algol68>BEGIN
MODE F = PROC(INT)INT; MODE Y = PROC(Y)F;
- compare python Y = lambda f: (lambda x: x(x)) (lambda y: f( lambda *args: y(y)(*args)))#
PROC y = (PROC(F)F f)F: ( (Y x)F: x(x)) ( (Y z)F: f((INT arg )INT: z(z)( arg )));
PROC fib = (F f)F: (INT n)INT: CASE n IN n,n OUT f(n-1) + f(n-2) ESAC;
FOR i TO 10 DO print(y(fib)(i)) OD
END</lang>
The version below works with Algol 68 Genie 2.8.4 tested with linux kernel release 5.16.11-200.fc35.x86_64
<lang algol68>BEGIN
- This version needs partial parameterisation in order to work #
- The commented code is JavaScript aka ECMAScript ES6 #
MODE F = PROC( INT ) INT ; MODE Y = PROC( Y ) F ;
- Y = f => ( y => y( y ) )( y => f( arg => y( y )( arg ) ) ) ; #
PROC y = ( PROC( F ) F pff ) F:
( ( Y y ) F: y( y ) )
(
(
( PROC( F ) F pff , Y y ) F:
pff( ( ( Y y , INT arg ) INT: y( y )( arg ) )( y , ) )
)( pff , )
)
- fibgen = fib => ( n => ( ( n === 0 ) ? 0 : ( n === 1 ) ? 1 : fib( n - 2 ) + fib( n - 1 ) ) ) ; #
PROC( F ) F fibgen = ( F fib ) F:
( ( F fib , INT n ) INT: CASE n IN 1 , 1 OUT fib( n - 2 ) + fib( n - 1 ) ESAC )( fib , )
- for ( let i = 1 ; i <= 12 ; i++) { console.log( " " +Y( fibgen )( i ) ) ; } #
INT nofibs = 12 ; print( ( "The first " , whole( nofibs , 0 ) , " fibonacci numbers." , newline ) ) ; FOR i TO nofibs DO
print( whole( y( fibgen )( i ) , -11 ) )
OD ; print( newline )
END</lang>
AppleScript
AppleScript is not particularly "functional" friendly. It can, however, support the Y combinator.
AppleScript does not have anonymous functions, but it does have anonymous objects. The code below implements the latter with the former (using a handler (i.e. function) named 'lambda' in each anonymous object).
Unfortunately, an anonymous object can only be created in its own statement ('script'...'end script' can not be in an expression). Thus, we have to apply Y to the automatic 'result' variable that holds the value of the previous statement.
The identifier used for Y uses "pipe quoting" to make it obviously distinct from the y used inside the definition. <lang AppleScript>-- Y COMBINATOR ---------------------------------------------------------------
on |Y|(f)
script
on |λ|(y)
script
on |λ|(x)
y's |λ|(y)'s |λ|(x)
end |λ|
end script
f's |λ|(result)
end |λ|
end script
result's |λ|(result)
end |Y|
-- TEST -----------------------------------------------------------------------
on run
-- Factorial
script fact
on |λ|(f)
script
on |λ|(n)
if n = 0 then return 1
n * (f's |λ|(n - 1))
end |λ|
end script
end |λ|
end script
-- Fibonacci
script fib
on |λ|(f)
script
on |λ|(n)
if n = 0 then return 0
if n = 1 then return 1
(f's |λ|(n - 2)) + (f's |λ|(n - 1))
end |λ|
end script
end |λ|
end script
{facts:map(|Y|(fact), enumFromTo(0, 11)), fibs:map(|Y|(fib), enumFromTo(0, 20))}
--> {facts:{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800},
--> fibs:{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,
-- 1597, 2584, 4181, 6765}}
end run
-- GENERIC FUNCTIONS FOR TEST -------------------------------------------------
-- map :: (a -> b) -> [a] -> [b] on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- enumFromTo :: Int -> Int -> [Int] on enumFromTo(m, n)
if n < m then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end enumFromTo
-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn</lang>
- Output:
<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>
ARM Assembly
<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:
Program start Function factorielle : Result value :1 Result value :2 Result value :6 Result value :24 Result value :120 Result value :720 Result value :5040 Result value :40320 Result value :362880 Result value :3628800 Function Fibonacci : Result value :1 Result value :2 Result value :3 Result value :5 Result value :8 Result value :13 Result value :21 Result value :34 Result value :55 Result value :89 Program normal end.
ATS
<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>
BlitzMax
BlitzMax doesn't support anonymous functions or classes, so everything needs to be explicitly named. <lang blitzmax>SuperStrict
'Boxed type so we can just use object arrays for argument lists Type Integer Field val:Int Function Make:Integer(_val:Int) Local i:Integer = New Integer i.val = _val Return i End Function End Type
'Higher-order function type - just a procedure attached to a scope
Type Func Abstract
Method apply:Object(args:Object[]) Abstract
End Type
'Function definitions - extend with fields as locals and implement apply as body Type Scope Extends Func Abstract Field env:Scope
'Constructor - bind an environment to a procedure Function lambda:Scope(env:Scope) Abstract
Method _init:Scope(_env:Scope) 'Helper to keep constructors small env = _env ; Return Self End Method End Type
'Based on the following definition:
'(define (Y f)
' (let ((_r (lambda (r) (f (lambda a (apply (r r) a))))))
' (_r _r)))
'Y (outer) Type Y Extends Scope Field f:Func 'Parameter - gets closed over
Function lambda:Scope(env:Scope) 'Necessary due to highly limited constructor syntax Return (New Y)._init(env) End Function
Method apply:Func(args:Object[]) f = Func(args[0]) Local _r:Func = YInner1.lambda(Self) Return Func(_r.apply([_r])) End Method End Type
'First lambda within Y Type YInner1 Extends Scope Field r:Func 'Parameter - gets closed over
Function lambda:Scope(env:Scope) Return (New YInner1)._init(env) End Function
Method apply:Func(args:Object[]) r = Func(args[0]) Return Func(Y(env).f.apply([YInner2.lambda(Self)])) End Method End Type
'Second lambda within Y Type YInner2 Extends Scope Field a:Object[] 'Parameter - not really needed, but good for clarity
Function lambda:Scope(env:Scope) Return (New YInner2)._init(env) End Function
Method apply:Object(args:Object[]) a = args Local r:Func = YInner1(env).r Return Func(r.apply([r])).apply(a) End Method End Type
'Based on the following definition:
'(define fac (Y (lambda (f)
' (lambda (x)
' (if (<= x 0) 1 (* x (f (- x 1)))))))
Type FacL1 Extends Scope Field f:Func 'Parameter - gets closed over
Function lambda:Scope(env:Scope) Return (New FacL1)._init(env) End Function
Method apply:Object(args:Object[]) f = Func(args[0]) Return FacL2.lambda(Self) End Method End Type
Type FacL2 Extends Scope Function lambda:Scope(env:Scope) Return (New FacL2)._init(env) End Function
Method apply:Object(args:Object[]) Local x:Int = Integer(args[0]).val If x <= 0 Then Return Integer.Make(1) ; Else Return Integer.Make(x * Integer(FacL1(env).f.apply([Integer.Make(x - 1)])).val) End Method End Type
'Based on the following definition:
'(define fib (Y (lambda (f)
' (lambda (x)
' (if (< x 2) x (+ (f (- x 1)) (f (- x 2)))))))
Type FibL1 Extends Scope Field f:Func 'Parameter - gets closed over
Function lambda:Scope(env:Scope) Return (New FibL1)._init(env) End Function
Method apply:Object(args:Object[]) f = Func(args[0]) Return FibL2.lambda(Self) End Method End Type
Type FibL2 Extends Scope Function lambda:Scope(env:Scope) Return (New FibL2)._init(env) End Function
Method apply:Object(args:Object[]) Local x:Int = Integer(args[0]).val If x < 2 Return Integer.Make(x) Else Local f:Func = FibL1(env).f Local x1:Int = Integer(f.apply([Integer.Make(x - 1)])).val Local x2:Int = Integer(f.apply([Integer.Make(x - 2)])).val Return Integer.Make(x1 + x2) EndIf End Method End Type
'Now test
Local _Y:Func = Y.lambda(Null)
Local fac:Func = Func(_Y.apply([FacL1.lambda(Null)])) Print Integer(fac.apply([Integer.Make(10)])).val
Local fib:Func = Func(_Y.apply([FibL1.lambda(Null)])) Print Integer(fib.apply([Integer.Make(10)])).val</lang>
Bracmat
The lambda abstraction
(λx.x)y
translates to
/('(x.$x))$y
in Bracmat code. Likewise, the fixed point combinator
Y := λg.(λx.g (x x)) (λx.g (x x))
the factorial
G := λr. λn.(1, if n = 0; else n × (r (n−1)))
the Fibonacci function
H := λr. λn.(1, if n = 1 or n = 2; else (r (n−1)) + (r (n−2)))
and the calls
(Y G) i
and
(Y H) i
where i varies between 1 and 10, are translated into Bracmat as shown below <lang bracmat>( ( Y
= /(
' ( g
. /('(x.$g'($x'$x)))
$ /('(x.$g'($x'$x)))
)
)
)
& ( G
= /(
' ( r
. /(
' ( n
. $n:~>0&1
| $n*($r)$($n+-1)
)
)
)
)
)
& ( H
= /(
' ( r
. /(
' ( n
. $n:(1|2)&1
| ($r)$($n+-1)+($r)$($n+-2)
)
)
)
)
)
& 0:?i
& whl
' ( 1+!i:~>10:?i
& out$(str$(!i "!=" (!Y$!G)$!i))
)
& 0:?i
& whl
' ( 1+!i:~>10:?i
& out$(str$("fib(" !i ")=" (!Y$!H)$!i))
)
&
)</lang>
- Output:
1!=1 2!=2 3!=6 4!=24 5!=120 6!=720 7!=5040 8!=40320 9!=362880 10!=3628800 fib(1)=1 fib(2)=1 fib(3)=2 fib(4)=3 fib(5)=5 fib(6)=8 fib(7)=13 fib(8)=21 fib(9)=34 fib(10)=55
C
C doesn't have first class functions, so we demote everything to second class to match.<lang C>#include <stdio.h>
- include <stdlib.h>
/* func: our one and only data type; it holds either a pointer to
a function call, or an integer. Also carry a func pointer to a potential parameter, to simulate closure */
typedef struct func_t *func; typedef struct func_t {
func (*fn) (func, func);
func _;
int num;
} func_t;
func new(func(*f)(func, func), func _) {
func x = malloc(sizeof(func_t));
x->fn = f;
x->_ = _; /* closure, sort of */
x->num = 0;
return x;
}
func call(func f, func n) {
return f->fn(f, n);
}
func Y(func(*f)(func, func)) {
func g = new(f, 0);
g->_ = g;
return g;
}
func num(int n) {
func x = new(0, 0);
x->num = n;
return x;
}
func fac(func self, func n) {
int nn = n->num;
return nn > 1 ? num(nn * call(self->_, num(nn - 1))->num)
: num(1);
}
func fib(func self, func n) {
int nn = n->num;
return nn > 1
? num( call(self->_, num(nn - 1))->num +
call(self->_, num(nn - 2))->num )
: num(1);
}
void show(func n) { printf(" %d", n->num); }
int main() {
int i;
func f = Y(fac);
printf("fac: ");
for (i = 1; i < 10; i++)
show( call(f, num(i)) );
printf("\n");
f = Y(fib);
printf("fib: ");
for (i = 1; i < 10; i++)
show( call(f, num(i)) );
printf("\n");
return 0;
} </lang>
- Output:
fac: 1 2 6 24 120 720 5040 40320 362880 fib: 1 2 3 5 8 13 21 34 55
C#
Like many other statically typed languages, this involves a recursive type, and like other strict languages, it is the Z-combinator instead.
The combinator here is expressed entirely as a lambda expression and is a static property of the generic YCombinator class. Both it and the RecursiveFunc type thus "inherit" the type parameters of the containing class—there effectively exists a separate specialized copy of both for each generic instantiation of YCombinator.
Note: in the code, Func<T, TResult> is a delegate type (the CLR equivalent of a function pointer) that has a parameter of type T and return type of TResult. See Higher-order functions#C# or 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 fac = YCombinator<int, int>.Fix(f => x => x < 2 ? 1 : x * f(x - 1));
var fib = YCombinator<int, int>.Fix(f => x => x < 2 ? x : f(x - 1) + f(x - 2));
Console.WriteLine(fac(10));
Console.WriteLine(fib(10));
}
} </lang>
- Output:
3628800 55
Alternatively, with a non-generic holder class (note that Fix 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 dynamic 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#.
C++
std::function<TResult(T)> in C++ corresponds to Func<T, TResult> 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>
Ceylon
TResult(T) in Ceylon corresponds to Func<T, TResult> in C#.
Since C# does not have local classes, RecursiveFunc and y1 are declared in a class of their own. Moving the type parameters to the class also prevents type parameter inference.
Verbatim <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>
Go
func(T) TResult in Go corresponds to Func<T, TResult> 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>
Java
Verbatim
Since Java uses interfaces and C# uses delegates, which are the only type that the C# compiler will coerce lambda expressions to, this code declares a Functions class for providing a means of converting CLR delegates to objects that implement the Function and RecursiveFunction interfaces.
<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 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>
Swift
T -> TResult in Swift corresponds to Func<T, TResult> 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>
C++
Known to work with GCC 4.7.2. Compile with
g++ --std=c++11 ycomb.cc
<lang cpp>#include <iostream>
- include <functional>
template <typename F> struct RecursiveFunc { std::function<F(RecursiveFunc)> o; };
template <typename A, typename B> std::function<B(A)> Y (std::function<std::function<B(A)>(std::function<B(A)>)> f) { RecursiveFunc<std::function<B(A)>> r = { std::function<std::function<B(A)>(RecursiveFunc<std::function<B(A)>>)>([f](RecursiveFunc<std::function<B(A)>> w) { return f(std::function<B(A)>([w](A x) { return w.o(w)(x); })); }) }; return r.o(r); }
typedef std::function<int(int)> Func; typedef std::function<Func(Func)> FuncFunc; FuncFunc almost_fac = [](Func f) { return Func([f](int n) { if (n <= 1) return 1; return n * f(n - 1); }); };
FuncFunc almost_fib = [](Func f) { return Func([f](int n) { if (n <= 2) return 1; return f(n - 1) + f(n - 2); }); };
int main() { auto fib = Y(almost_fib); auto fac = Y(almost_fac); std::cout << "fib(10) = " << fib(10) << std::endl; std::cout << "fac(10) = " << fac(10) << std::endl; return 0; }</lang>
- Output:
fib(10) = 55 fac(10) = 3628800
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? 1: 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>
- Output:
fib(10) = 55 fac(10) = 3628800
The usual version using recursion, disallowed by the task: <lang cpp>template <typename A, typename B> std::function<B(A)> Y (std::function<std::function<B(A)>(std::function<B(A)>)> f) { return [f](A x) { return f(Y(f))(x); }; }</lang>
Another version which is disallowed because a function passes itself, which is also a kind of recursion: <lang cpp>template <typename A, typename B> struct YFunctor {
const std::function<std::function<B(A)>(std::function<B(A)>)> f;
YFunctor(std::function<std::function<B(A)>(std::function<B(A)>)> _f) : f(_f) {}
B operator()(A x) const {
return f(*this)(x);
}
};
template <typename A, typename B> std::function<B(A)> Y (std::function<std::function<B(A)>(std::function<B(A)>)> f) {
return YFunctor<A,B>(f);
}</lang>
Ceylon
Using a class for the recursive type: <lang ceylon>Result(*Args) y1<Result,Args>(
Result(*Args)(Result(*Args)) f)
given Args satisfies Anything[] {
class RecursiveFunction(o) {
shared Result(*Args)(RecursiveFunction) o;
}
value r = RecursiveFunction((RecursiveFunction w)
=> f(flatten((Args args) => w.o(w)(*args))));
return r.o(r);
}
value factorialY1 = y1((Integer(Integer) fact)(Integer x)
=> if (x > 1) then x * fact(x - 1) else 1);
value fibY1 = y1((Integer(Integer) fib)(Integer x)
=> if (x > 2) then fib(x - 1) + fib(x - 2) else 2);
print(factorialY1(10)); // 3628800 print(fibY1(10)); // 110</lang>
Using Anything to erase the function type: <lang ceylon>Result(*Args) y2<Result,Args>(
Result(*Args)(Result(*Args)) f)
given Args satisfies Anything[] {
function r(Anything w) {
assert (is Result(*Args)(Anything) w);
return f(flatten((Args args) => w(w)(*args)));
}
return r(r);
}</lang>
Using recursion, this does not satisfy the task requirements, but is included here for illustrative purposes: <lang ceylon>Result(*Args) y3<Result, Args>(
Result(*Args)(Result(*Args)) f)
given Args satisfies Anything[]
=> flatten((Args args) => f(y3(f))(*args));</lang>
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:
<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>
- Output:
3628800 55
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:
<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.
Clojure
<lang lisp>(defn Y [f]
((fn [x] (x x))
(fn [x]
(f (fn [& args]
(apply (x x) args))))))
(def fac
(fn [f]
(fn [n]
(if (zero? n) 1 (* n (f (dec n)))))))
(def fib
(fn [f]
(fn [n]
(condp = n
0 0
1 1
(+ (f (dec n))
(f (dec (dec n))))))))</lang>
- Output:
user> ((Y fac) 10) 3628800 user> ((Y fib) 10) 55
Y can be written slightly more concisely via syntax sugar:
<lang lisp>(defn Y [f]
(#(% %) #(f (fn [& args] (apply (% %) args)))))</lang>
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>
Common Lisp
<lang lisp>(defun Y (f)
((lambda (g) (funcall g g))
(lambda (g)
(funcall f (lambda (&rest a)
(apply (funcall g g) a))))))
(defun fac (n)
(funcall
(Y (lambda (f)
(lambda (n)
(if (zerop n)
1 (* n (funcall f (1- n)))))))
n))
(defun fib (n)
(funcall
(Y (lambda (f)
(lambda (n a b)
(if (< n 1)
a
(funcall f (1- n) b (+ a b))))))
n 0 1))
? (mapcar #'fac '(1 2 3 4 5 6 7 8 9 10)) (1 2 6 24 120 720 5040 40320 362880 3628800))
? (mapcar #'fib '(1 2 3 4 5 6 7 8 9 10)) (1 1 2 3 5 8 13 21 34 55)</lang>
Crystal
Although Crystal is very much an OOP language, it does have "Proc"'s that can be used as lambda functions and even as closures where they capture state from the environment external to the body, and these can be used to implement the Y-Combinator. Note that many of the other static strict languages don't implement the true Y-Combinator but rather the Z-Combinator, which lacks one Beta reduction from the Y-Combinator and is more limiting in use. For strict languages such as Crystal, all that is needed to implement the true Y-Combinator is to inject some laziness by deferring execution using a "Thunk" - a function without any arguments returning a deferred value, which requires that functions can also be closures.
The following Crystal code implements the name-recursion Y-Combinator without assuming that functions are recursive (which in Crystal they actually are):
<lang crystal>require "big"
struct RecursiveFunc(T) # a generic recursive function wrapper...
getter recfnc : RecursiveFunc(T) -> T def initialize(@recfnc) end
end
struct YCombo(T) # a struct or class needs to be used so as to be generic...
def initialize (@fnc : Proc(T) -> T) end
def fixy
g = -> (x : RecursiveFunc(T)) {
@fnc.call(-> { x.recfnc.call(x) }) }
g.call(RecursiveFunc(T).new(g))
end
end
def fac(x) # horrendouly inefficient not using tail calls...
facp = -> (fn : Proc(BigInt -> BigInt)) {
-> (n : BigInt) { n < 2 ? n : n * fn.call.call(n - 1) } }
YCombo.new(facp).fixy.call(BigInt.new(x))
end
def fib(x) # horrendouly inefficient not using tail calls...
facp = -> (fn : Proc(BigInt -> BigInt)) {
-> (n : BigInt) {
n < 3 ? n - 1 : fn.call.call(n - 2) + fn.call.call(n - 1) } }
YCombo.new(facp).fixy.call(BigInt.new(x))
end
puts fac(10) puts fib(11) # starts from 0 not 1!</lang>
The "horrendously inefficient" massively repetitious implementations can be made much more efficient by changing the implementation for the two functions as follows:
<lang crystal>def fac(x) # the more efficient tail recursive version...
facp = -> (fn : Proc(BigInt -> (Int32 -> BigInt))) {
-> (n : BigInt) { -> (i : Int32) {
i < 2 ? n : fn.call.call(i * n).call(i - 1) } } }
YCombo.new(facp).fixy.call(BigInt.new(1)).call(x)
end
def fib(x) # the more efficient tail recursive version...
fibp = -> (fn : Proc(BigInt -> (BigInt -> (Int32 -> BigInt)))) {
-> (f : BigInt) { -> (s : BigInt) { -> (i : Int32) {
i < 2 ? f : fn.call.call(s).call(f + s).call(i - 1) } } } }
YCombo.new(fibp).fixy.call(BigInt.new).call(BigInt.new(1)).call(x)
end</lang>
Finally, since Crystal function's/"def"'s can call themselves recursively, the implementation of the Y-Combinator can be changed to use this while still being "call by name" (not value/variable recursion) as follows; this uses the identical lambda "Proc"'s internally with just the application to the Y-Combinator changed:
<lang crystal>def ycombo(f)
f.call(-> { ycombo(f) })
end
def fac(x) # the more efficient tail recursive version...
facp = -> (fn : Proc(BigInt -> (Int32 -> BigInt))) {
-> (n : BigInt) { -> (i : Int32) {
i < 2 ? n : fn.call.call(i * n).call(i - 1) } } }
ycombo(facp).call(BigInt.new(1)).call(x)
end
def fib(x) # the more efficient tail recursive version...
fibp = -> (fn : Proc(BigInt -> (BigInt -> (Int32 -> BigInt)))) {
-> (f : BigInt) { -> (s : BigInt) { -> (i : Int32) {
i < 2 ? f : fn.call.call(s).call(f + s).call(i - 1) } } } }
ycombo(fibp).call(BigInt.new).call(BigInt.new(1)).call(x)
end</lang>
All versions produce the same output:
- Output:
3628800 55
D
A stateless generic Y combinator: <lang d>import std.stdio, std.traits, std.algorithm, std.range;
auto Y(S, T...)(S delegate(T) delegate(S delegate(T)) f) {
static struct F {
S delegate(T) delegate(F) f;
alias f this;
}
return (x => x(x))(F(x => f((T v) => x(x)(v))));
}
void main() { // Demo code:
auto factorial = Y((int delegate(int) self) =>
(int n) => 0 == n ? 1 : n * self(n - 1)
);
auto ackermann = Y((ulong delegate(ulong, ulong) self) =>
(ulong m, ulong n) {
if (m == 0) return n + 1;
if (n == 0) return self(m - 1, 1);
return self(m - 1, self(m, n - 1));
});
writeln("factorial: ", 10.iota.map!factorial);
writeln("ackermann(3, 5): ", ackermann(3, 5));
}</lang>
- Output:
factorial: [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880] ackermann(3, 5): 253
Delphi
May work with Delphi 2009 and 2010 too.
(The translation is not literal; e.g. the function argument type is left unspecified to increase generality.) <lang delphi>program Y;
{$APPTYPE CONSOLE}
uses
SysUtils;
type
YCombinator = class sealed class function Fix<T> (F: TFunc<TFunc<T, T>, TFunc<T, T>>): TFunc<T, T>; static; end;
TRecursiveFuncWrapper<T> = record // workaround required because of QC #101272 (http://qc.embarcadero.com/wc/qcmain.aspx?d=101272) type TRecursiveFunc = reference to function (R: TRecursiveFuncWrapper<T>): TFunc<T, T>; var O: TRecursiveFunc; end;
class function YCombinator.Fix<T> (F: TFunc<TFunc<T, T>, TFunc<T, T>>): TFunc<T, T>; var
R: TRecursiveFuncWrapper<T>;
begin
R.O := function (W: TRecursiveFuncWrapper<T>): TFunc<T, T>
begin
Result := F (function (I: T): T
begin
Result := W.O (W) (I);
end);
end;
Result := R.O (R);
end;
type
IntFunc = TFunc<Integer, Integer>;
function AlmostFac (F: IntFunc): IntFunc; begin
Result := function (N: Integer): Integer
begin
if N <= 1 then
Result := 1
else
Result := N * F (N - 1);
end;
end;
function AlmostFib (F: TFunc<Integer, Integer>): TFunc<Integer, Integer>; begin
Result := function (N: Integer): Integer
begin
if N <= 2 then
Result := 1
else
Result := F (N - 1) + F (N - 2);
end;
end;
var
Fib, Fac: IntFunc;
begin
Fib := YCombinator.Fix<Integer> (AlmostFib);
Fac := YCombinator.Fix<Integer> (AlmostFac);
Writeln ('Fib(10) = ', Fib (10));
Writeln ('Fac(10) = ', Fac (10));
end.</lang>
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>
Déjà Vu
<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>
- Output:
720 13
E
<lang e>def y := fn f { fn x { x(x) }(fn y { f(fn a { y(y)(a) }) }) } def fac := fn f { fn n { if (n<2) {1} else { n*f(n-1) } }} def fib := fn f { fn n { if (n == 0) {0} else if (n == 1) {1} else { f(n-1) + f(n-2) } }}</lang>
<lang e>? pragma.enable("accumulator") ? accum [] for i in 0..!10 { _.with(y(fac)(i)) } [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
? accum [] for i in 0..!10 { _.with(y(fib)(i)) } [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]</lang>
EchoLisp
<lang scheme>
(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>
Eero
Translated from Objective-C example on this page. <lang objc>#import <Foundation/Foundation.h>
typedef int (^Func)(int) typedef Func (^FuncFunc)(Func) typedef Func (^RecursiveFunc)(id) // hide recursive typing behind dynamic typing
Func fix(FuncFunc f)
Func r(RecursiveFunc g)
int s(int x)
return g(g)(x)
return f(s)
return r(r)
int main(int argc, const char *argv[])
autoreleasepool
Func almost_fac(Func f)
return (int n | return n <= 1 ? 1 : n * f(n - 1))
Func almost_fib(Func f)
return (int n | return n <= 2 ? 1 : f(n - 1) + f(n - 2))
fib := fix(almost_fib) fac := fix(almost_fac)
Log('fib(10) = %d', fib(10))
Log('fac(10) = %d', fac(10))
return 0</lang>
Ela
<lang ela>fix = \f -> (\x -> & f (x x)) (\x -> & f (x x))
fac _ 0 = 1 fac f n = n * f (n - 1)
fib _ 0 = 0 fib _ 1 = 1 fib f n = f (n - 1) + f (n - 2)
(fix fac 12, fix fib 12)</lang>
- Output:
(479001600,144)
Elena
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>
- Output:
fib(10)=55 fact(10)=3628800
Elixir
<lang elixir> iex(1)> yc = fn f -> (fn x -> x.(x) end).(fn y -> f.(fn arg -> y.(y).(arg) end) end) end
- Function<6.90072148/1 in :erl_eval.expr/5>
iex(2)> fac = fn f -> fn n -> if n < 2 do 1 else n * f.(n-1) end end end
- Function<6.90072148/1 in :erl_eval.expr/5>
iex(3)> for i <- 0..9, do: yc.(fac).(i) [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880] iex(4)> fib = fn f -> fn n -> if n == 0 do 0 else (if n == 1 do 1 else f.(n-1) + f.(n-2) end) end end end
- Function<6.90072148/1 in :erl_eval.expr/5>
iex(5)> for i <- 0..9, do: yc.(fib).(i) [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] </lang>
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>
- Output:
3628800 55 3628800 55 ( 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 )
Erlang
<lang erlang>Y = fun(M) -> (fun(X) -> X(X) end)(fun (F) -> M(fun(A) -> (F(F))(A) end) end) end.
Fac = fun (F) ->
fun (0) -> 1;
(N) -> N * F(N-1)
end
end.
Fib = fun(F) ->
fun(0) -> 0;
(1) -> 1;
(N) -> F(N-1) + F(N-2)
end
end.
(Y(Fac))(5). %% 120 (Y(Fib))(8). %% 21</lang>
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:
3628800 55
Note that the first `fac` definition isn't really very good as the recursion is not in tail call position and thus will build stack, although for these functions one will likely never use it to stack overflow as the result would be exceedingly large; it is better defined as per the second definition as a steadily increasing function controlled by an `int` indexing argument and thus be in tail call position as is done for the `fib` function.
Also note that the above isn't the true fix point Y-combinator which would race without the beta conversion to the Z-combinator with the included `a` argument; the Z-combinator can't be used in all cases that require a true Y-combinator such as in the formation of deferred execution sequences in the last example, as follows:
<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... let fac = fix (fun f n i -> if i < 2 then n else f () (bigint i * n) (i - 1)) <| 1I // val fac : (int -> BigInteger) = <fun>
// same efficient version of 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 fib : (int -> BigInteger) = <fun>
// given the following definition for an infinite Co-Inductive Stream (CIS)... type CIS<'a> = CIS of 'a * (unit -> CIS<'a>) // ' fix formatting
// 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:
3628800 55 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
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.
Factor
In rosettacode/Y.factor <lang factor>USING: fry kernel math ; IN: rosettacode.Y
- Y ( quot -- quot )
'[ [ dup call call ] curry @ ] dup call ; inline
- almost-fac ( quot -- quot )
'[ dup zero? [ drop 1 ] [ dup 1 - @ * ] if ] ;
- almost-fib ( quot -- quot )
'[ dup 2 >= [ 1 2 [ - @ ] bi-curry@ bi + ] when ] ;</lang>
In rosettacode/Y-tests.factor <lang factor>USING: kernel tools.test rosettacode.Y ; IN: rosettacode.Y.tests
[ 120 ] [ 5 [ almost-fac ] Y call ] unit-test [ 8 ] [ 6 [ almost-fib ] Y call ] unit-test</lang> running the tests :
( scratchpad - auto ) "rosettacode.Y" test
Loading resource:work/rosettacode/Y/Y-tests.factor
Unit Test: { [ 120 ] [ 5 [ almost-fac ] Y call ] }
Unit Test: { [ 8 ] [ 6 [ almost-fib ] Y call ] }
Falcon
<lang Falcon> Y = { f => {x=> {n => f(x(x))(n)}} ({x=> {n => f(x(x))(n)}}) } facStep = { f => {x => x < 1 ? 1 : x*f(x-1) }} fibStep = { f => {x => x == 0 ? 0 : (x == 1 ? 1 : f(x-1) + f(x-2))}}
YFac = Y(facStep) YFib = Y(fibStep)
> "Factorial 10: ", YFac(10) > "Fibonacci 10: ", YFib(10) </lang>
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>
GAP
<lang gap>Y := function(f)
local u; u := x -> x(x); return u(y -> f(a -> y(y)(a)));
end;
fib := function(f)
local u;
u := function(n)
if n < 2 then
return n;
else
return f(n-1) + f(n-2);
fi;
end;
return u;
end;
Y(fib)(10);
- 55
fac := function(f)
local u;
u := function(n)
if n < 2 then
return 1;
else
return n*f(n-1);
fi;
end;
return u;
end;
Y(fac)(8);
- 40320</lang>
Genyris
<lang genyris>def fac (f)
function (n)
if (equal? n 0) 1
* n (f (- n 1))
def fib (f)
function (n)
cond
(equal? n 0) 0
(equal? n 1) 1
else (+ (f (- n 1)) (f (- n 2)))
def Y (f)
(function (x) (x x))
function (y)
f
function (&rest args) (apply (y y) args)
assertEqual ((Y fac) 5) 120 assertEqual ((Y fib) 8) 21</lang>
Go
<lang go>package main
import "fmt"
type Func func(int) int type FuncFunc func(Func) Func type RecursiveFunc func (RecursiveFunc) Func
func main() { fac := Y(almost_fac) fib := Y(almost_fib) fmt.Println("fac(10) = ", fac(10)) fmt.Println("fib(10) = ", fib(10)) }
func Y(f FuncFunc) Func { g := func(r RecursiveFunc) Func { return f(func(x int) int { return r(r)(x) }) } return g(g) }
func almost_fac(f Func) Func { return func(x int) int { if x <= 1 { return 1 } return x * f(x-1) } }
func almost_fib(f Func) Func { return func(x int) int { if x <= 2 { return 1 } return f(x-1)+f(x-2) } }</lang>
- Output:
fac(10) = 3628800 fib(10) = 55
The usual version using recursion, disallowed by the task: <lang go>func Y(f FuncFunc) Func { return func(x int) int { return f(Y(f))(x) } }</lang>
Groovy
Here is the simplest (unary) form of applicative order Y: <lang groovy>def Y = { le -> ({ f -> f(f) })({ f -> le { x -> f(f)(x) } }) }
def factorial = Y { fac ->
{ n -> n <= 2 ? n : n * fac(n - 1) }
}
assert 2432902008176640000 == factorial(20G)
def fib = Y { fibStar ->
{ n -> n <= 1 ? n : fibStar(n - 1) + fibStar(n - 2) }
}
assert fib(10) == 55</lang> This version was translated from the one in The Little Schemer by Friedman and Felleisen. The use of the variable name le is due to the fact that the authors derive Y from an ordinary recursive length function.
A variadic version of Y in Groovy looks like this: <lang groovy>def Y = { le -> ({ f -> f(f) })({ f -> le { Object[] args -> f(f)(*args) } }) }
def mul = Y { mulStar -> { a, b -> a ? b + mulStar(a - 1, b) : 0 } }
1.upto(10) {
assert mul(it, 10) == it * 10
}</lang>
Haskell
The obvious definition of the Y combinator (\f-> (\x -> f (x x)) (\x-> f (x x))) cannot be used in Haskell because it contains an infinite recursive type (a = a -> b). Defining a data type (Mu) allows this recursion to be broken.
<lang haskell>newtype Mu a = Roll
{ unroll :: Mu a -> a }
fix :: (a -> a) -> a fix = g <*> (Roll . g)
where g = (. (>>= id) unroll)
- this version is not in tail call position... -- fac :: Integer -> Integer -- fac = -- fix $ \f n -> if n <= 0 then 1 else n * f (n - 1)
-- this version builds a progression from tail call position and is more efficient... fac :: Integer -> Integer fac =
(fix $ \f n i -> if i <= 0 then n else f (i * n) (i - 1)) 1
-- make fibs a function, else memory leak as -- head of the list can never be released as per: -- https://wiki.haskell.org/Memory_leak, type 1.1 -- overly complex version... {-- fibs :: () -> [Integer] fibs() =
fix $
(0 :) . (1 :) .
(fix
(\f (x:xs) (y:ys) ->
case x + y of n -> n `seq` n : f xs ys) <*> tail)
--}
-- easier to read, simpler (faster) version... fibs :: () -> [Integer] fibs() = 0 : 1 : fix fibs_ 0 1
where
fibs_ fnc f s =
case f + s of n -> n `seq` n : fnc s n
main :: IO () main =
mapM_ print [ map fac [1 .. 20] , take 20 $ fibs() ]</lang>
The usual version uses recursion on a binding, disallowed by the task, to define the fix itself; but the definitions produced by this fix does not use recursion on value bindings although it does use recursion when defining a function (not possible in all languages), so it can be viewed as a true Y-combinator too:
<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 $
\fnc f s i ->
if i <= 1 then f
else case f + s of n -> n `seq` fnc s n (i - 1)) 0 1
{-- -- compute a lazy infinite list. This is -- a Y-combinator version of: fibs() = 0:1:zipWith (+) fibs (tail fibs) -- which is the same as the above version but easier to read... fibs :: () -> [Integer] fibs() = fix fibs_
where
zipP f (x:xs) (y:ys) =
case x + y of n -> n `seq` n : f xs ys
fibs_ a = 0 : 1 : fix zipP a (tail a)
--}
-- easier to read, simpler (faster) version... fibs :: () -> [Integer] fibs() = 0 : 1 : fix fibs_ 0 1
where
fibs_ fnc f s =
case f + s of n -> n `seq` n : fnc s n
-- This code shows how the functions can be used: main :: IO () main =
mapM_ print [ map fac [1 .. 20] , map fib [1 .. 20] , take 20 fibs() ]</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.
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=. [ apply f. ,&< NB. Self referring fac=. (1:`(n * u sr n - 1:)) @. (0 < n) fac f. Y 10
3628800
Fib=. ((u sr n - 2:) + u sr n - 1:) ^: (1 < n) Fib f. Y 10
55</lang> The stateless functions are shown next (the f. adverb replaces all embedded names by its referents): <lang j> fac f. Y NB. Factorial... '1:`(] * [ ([ 128!:2 ,&<) ] - 1:)@.(0 < ])&>/'&([ 128!:2 ,&<)
fac f. NB. Factorial step...
1:`(] * [ ([ 128!:2 ,&<) ] - 1:)@.(0 < ])
Fib f. Y NB. Fibonacci...
'(([ ([ 128!:2 ,&<) ] - 2:) + [ ([ 128!:2 ,&<) ] - 1:)^:(1 < ])&>/'&([ 128!:2 ,&<)
Fib f. NB. Fibonacci step...
(([ ([ 128!:2 ,&<) ] - 2:) + [ ([ 128!:2 ,&<) ] - 1:)^:(1 < ])</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:
<lang j>Y=:1 :0
f=. u Defer (5!:1<'f') f y
)
Defer=: 1 :0
g=. x&(x`:6) (5!:1<'g') u y
)
almost_factorial=: 4 :0
if. 0 >: y do. 1 else. y * x`:6 y-1 end.
)
almost_fibonacci=: 4 :0
if. 2 > y do. y else. (x`:6 y-1) + x`:6 y-2 end.
)</lang>
Example use:
<lang J> almost_factorial Y 9 362880
almost_fibonacci Y 9
34
almost_fibonacci Y"0 i. 10
0 1 1 2 3 5 8 13 21 34</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>
Java
<lang java5>import java.util.function.Function;
public interface YCombinator {
interface RecursiveFunction<F> extends Function<RecursiveFunction<F>, F> { }
public static <A,B> Function<A,B> Y(Function<Function<A,B>, Function<A,B>> f) {
RecursiveFunction<Function<A,B>> r = w -> f.apply(x -> w.apply(w).apply(x));
return r.apply(r);
}
public static void main(String... arguments) {
Function<Integer,Integer> fib = Y(f -> n ->
(n <= 2)
? 1
: (f.apply(n - 1) + f.apply(n - 2))
);
Function<Integer,Integer> fac = Y(f -> n ->
(n <= 1)
? 1
: (n * f.apply(n - 1))
);
System.out.println("fib(10) = " + fib.apply(10));
System.out.println("fac(10) = " + fac.apply(10));
}
}</lang>
- Output:
fib(10) = 55 fac(10) = 3628800
The usual version using recursion, disallowed by the task: <lang java5> public static <A,B> Function<A,B> Y(Function<Function<A,B>, Function<A,B>> f) {
return x -> f.apply(Y(f)).apply(x); }</lang>
Another version which is disallowed because a function passes itself, which is also a kind of recursion: <lang java5> public static <A,B> Function<A,B> Y(Function<Function<A,B>, Function<A,B>> f) {
return new Function<A,B>() {
public B apply(A x) { return f.apply(this).apply(x); } };
}</lang>
We define a generic function interface like Java 8's Function.
<lang java5>interface Function<A, B> {
public B call(A x);
}
public class YCombinator {
interface RecursiveFunc<F> extends Function<RecursiveFunc<F>, F> { }
public static <A,B> Function<A,B> fix(final Function<Function<A,B>, Function<A,B>> f) {
RecursiveFunc<Function<A,B>> r =
new RecursiveFunc<Function<A,B>>() {
public Function<A,B> call(final RecursiveFunc<Function<A,B>> w) {
return f.call(new Function<A,B>() {
public B call(A x) {
return w.call(w).call(x);
}
});
}
};
return r.call(r);
}
public static void main(String[] args) {
Function<Function<Integer,Integer>, Function<Integer,Integer>> almost_fib =
new Function<Function<Integer,Integer>, Function<Integer,Integer>>() {
public Function<Integer,Integer> call(final Function<Integer,Integer> f) {
return new Function<Integer,Integer>() {
public Integer call(Integer n) {
if (n <= 2) return 1;
return f.call(n - 1) + f.call(n - 2);
}
};
}
};
Function<Function<Integer,Integer>, Function<Integer,Integer>> almost_fac =
new Function<Function<Integer,Integer>, Function<Integer,Integer>>() {
public Function<Integer,Integer> call(final Function<Integer,Integer> f) {
return new Function<Integer,Integer>() {
public Integer call(Integer n) {
if (n <= 1) return 1;
return n * f.call(n - 1);
}
};
}
};
Function<Integer,Integer> fib = fix(almost_fib);
Function<Integer,Integer> fac = fix(almost_fac);
System.out.println("fib(10) = " + fib.call(10));
System.out.println("fac(10) = " + fac.call(10));
}
}</lang>
The following code modifies the Function interface such that multiple parameters (via varargs) are supported, simplifies the y function considerably, and the Ackermann function has been included in this implementation (mostly because both D and PicoLisp include it in their own implementations).
<lang java5>import java.util.function.Function;
@FunctionalInterface public interface SelfApplicable<OUTPUT> extends Function<SelfApplicable<OUTPUT>, OUTPUT> {
public default OUTPUT selfApply() {
return apply(this);
}
}</lang>
<lang java5>import java.util.function.Function; import java.util.function.UnaryOperator;
@FunctionalInterface public interface FixedPoint<FUNCTION> extends Function<UnaryOperator<FUNCTION>, FUNCTION> {}</lang>
<lang java5>import java.util.Arrays; import java.util.Optional; import java.util.function.Function; import java.util.function.BiFunction;
@FunctionalInterface public interface VarargsFunction<INPUTS, OUTPUT> extends Function<INPUTS[], OUTPUT> {
@SuppressWarnings("unchecked")
public OUTPUT apply(INPUTS... inputs);
public static <INPUTS, OUTPUT> VarargsFunction<INPUTS, OUTPUT> from(Function<INPUTS[], OUTPUT> function) {
return function::apply;
}
public static <INPUTS, OUTPUT> VarargsFunction<INPUTS, OUTPUT> upgrade(Function<INPUTS, OUTPUT> function) {
return inputs -> function.apply(inputs[0]);
}
public static <INPUTS, OUTPUT> VarargsFunction<INPUTS, OUTPUT> upgrade(BiFunction<INPUTS, INPUTS, OUTPUT> function) {
return inputs -> function.apply(inputs[0], inputs[1]);
}
@SuppressWarnings("unchecked")
public default <POST_OUTPUT> VarargsFunction<INPUTS, POST_OUTPUT> andThen(
VarargsFunction<OUTPUT, POST_OUTPUT> after) {
return inputs -> after.apply(apply(inputs));
}
@SuppressWarnings("unchecked")
public default Function<INPUTS, OUTPUT> toFunction() {
return input -> apply(input);
}
@SuppressWarnings("unchecked")
public default BiFunction<INPUTS, INPUTS, OUTPUT> toBiFunction() {
return (input, input2) -> apply(input, input2);
}
@SuppressWarnings("unchecked")
public default <PRE_INPUTS> VarargsFunction<PRE_INPUTS, OUTPUT> transformArguments(Function<PRE_INPUTS, INPUTS> transformer) {
return inputs -> apply((INPUTS[]) Arrays.stream(inputs).parallel().map(transformer).toArray());
}
}</lang>
<lang java5>import java.math.BigDecimal; import java.math.BigInteger; import java.util.Arrays; import java.util.HashMap; import java.util.Map; import java.util.function.Function; import java.util.function.UnaryOperator; import java.util.stream.Collectors; import java.util.stream.LongStream;
@FunctionalInterface public interface Y<FUNCTION> extends SelfApplicable<FixedPoint<FUNCTION>> {
public static void main(String... arguments) {
BigInteger TWO = BigInteger.ONE.add(BigInteger.ONE);
Function<Number, Long> toLong = Number::longValue; Function<Number, BigInteger> toBigInteger = toLong.andThen(BigInteger::valueOf);
/* Based on https://gist.github.com/aruld/3965968/#comment-604392 */ Y<VarargsFunction<Number, Number>> combinator = y -> f -> x -> f.apply(y.selfApply().apply(f)).apply(x); FixedPoint<VarargsFunction<Number, Number>> fixedPoint = combinator.selfApply();
VarargsFunction<Number, Number> fibonacci = fixedPoint.apply(
f -> VarargsFunction.upgrade(
toBigInteger.andThen(
n -> (n.compareTo(TWO) <= 0)
? 1
: new BigInteger(f.apply(n.subtract(BigInteger.ONE)).toString())
.add(new BigInteger(f.apply(n.subtract(TWO)).toString()))
)
)
);
VarargsFunction<Number, Number> factorial = fixedPoint.apply(
f -> VarargsFunction.upgrade(
toBigInteger.andThen(
n -> (n.compareTo(BigInteger.ONE) <= 0)
? 1
: n.multiply(new BigInteger(f.apply(n.subtract(BigInteger.ONE)).toString()))
)
)
);
VarargsFunction<Number, Number> ackermann = fixedPoint.apply(
f -> VarargsFunction.upgrade(
(BigInteger m, BigInteger n) -> m.equals(BigInteger.ZERO)
? n.add(BigInteger.ONE)
: f.apply(
m.subtract(BigInteger.ONE),
n.equals(BigInteger.ZERO)
? BigInteger.ONE
: f.apply(m, n.subtract(BigInteger.ONE))
)
).transformArguments(toBigInteger)
);
Map<String, VarargsFunction<Number, Number>> functions = new HashMap<>();
functions.put("fibonacci", fibonacci);
functions.put("factorial", factorial);
functions.put("ackermann", ackermann);
Map<VarargsFunction<Number, Number>, Number[]> parameters = new HashMap<>();
parameters.put(functions.get("fibonacci"), new Number[]{20});
parameters.put(functions.get("factorial"), new Number[]{10});
parameters.put(functions.get("ackermann"), new Number[]{3, 2});
functions.entrySet().stream().parallel().map(
entry -> entry.getKey()
+ Arrays.toString(parameters.get(entry.getValue()))
+ " = "
+ entry.getValue().apply(parameters.get(entry.getValue()))
).forEach(System.out::println);
}
}</lang>
- Output:
(may depend on which function gets processed first)
<lang>factorial[10] = 3628800 ackermann[3, 2] = 29 fibonacci[20] = 6765</lang>
JavaScript
The standard version of the Y combinator does not use lexically bound local variables (or any local variables at all), which necessitates adding a wrapper function and some code duplication - the remaining locale variables are only there to make the relationship to the previous implementation more explicit: <lang javascript>function Y(f) {
var g = f((function(h) {
return function() {
var g = f(h(h));
return g.apply(this, arguments);
}
})(function(h) {
return function() {
var g = f(h(h));
return g.apply(this, arguments);
}
}));
return g;
}
var fac = Y(function(f) {
return function (n) {
return n > 1 ? n * f(n - 1) : 1;
};
});
var fib = Y(function(f) {
return function(n) {
return n > 1 ? f(n - 1) + f(n - 2) : n;
};
});</lang>
Changing the order of function application (i.e. the place where f gets called) and making use of the fact that we're generating a fixed-point, this can be reduced to
<lang javascript>function Y(f) {
return (function(h) {
return h(h);
})(function(h) {
return f(function() {
return h(h).apply(this, arguments);
});
});
}</lang>
A functionally equivalent version using the implicit this parameter is also possible:
<lang javascript>function pseudoY(f) {
return (function(h) {
return h(h);
})(function(h) {
return f.bind(function() {
return h(h).apply(null, arguments);
});
});
}
var fac = pseudoY(function(n) {
return n > 1 ? n * this(n - 1) : 1;
});
var fib = pseudoY(function(n) {
return n > 1 ? this(n - 1) + this(n - 2) : n;
});</lang>
However, pseudoY() is not a fixed-point combinator.
The usual version using recursion, disallowed by the task: <lang javascript>function Y(f) {
return function() {
return f(Y(f)).apply(this, arguments);
};
}</lang>
Another version which is disallowed because it uses arguments.callee for a function to get itself recursively:
<lang javascript>function Y(f) {
return function() {
return f(arguments.callee).apply(this, arguments);
};
}</lang>
ECMAScript 2015 (ES6) variants
Since ECMAScript 2015 (ES6) just reached final draft, there are new ways to encode the applicative order Y combinator. These use the new fat arrow function expression syntax, and are made to allow functions of more than one argument through the use of new rest parameters syntax and the corresponding new spread operator syntax. Also showcases new default parameter value syntax: <lang javascript>let
Y= // Except for the η-abstraction necessary for applicative order languages, this is the formal Y combinator.
f=>((g=>(f((...x)=>g(g)(...x))))
(g=>(f((...x)=>g(g)(...x))))),
Y2= // Using β-abstraction to eliminate code repetition.
f=>((f=>f(f))
(g=>(f((...x)=>g(g)(...x))))),
Y3= // Using β-abstraction to separate out the self application combinator δ.
((δ=>f=>δ(g=>(f((...x)=>g(g)(...x)))))
((f=>f(f)))),
fix= // β/η-equivalent fix point combinator. Easier to convert to memoise than the Y combinator.
(((f)=>(g)=>(h)=>(f(h)(g(h)))) // The Substitute combinator out of SKI calculus
((f)=>(g)=>(...x)=>(f(g(g)))(...x)) // S((S(KS)K)S(S(KS)K))(KI)
((f)=>(g)=>(...x)=>(f(g(g)))(...x))),
fix2= // β/η-converted form of fix above into a more compact form
f=>(f=>f(f))(g=>(...x)=>f(g(g))(...x)),
opentailfact= // Open version of the tail call variant of the factorial function
fact=>(n,m=1)=>n<2?m:fact(n-1,n*m);
tailfact= // Tail call version of factorial function
Y(opentailfact);</lang>
ECMAScript 2015 (ES6) also permits a really compact polyvariadic variant for mutually recursive functions: <lang javascript>let
polyfix= // A version that takes an array instead of multiple arguments would simply use l instead of (...l) for parameter
(...l)=>(
(f=>f(f))
(g=>l.map(f=>(...x)=>f(...g(g))(...x)))),
[even,odd]= // The new destructive assignment syntax for arrays
polyfix(
(even,odd)=>n=>(n===0)||odd(n-1),
(even,odd)=>n=>(n!==0)&&even(n-1));</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>
Joy
<lang joy>DEFINE y == [dup cons] swap concat dup cons i;
fac == [ [pop null] [pop succ] [[dup pred] dip i *] ifte ] y.</lang>
Julia
<lang julia>
_ _ _ _(_)_ | Documentation: https://docs.julialang.org (_) | (_) (_) | _ _ _| |_ __ _ | Type "?" for help, "]?" for Pkg help. | | | | | | |/ _` | | | | |_| | | | (_| | | Version 1.6.3 (2021-09-23) _/ |\__'_|_|_|\__'_| | Official https://julialang.org/ release
|__/ |
julia> using Markdown
julia> @doc md"""
# Y Combinator
$λf. (λx. f (x x)) (λx. f (x x))$
""" ->
Y = f -> (x -> x(x))(y -> f((t...) -> y(y)(t...)))
Y </lang>
Usage:
<lang julia> julia> fac = f -> (n -> n < 2 ? 1 : n * f(n - 1))
- 9 (generic function with 1 method)
julia> fib = f -> (n -> n == 0 ? 0 : (n == 1 ? 1 : f(n - 1) + f(n - 2)))
- 13 (generic function with 1 method)
julia> Y(fac).(1:10) 10-element Vector{Int64}:
1
2
6
24
120
720
5040
40320
362880
3628800
julia> Y(fib).(1:10) 10-element Vector{Int64}:
1 1 2 3 5 8 13 21 34 55
</lang>
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>
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>
- Output:
fib: 1 1 2 3 5 8 13 21 34 55 fac: 1 2 6 24 120 720 5040 40320 362880 3628800 End
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>
- Output:
Factorial(1..10) : 1 2 6 24 120 720 5040 40320 362880 3628800 Fibonacci(1..10) : 1 1 2 3 5 8 13 21 34 55
Lambdatalk
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>
Lua
<lang lua>Y = function (f)
return function(...)
return (function(x) return x(x) end)(function(x) return f(function(y) return x(x)(y) end) end)(...)
end
end </lang>
Usage:
<lang lua>almostfactorial = function(f) return function(n) return n > 0 and n * f(n-1) or 1 end end almostfibs = function(f) return function(n) return n < 2 and n or f(n-1) + f(n-2) end end factorial, fibs = Y(almostfactorial), Y(almostfibs) print(factorial(7))</lang>
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 as decimal)->if(n=0->1, n*g(g, n-1)), 10)=3628800 ' true Print y(lambda (g, n)->if(n<=1->n,g(g, n-1)+g(g, n-2)), 10)=55 ' true
\\ Using closure in y, y() return function y=lambda (g)->lambda g (x) -> g(g, x) fact=y((lambda (g, n as decimal)-> if(n=0->1, n*g(g, n-1)))) Print fact(6)=720, fact(24)=620448401733239439360000@ fib=y(lambda (g, n)->if(n<=1->n, g(g, n-1)+g(g, n-2))) Print fib(10)=55 } Ycombinator </lang>
<lang M2000 Interpreter> Module Checkit { Rem { 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) } } 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 } CheckRecursion </lang>
MANOOL
Here one additional technique is demonstrated: the Y combinator is applied to a function during compilation due to the $ 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:
0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 9! = 362880 Fib 0 = 0 Fib 1 = 1 Fib 2 = 1 Fib 3 = 2 Fib 4 = 3 Fib 5 = 5 Fib 6 = 8 Fib 7 = 13 Fib 8 = 21 Fib 9 = 34
Maple
<lang Maple> > Y:=f->(x->x(x))(g->f((()->g(g)(args)))): > Yfac:=Y(f->(x->`if`(x<2,1,x*f(x-1)))): > seq( Yfac( i ), i = 1 .. 10 );
1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800
> Yfib:=Y(f->(x->`if`(x<2,x,f(x-1)+f(x-2)))): > seq( Yfib( i ), i = 1 .. 10 );
1, 1, 2, 3, 5, 8, 13, 21, 34, 55
</lang>
Mathematica / Wolfram Language
<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>
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>
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>
- Output:
3628800 55 3628800 55 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
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.
Objective-C
<lang objc>#import <Foundation/Foundation.h>
typedef int (^Func)(int); typedef Func (^FuncFunc)(Func); typedef Func (^RecursiveFunc)(id); // hide recursive typing behind dynamic typing
Func Y(FuncFunc f) {
RecursiveFunc r =
^(id y) {
RecursiveFunc w = y; // cast value back into desired type
return f(^(int x) {
return w(w)(x);
});
};
return r(r);
}
int main (int argc, const char *argv[]) {
@autoreleasepool {
Func fib = Y(^Func(Func f) {
return ^(int n) {
if (n <= 2) return 1;
return f(n - 1) + f(n - 2);
};
});
Func fac = Y(^Func(Func f) {
return ^(int n) {
if (n <= 1) return 1;
return n * f(n - 1);
};
});
Func fib = fix(almost_fib); Func fac = fix(almost_fac); NSLog(@"fib(10) = %d", fib(10)); NSLog(@"fac(10) = %d", fac(10));
} return 0;
}</lang>
The usual version using recursion, disallowed by the task: <lang objc>Func Y(FuncFunc f) {
return ^(int x) {
return f(Y(f))(x);
};
}</lang>
OCaml
The Y-combinator over functions may be written directly in OCaml provided rectypes are enabled: <lang ocaml>let fix f g = (fun x a -> f (x x) a) (fun x a -> f (x x) a) g</lang> Polymorphic variants are the simplest workaround in the absence of rectypes: <lang ocaml>let fix f = (fun (`X x) -> f(x (`X x))) (`X(fun (`X x) y -> f(x (`X x)) y));;</lang> Otherwise, an ordinary variant can be defined and used: <lang ocaml>type 'a mu = Roll of ('a mu -> 'a);;
let unroll (Roll x) = x;;
let fix f = (fun x a -> f (unroll x x) a) (Roll (fun x a -> f (unroll x x) a));;
let fac f = function
0 -> 1 | n -> n * f (n-1)
let fib f = function
0 -> 0 | 1 -> 1 | n -> f (n-1) + f (n-2)
(* val unroll : 'a mu -> 'a mu -> 'a = <fun> val fix : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b = <fun> val fac : (int -> int) -> int -> int = <fun> val fib : (int -> int) -> int -> int = <fun> *)
fix fac 5;; (* - : int = 120 *)
fix fib 8;; (* - : int = 21 *)</lang>
The usual version using recursion, disallowed by the task: <lang ocaml>let rec fix f x = f (fix f) x;;</lang>
Oforth
These combinators work for any number of parameters (see Ackermann usage)
With recursion into Y definition (so non stateless Y) : <lang Oforth>: Y(f) #[ f Y f perform ] ;</lang>
Without recursion into Y definition (stateless Y). <lang Oforth>: X(me, f) #[ me f me perform f perform ] ;
- Y(f) #X f X ;</lang>
Usage : <lang Oforth>: almost-fact(n, f) n ifZero: [ 1 ] else: [ n n 1 - f perform * ] ;
- almost-fact Y => fact
- almost-fib(n, f) n 1 <= ifTrue: [ n ] else: [ n 1 - f perform n 2 - f perform + ] ;
- almost-fib Y => fib
- almost-Ackermann(m, n, f)
m 0 == ifTrue: [ n 1 + return ] n 0 == ifTrue: [ 1 m 1 - f perform return ] n 1 - m f perform m 1 - f perform ;
- almost-Ackermann Y => Ackermann </lang>
Order
<lang c>#include <order/interpreter.h>
- define ORDER_PP_DEF_8y \
ORDER_PP_FN(8fn(8F, \
8let((8R, 8fn(8G, \
8ap(8F, 8fn(8A, 8ap(8ap(8G, 8G), 8A))))), \
8ap(8R, 8R))))
- define ORDER_PP_DEF_8fac \
ORDER_PP_FN(8fn(8F, 8X, \
8if(8less_eq(8X, 0), 1, 8times(8X, 8ap(8F, 8minus(8X, 1))))))
- define ORDER_PP_DEF_8fib \
ORDER_PP_FN(8fn(8F, 8X, \
8if(8less(8X, 2), 8X, 8plus(8ap(8F, 8minus(8X, 1)), \
8ap(8F, 8minus(8X, 2))))))
ORDER_PP(8to_lit(8ap(8y(8fac), 10))) // 3628800 ORDER_PP(8ap(8y(8fib), 10)) // 55</lang>
Oz
<lang oz>declare
Y = fun {$ F}
{fun {$ X} {X X} end
fun {$ X} {F fun {$ Z} {{X X} Z} end} end}
end
Fac = {Y fun {$ F}
fun {$ N}
if N == 0 then 1 else N*{F N-1} end
end
end}
Fib = {Y fun {$ F}
fun {$ N}
case N of 0 then 0
[] 1 then 1
else {F N-1} + {F N-2}
end
end
end}
in
{Show {Fac 5}}
{Show {Fib 8}}</lang>
PARI/GP
As of 2.8.0, GP cannot make general self-references in closures declared inline, so the Y combinator is required to implement these functions recursively in that environment, e.g., for use in parallel processing. <lang parigp>Y(f)=x->f(f,x); fact=Y((f,n)->if(n,n*f(f,n-1),1)); fib=Y((f,n)->if(n>1,f(f,n-1)+f(f,n-2),n)); apply(fact, [1..10]) apply(fib, [1..10])</lang>
- Output:
%1 = [1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800] %2 = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
Perl
<lang perl>sub Y { my $f = shift; # λf.
sub { my $x = shift; $x->($x) }->( # (λx.x x)
sub {my $y = shift; $f->(sub {$y->($y)(@_)})} # λy.f λz.y y z
)
} my $fac = sub {my $f = shift;
sub {my $n = shift; $n < 2 ? 1 : $n * $f->($n-1)}
}; my $fib = sub {my $f = shift;
sub {my $n = shift; $n == 0 ? 0 : $n == 1 ? 1 : $f->($n-1) + $f->($n-2)}
}; for my $f ($fac, $fib) {
print join(' ', map Y($f)->($_), 0..9), "\n";
}</lang>
- Output:
1 1 2 6 24 120 720 5040 40320 362880 0 1 1 2 3 5 8 13 21 34
The usual version using recursion, disallowed by the task: <lang perl>sub Y { my $f = shift;
sub {$f->(Y($f))->(@_)}
}</lang>
Phix
After (over) simplifying things, the Y function has become a bit of a joke, but at least the recursion has been shifted out of fib/fac
Before saying anything too derogatory about Y(f)=f, it is clearly a fixed-point combinator, and I feel compelled to quote from the Mike Vanier link above:
"It doesn't matter whether you use cos or (lambda (x) (cos x)) as your cosine function; they will both do the same thing."
Anyone thinking they can do better may find some inspiration at
Currying,
Closures/Value_capture,
Partial_function_application,
and/or Function_composition
<lang Phix>function call_fn(integer f, n)
return call_func(f,{f,n})
end function
function Y(integer f)
return f
end function
function fac(integer self, integer n)
return iff(n>1?n*call_fn(self,n-1):1)
end function
function fib(integer self, integer n)
return iff(n>1?call_fn(self,n-1)+call_fn(self,n-2):n)
end function
procedure test(string name, integer rid=routine_id(name))
integer f = Y(rid)
printf(1,"%s: ",{name})
for i=1 to 10 do
printf(1," %d",call_fn(f,i))
end for
printf(1,"\n");
end procedure test("fac") test("fib")</lang>
- Output:
fac: 1 2 6 24 120 720 5040 40320 362880 3628800 fib: 1 1 2 3 5 8 13 21 34 55
Phixmonti
<lang Phixmonti>0 var subr
def fac
dup 1 > if
dup 1 - subr exec *
endif
enddef
def fib
dup 1 > if
dup 1 - subr exec swap 2 - subr exec +
endif
enddef
def test
print ": " print
var subr
10 for
subr exec print " " print
endfor
nl
enddef
getid fac "fac" test getid fib "fib" test</lang>
PHP
<lang php><?php function Y($f) {
$g = function($w) use($f) {
return $f(function() use($w) {
return call_user_func_array($w($w), func_get_args());
});
};
return $g($g);
}
$fibonacci = Y(function($f) {
return function($i) use($f) { return ($i <= 1) ? $i : ($f($i-1) + $f($i-2)); };
});
echo $fibonacci(10), "\n";
$factorial = Y(function($f) {
return function($i) use($f) { return ($i <= 1) ? 1 : ($f($i - 1) * $i); };
});
echo $factorial(10), "\n"; ?></lang> The usual version using recursion, disallowed by the task: <lang php>function Y($f) {
return function() use($f) {
return call_user_func_array($f(Y($f)), func_get_args());
};
}</lang>
with create_function instead of real closures. A little far-fetched, but... <lang php><?php function Y($f) {
$g = create_function('$w', '$f = '.var_export($f,true).';
return $f(create_function(\'\', \'$w = \'.var_export($w,true).\';
return call_user_func_array($w($w), func_get_args());
\'));
');
return $g($g);
}
function almost_fib($f) {
return create_function('$i', '$f = '.var_export($f,true).';
return ($i <= 1) ? $i : ($f($i-1) + $f($i-2));
');
}; $fibonacci = Y('almost_fib'); echo $fibonacci(10), "\n";
function almost_fac($f) {
return create_function('$i', '$f = '.var_export($f,true).';
return ($i <= 1) ? 1 : ($f($i - 1) * $i);
');
}; $factorial = Y('almost_fac'); echo $factorial(10), "\n"; ?></lang>
A functionally equivalent version using the $this parameter in closures is also possible:
<lang php><?php function pseudoY($f) {
$g = function($w) use ($f) {
return $f->bindTo(function() use ($w) {
return call_user_func_array($w($w), func_get_args());
});
};
return $g($g);
}
$factorial = pseudoY(function($n) {
return $n > 1 ? $n * $this($n - 1) : 1;
}); echo $factorial(10), "\n";
$fibonacci = pseudoY(function($n) {
return $n > 1 ? $this($n - 1) + $this($n - 2) : $n;
});
echo $fibonacci(10), "\n";
?></lang>
However, pseudoY() is not a fixed-point combinator.
PicoLisp
<lang PicoLisp>(de Y (F)
(let X (curry (F) (Y) (F (curry (Y) @ (pass (Y Y)))))
(X X) ) )</lang>
Factorial
<lang PicoLisp># Factorial (de fact (F)
(curry (F) (N)
(if (=0 N)
1
(* N (F (dec N))) ) ) )
- ((Y fact) 6)
-> 720</lang>
Fibonacci sequence
<lang PicoLisp># Fibonacci (de fibo (F)
(curry (F) (N)
(if (> 2 N)
1
(+ (F (dec N)) (F (- N 2))) ) ) )
- ((Y fibo) 22)
-> 28657</lang>
Ackermann function
<lang PicoLisp># Ackermann (de ack (F)
(curry (F) (X Y)
(cond
((=0 X) (inc Y))
((=0 Y) (F (dec X) 1))
(T (F (dec X) (F X (dec Y)))) ) ) )
- ((Y ack) 3 4)
-> 125</lang>
Pop11
<lang pop11>define Y(f);
procedure (x); x(x) endprocedure(
procedure (y);
f(procedure(z); (y(y))(z) endprocedure)
endprocedure
)
enddefine;
define fac(h);
procedure (n);
if n = 0 then 1 else n * h(n - 1) endif
endprocedure
enddefine;
define fib(h);
procedure (n);
if n < 2 then 1 else h(n - 1) + h(n - 2) endif
endprocedure
enddefine;
Y(fac)(5) => Y(fib)(5) =></lang>
- Output:
** 120 ** 8
PostScript
<lang postscript>y {
{dup cons} exch concat dup cons i
}.
/fac {
{ {pop zero?} {pop succ} {{dup pred} dip i *} ifte }
y
}.</lang>
PowerShell
PowerShell Doesn't have true closure, in order to fake it, the script-block is converted to text and inserted whole into the next function using variable expansion in double-quoted strings. For simple translation of lambda calculus, translates as param inside of a ScriptBlock, translates as Invoke-Expression "{}", invocation (written as a space) translates to InvokeReturnAsIs. <lang PowerShell>$fac = {
param([ScriptBlock] $f)
invoke-expression @"
{
param([int] `$n)
if (`$n -le 0) {1}
else {`$n * {$f}.InvokeReturnAsIs(`$n - 1)}
}
"@
}
$fib = { param([ScriptBlock] $f) invoke-expression @" { param([int] `$n) switch (`$n)
{
0 {1}
1 {1}
default {{$f}.InvokeReturnAsIs(`$n-1)+{$f}.InvokeReturnAsIs(`$n-2)}
}
} "@ }
$Z = {
param([ScriptBlock] $f)
invoke-expression @"
{
param([ScriptBlock] `$x)
{$f}.InvokeReturnAsIs(`$(invoke-expression @`"
{
param(```$y)
{`$x}.InvokeReturnAsIs({`$x}).InvokeReturnAsIs(```$y)
}
`"@))
}.InvokeReturnAsIs({
param([ScriptBlock] `$x)
{$f}.InvokeReturnAsIs(`$(invoke-expression @`"
{
param(```$y)
{`$x}.InvokeReturnAsIs({`$x}).InvokeReturnAsIs(```$y)
}
`"@))
})
"@ }
$Z.InvokeReturnAsIs($fac).InvokeReturnAsIs(5) $Z.InvokeReturnAsIs($fib).InvokeReturnAsIs(5)</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>
Prolog
Works with SWI-Prolog and module lambda, written by Ulrich Neumerkel found there http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl.
The code is inspired from this page : http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/ISO-Hiord#Hiord (p 106).
Original code is from Hermenegildo and al : Hiord: A Type-Free Higher-Order Logic Programming Language with Predicate Abstraction, pdf accessible here http://www.stups.uni-duesseldorf.de/asap/?id=129.
<lang Prolog>:- use_module(lambda).
% The Y combinator y(P, Arg, R) :- Pred = P +\Nb2^F2^call(P,Nb2,F2,P), call(Pred, Arg, R).
test_y_combinator :-
% code for Fibonacci function Fib = \NFib^RFib^RFibr1^(NFib < 2 ->
RFib = NFib ; NFib1 is NFib - 1, NFib2 is NFib - 2, call(RFibr1,NFib1,RFib1,RFibr1), call(RFibr1,NFib2,RFib2,RFibr1), RFib is RFib1 + RFib2 ),
y(Fib, 10, FR), format('Fib(~w) = ~w~n', [10, FR]),
% code for Factorial function Fact = \NFact^RFact^RFactr1^(NFact = 1 ->
RFact = NFact
;
NFact1 is NFact - 1, call(RFactr1,NFact1,RFact1,RFactr1), RFact is NFact * RFact1 ),
y(Fact, 10, FF), format('Fact(~w) = ~w~n', [10, FF]).</lang>
- Output:
?- test_y_combinator. Fib(10) = 55 Fact(10) = 3628800 true.
Python
<lang python>>>> Y = lambda f: (lambda x: x(x))(lambda y: f(lambda *args: y(y)(*args))) >>> fac = lambda f: lambda n: (1 if n<2 else n*f(n-1)) >>> [ Y(fac)(i) for i in range(10) ] [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880] >>> fib = lambda f: lambda n: 0 if n == 0 else (1 if n == 1 else f(n-1) + f(n-2)) >>> [ Y(fib)(i) for i in range(10) ] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]</lang>
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>
Q
<lang Q>> Y: {{x x} {({y {(x x) y} x} y) x} x} > fac: {{$[y<2; 1; y*x y-1]} x} > (Y fac) 6 720j > fib: {{$[y<2; 1; (x y-1) + (x y-2)]} x} > (Y fib) each til 20 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 </lang>
R
<lang R>Y <- function(f) {
(function(x) { (x)(x) })( function(y) { f( (function(a) {y(y)})(a) ) } )
}</lang>
<lang R>fac <- function(f) {
function(n) {
if (n<2)
1
else
n*f(n-1)
}
}
fib <- function(f) {
function(n) {
if (n <= 1)
n
else
f(n-1) + f(n-2)
}
}</lang>
<lang R>for(i in 1:9) print(Y(fac)(i)) for(i in 1:9) print(Y(fib)(i))</lang>
Racket
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>
- Output:
> (!! (map Fact '(1 2 4 8 16))) '(1 2 24 40320 20922789888000) > (!! (map Fib '(1 2 4 8 16))) '(0 1 2 13 610)
Strict realization: <lang racket>#lang racket (define Y (λ (b) ((λ (f) (b (λ (x) ((f f) x))))
(λ (f) (b (λ (x) ((f f) x)))))))</lang>
Definitions of Fact and Fib functions will be the same as in Lazy Racket.
Finally, a definition in Typed Racket is a little difficult as in other statically typed languages: <lang racket>#lang typed/racket
(: make-recursive : (All (S T) ((S -> T) -> (S -> T)) -> (S -> T))) (define-type Tau (All (S T) (Rec this (this -> (S -> T))))) (define (make-recursive f)
((lambda: ([x : (Tau S T)]) (f (lambda (z) ((x x) z)))) (lambda: ([x : (Tau S T)]) (f (lambda (z) ((x x) z))))))
(: fact : Number -> Number) (define fact (make-recursive
(lambda: ([fact : (Number -> Number)])
(lambda: ([n : Number])
(if (zero? n)
1
(* n (fact (- n 1))))))))
(fact 5)</lang>
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>
- Output:
(1 1 2 6 24 120 720 5040 40320 362880) (0 1 1 2 3 5 8 13 21 34)
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>
REBOL
<lang rebol>Y: closure [g] [do func [f] [f :f] closure [f] [g func [x] [do f :f :x]]]</lang>
- usage example
<lang rebol>fact*: closure [h] [func [n] [either n <= 1 [1] [n * h n - 1]]] fact: Y :fact*</lang>
REXX
Programming note: length, reverse, sign, trunc, b2x, d2x, and x2d are REXX BIFs (Built In Functions). <lang rexx>/*REXX program implements and displays a stateless Y combinator. */ numeric digits 1000 /*allow big numbers. */ say ' fib' Y(fib (50) ) /*Fibonacci series. */ say ' fib' Y(fib (12 11 10 9 8 7 6 5 4 3 2 1 0) ) /*Fibonacci series. */ say ' fact' Y(fact (60) ) /*single factorial.*/ say ' fact' Y(fact (0 1 2 3 4 5 6 7 8 9 10 11) ) /*single factorial.*/ say ' Dfact' Y(dfact (4 5 6 7 8 9 10 11 12 13) ) /*double factorial.*/ say ' Tfact' Y(tfact (4 5 6 7 8 9 10 11 12 13) ) /*triple factorial.*/ say ' Qfact' Y(qfact (4 5 6 7 8 40) ) /*quadruple factorial.*/ say ' length' Y(length (when for to where whenceforth) ) /*lengths of words. */ say 'reverse' Y(reverse (123 66188 3007 45.54 MAS I MA) ) /*reverses strings. */ say ' sign' Y(sign (-8 0 8) ) /*sign of the numbers.*/ say ' trunc' Y(trunc (-7.0005 12 3.14159 6.4 78.999) ) /*truncates numbers. */ 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(x2d (8 9 10 11 12 88 89 90 91 6789) ) /*converts HEX──►DEC. */ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ Y: parse arg Y _; $=; do j=1 for words(_); interpret '$=$' Y"("word(_,j)')'; end; return $ /*──────────────────────────────────────────────────────────────────────────────────────*/ fib: procedure; parse arg x; if x<2 then return x; s= 0; a= 0; b= 1
do j=2 to x; s= a+b; a= b; b= s; end; return s
/*──────────────────────────────────────────────────────────────────────────────────────*/ dfact: procedure; parse arg x; != 1; do j=x to 2 by -2; != !*j; end; return ! tfact: procedure; parse arg x; != 1; do j=x to 2 by -3; != !*j; end; return ! qfact: procedure; parse arg x; != 1; do j=x to 2 by -4; != !*j; end; return !
fact: procedure; parse arg x; != 1; do j=2 to x ; != !*j; end; return !</lang>
- output when using the internal default input:
fib 12586269025
fib 144 89 55 34 21 13 8 5 3 2 1 1 0
fact 8320987112741390144276341183223364380754172606361245952449277696409600000000000000
fact 1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800
Dfact 8 15 48 105 384 945 3840 10395 46080 135135
Tfact 4 10 18 28 80 162 280 880 1944 3640
Qfact 4 5 12 21 32 3805072588800
length 4 3 2 5 11
reverse 321 88166 7003 45.54 SAM I AM
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
Ruby
Using a lambda:
<lang ruby>y = lambda do |f|
lambda {|g| g[g]}[lambda do |g|
f[lambda {|*args| g[g][*args]}]
end]
end
fac = lambda{|f| lambda{|n| n < 2 ? 1 : n * f[n-1]}} p Array.new(10) {|i| y[fac][i]} #=> [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
fib = lambda{|f| lambda{|n| n < 2 ? n : f[n-1] + f[n-2]}} p Array.new(10) {|i| y[fib][i]} #=> [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]</lang>
Same as the above, using the new short lambda syntax:
<lang ruby>y = ->(f) {->(g) {g.(g)}.(->(g) { f.(->(*args) {g.(g).(*args)})})}
fac = ->(f) { ->(n) { n < 2 ? 1 : n * f.(n-1) } }
p 10.times.map {|i| y.(fac).(i)}
fib = ->(f) { ->(n) { n < 2 ? n : f.(n-2) + f.(n-1) } }
p 10.times.map {|i| y.(fib).(i)}</lang>
Using a method:
<lang ruby>def y(&f)
lambda do |g|
f.call {|*args| g[g][*args]}
end.tap {|g| break g[g]}
end
fac = y {|&f| lambda {|n| n < 2 ? 1 : n * f[n - 1]}} fib = y {|&f| lambda {|n| n < 2 ? n : f[n - 1] + f[n - 2]}}
p Array.new(10) {|i| fac[i]}
- => [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
p Array.new(10) {|i| fib[i]}
- => [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]</lang>
The usual version using recursion, disallowed by the task: <lang ruby>y = lambda do |f|
lambda {|*args| f[y[f]][*args]}
end</lang>
Rust
<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:
fac(10) = 3628800 fib(10) = 89
Scala
Credit goes to the thread in scala blog <lang scala> def Y[A, B](f: (A => B) => (A => B)): A => B = {
case class W(wf: W => (A => B)) {
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>
Scheme
<lang scheme>(define Y ; (Y f) = (g g) where
(lambda (f) ; (g g) = (f (lambda a (apply (g g) a)))
((lambda (g) (g g)) ; (Y f) == (f (lambda a (apply (Y f) a)))
(lambda (g)
(f (lambda a (apply (g g) a)))))))
- head-recursive factorial
(define fac ; fac = (Y f) = (f (lambda a (apply (Y f) a)))
(Y (lambda (r) ; = (lambda (x) ... (r (- x 1)) ... )
(lambda (x) ; where r = (lambda a (apply (Y f) a))
(if (< x 2) ; (r ... ) == ((Y f) ... )
1 ; == (lambda (x) ... (fac (- x 1)) ... )
(* x (r (- x 1))))))))
- tail-recursive factorial
(define fac2
(lambda (x)
((Y (lambda (r) ; (Y f) == (f (lambda a (apply (Y f) a)))
(lambda (x acc) ; r == (lambda a (apply (Y f) a))
(if (< x 2) ; (r ... ) == ((Y f) ... )
acc
(r (- x 1) (* x acc))))))
x 1)))
- double-recursive Fibonacci
(define fib
(Y (lambda (f)
(lambda (x)
(if (< x 2)
x
(+ (f (- x 1)) (f (- x 2))))))))
- tail-recursive Fibonacci
(define fib2
(lambda (x)
((Y (lambda (f)
(lambda (x a b)
(if (< x 1)
a
(f (- x 1) b (+ a b))))))
x 0 1)))
(display (fac 6)) (newline)
(display (fib2 134)) (newline)</lang>
- Output:
720 4517090495650391871408712937
If we were allowed to use recursion (with Y referring to itself by name in its body) we could define the equivalent to the above as
<lang scheme>(define Yr ; (Y f) == (f (lambda a (apply (Y f) a)))
(lambda (f) (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>
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>
- Output:
1 120 3628800
Sidef
<lang ruby>var y = ->(f) {->(g) {g(g)}(->(g) { f(->(*args) {g(g)(args...)})})}
var fac = ->(f) { ->(n) { n < 2 ? 1 : (n * f(n-1)) } } say 10.of { |i| y(fac)(i) }
var fib = ->(f) { ->(n) { n < 2 ? n : (f(n-2) + f(n-1)) } } say 10.of { |i| y(fib)(i) }</lang>
- Output:
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
Slate
The Y combinator is already defined in slate as: <lang slate>Method traits define: #Y &builder:
[[| :f | [| :x | f applyWith: (x applyWith: x)]
applyWith: [| :x | f applyWith: (x applyWith: x)]]].</lang>
Smalltalk
<lang smalltalk>Y := [:f| [:x| x value: x] value: [:g| f value: [:x| (g value: g) value: x] ] ].
fib := Y value: [:f| [:i| i <= 1 ifTrue: [i] ifFalse: [(f value: i-1) + (f value: i-2)] ] ].
(fib value: 10) displayNl.
fact := Y value: [:f| [:i| i = 0 ifTrue: [1] ifFalse: [(f value: i-1) * i] ] ].
(fact value: 10) displayNl.</lang>
- Output:
55 3628800
The usual version using recursion, disallowed by the task: <lang smalltalk>Y := [:f| [:x| (f value: (Y value: f)) value: x] ].</lang>
Standard ML
<lang sml>- datatype 'a mu = Roll of ('a mu -> 'a)
fun unroll (Roll x) = x
fun fix f = (fn x => fn a => f (unroll x x) a) (Roll (fn x => fn a => f (unroll x x) a))
fun fac f 0 = 1 | fac f n = n * f (n-1)
fun fib f 0 = 0 | fib f 1 = 1 | fib f n = f (n-1) + f (n-2)
datatype 'a mu = Roll of 'a mu -> 'a val unroll = fn : 'a mu -> 'a mu -> 'a val fix = fn : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b val fac = fn : (int -> int) -> int -> int val fib = fn : (int -> int) -> int -> int - List.tabulate (10, fix fac); val it = [1,1,2,6,24,120,720,5040,40320,362880] : int list - List.tabulate (10, fix fib); val it = [0,1,1,2,3,5,8,13,21,34] : int list</lang>
The usual version using recursion, disallowed by the task: <lang sml>fun fix f x = f (fix f) x</lang>
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>
Swift
Using a recursive type: <lang swift>struct RecursiveFunc<F> {
let o : RecursiveFunc<F> -> F
}
func Y<A, B>(f: (A -> B) -> A -> B) -> A -> B {
let r = RecursiveFunc<A -> B> { w in f { w.o(w)($0) } }
return r.o(r)
}
let fac = Y { (f: Int -> Int) in
{ $0 <= 1 ? 1 : $0 * f($0-1) }
} let fib = Y { (f: Int -> Int) in
{ $0 <= 2 ? 1 : f($0-1)+f($0-2) }
} println("fac(5) = \(fac(5))") println("fib(9) = \(fib(9))")</lang>
- Output:
fac(5) = 120 fib(9) = 34
Without a recursive type, and instead using Any to erase the type:
(for Swift 1.1 replace as! with as)
<lang swift>func Y<A, B>(f: (A -> B) -> A -> B) -> A -> B {
typealias RecursiveFunc = Any -> A -> B
let r : RecursiveFunc = { (z: Any) in let w = z as! RecursiveFunc; return f { w(w)($0) } }
return r(r)
}</lang>
The usual version using recursion, disallowed by the task: <lang swift>func Y<In, Out>( f: (In->Out) -> (In->Out) ) -> (In->Out) {
return { x in f(Y(f))(x) }
}</lang>
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>
- Output:
factorial 5: 120 fibonacci 5: 5
Tcl
Y combinator is derived in great detail here.
TXR
This prints out 24, the factorial of 4:
<lang txrlisp>;; The Y combinator: (defun y (f)
[(op @1 @1) (op f (op [@@1 @@1]))])
- The Y-combinator-based factorial
(defun fac (f)
(do if (zerop @1)
1
(* @1 [f (- @1 1)])))
- Test
(format t "~s\n" [[y fac] 4])</lang>
Both the op and do operators are a syntactic sugar for currying, in two different flavors. The forms within do that are symbols are evaluated in the normal Lisp-2 style and the first symbol can be an operator. Under op, any forms that are symbols are evaluated in the Lisp-2 style, and the first form is expected to evaluate to a function. The name do stems from the fact that the operator is used for currying over special forms like if in the above example, where there is evaluation control. Operators can have side effects: they can "do" something. Consider (do set a @1) which yields a function of one argument which assigns that argument to a.
The compounded @@... notation allows for inner functions to refer to outer parameters, when the notation is nested. Consider <lang txrlisp>(op foo @1 (op bar @2 @@2))</lang>. Here the @2 refers to the second argument of the anonymous function denoted by the inner op. The @@2 refers to the second argument of the outer op.
Ursala
The standard y combinator doesn't work in Ursala due to eager
evaluation, but an alternative is easily defined as shown
<lang Ursala>(r "f") "x" = "f"("f","x")
my_fix "h" = r ("f","x"). ("h" r "f") "x"</lang>
or by this shorter expression for the same thing in point free form.
<lang Ursala>my_fix = //~&R+ ^|H\~&+ ; //~&R</lang>
Normally you'd like to define a function recursively by writing
, where is just the body of the
function with recursive calls to in it. With a fixed point
combinator such as my_fix as defined above, you do almost the same thing, except it's my_fix
"f". ("f"), where the dot represents lambda abstraction and the
quotes signify a dummy variable. Using this
method, the definition of the factorial function becomes
<lang Ursala>#import nat
fact = my_fix "f". ~&?\1! product^/~& "f"+ predecessor</lang>
To make it easier, the compiler has a directive to let you install
your own fixed point combinator for it to use, which looks like
this,
<lang Ursala>#fix my_fix</lang>
with your choice of function to be used in place of my_fix.
Having done that, you may express recursive functions per convention by circular definitions,
as in this example of a Fibonacci function.
<lang Ursala>fib = {0,1}?</1! sum+ fib~~+ predecessor^~/~& predecessor</lang>
Note that this way is only syntactic sugar for the for explicit way
shown above. Without a fixed point combinator given in the #fix
directive, this definition of fib
would not have compiled. (Ursala allows user defined fixed point
combinators because they're good for other things besides
functions.)
To confirm that all this works, here is a test program applying
both of the functions defined above to the numbers from 1 to 8.
<lang Ursala>#cast %nLW
examples = (fact* <1,2,3,4,5,6,7,8>,fib* <1,2,3,4,5,6,7,8>)</lang>
- Output:
( <1,2,6,24,120,720,5040,40320>, <1,2,3,5,8,13,21,34>)
The fixed point combinator defined above is theoretically correct
but inefficient and limited to first order functions,
whereas the standard distribution includes a library (sol)
providing a hierarchy of fixed point combinators
suitable for production use and with higher order functions.
A more efficient alternative implementation of my_fix
would be general_function_fixer 0
(with 0 signifying the lowest order of fixed point combinators),
or if that's too easy, then by this definition.
<lang Ursala>#import sol
- fix general_function_fixer 1
my_fix "h" = "h" my_fix "h"</lang> Note that this equation is solved using the next fixed point combinator in the hierarchy.
VBA
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>
- Output:
fac 1 2 6 24 120 720 5040 40320 362880 3628800 fib 1 1 2 3 5 8 13 21 34 55
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>
Vim Script
There is no lambda in Vim (yet?), so here is a way to fake it using a Dictionary. This also provides garbage collection. <lang vim>" Translated from Python. Works with: Vim 7.0
func! Lambx(sig, expr, dict)
let fanon = {'d': a:dict}
exec printf("
\func fanon.f(%s) dict\n \ return %s\n \endfunc", \ a:sig, a:expr)
return fanon
endfunc
func! Callx(fanon, arglist)
return call(a:fanon.f, a:arglist, a:fanon.d)
endfunc
let g:Y = Lambx('f', 'Callx(Lambx("x", "Callx(a:x, [a:x])", {}), [Lambx("y", Callx(self.f, [Lambx("...", "Callx(Callx(self.y, [self.y]), a:000)", {"y": a:y})]), {"f": a:f})])', {})
let g:fac = Lambx('f', 'Lambx("n", "a:n<2 ? 1 : a:n * Callx(self.f, [a:n-1])", {"f": a:f})', {})
echo Callx(Callx(g:Y, [g:fac]), [5]) echo map(range(10), 'Callx(Callx(Y, [fac]), [v:val])') </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:
120 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
Wart
<lang python># Better names due to Jim Weirich: http://vimeo.com/45140590 def (Y improver)
((fn(gen) gen.gen)
(fn(gen)
(fn(n)
((improver gen.gen) n))))
factorial <- (Y (fn(f)
(fn(n)
(if zero?.n
1
(n * (f n-1))))))
prn factorial.5</lang>
Wren
<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>
- Output:
fac(10) = 3628800 fib(10) = 55
XQuery
Version 3.0 of the XPath and XQuery specifications added support for function items.
<lang XQuery>let $Y := function($f) {
(function($x) { ($x)($x) })( function($g) { $f( (function($a) { $g($g) ($a)}) ) } )
}
let $fac := $Y(function($f) { function($n) { if($n < 2) then 1 else $n * $f($n - 1) } }) let $fib := $Y(function($f) { function($n) { if($n <= 1) then $n else $f($n - 1) + $f($n - 2) } }) return (
$fac(6), $fib(6)
) </lang>
- Output:
<lang XQuery>720 8</lang>
Yabasic
<lang Yabasic>sub fac(self$, 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>
zkl
<lang zkl>fcn Y(f){ fcn(g){ g(g) }( 'wrap(h){ f( 'wrap(a){ h(h)(a) }) }) }</lang> Functions don't get to look outside of their scope so data in enclosing scopes needs to be bound to a function, the fp (function application/cheap currying) method does this. 'wrap is syntactic sugar for fp. <lang zkl>fcn almost_factorial(f){ fcn(n,f){ if(n<=1) 1 else n*f(n-1) }.fp1(f) } Y(almost_factorial)(6).println(); [0..10].apply(Y(almost_factorial)).println();</lang>
- Output:
720 L(1,1,2,6,24,120,720,5040,40320,362880,3628800)
<lang zkl>fcn almost_fib(f){ fcn(n,f){ if(n<2) 1 else f(n-1)+f(n-2) }.fp1(f) } Y(almost_fib)(9).println(); [0..10].apply(Y(almost_fib)).println();</lang>
- Output:
55 L(1,1,2,3,5,8,13,21,34,55,89)
- Programming Tasks
- Classic CS problems and programs
- Recursion
- AArch64 Assembly
- ALGOL 68
- AppleScript
- ARM Assembly
- ATS
- BlitzMax
- Bracmat
- C
- C sharp
- C++
- Ceylon
- Chapel
- Clojure
- CoffeeScript
- Common Lisp
- Crystal
- D
- Delphi
- Dhall
- Déjà Vu
- E
- EchoLisp
- Eero
- Ela
- Elena
- Elixir
- Elm
- Erlang
- F Sharp
- Factor
- Falcon
- Forth
- GAP
- Genyris
- Go
- Groovy
- Haskell
- J
- Java
- JavaScript
- Joy
- Julia
- Kitten
- Klingphix
- Kotlin
- Lambdatalk
- Lua
- M2000 Interpreter
- MANOOL
- Maple
- Mathematica
- Wolfram Language
- Moonscript
- Nim
- Objective-C
- OCaml
- Oforth
- Order
- Oz
- PARI/GP
- Perl
- Phix
- Phixmonti
- PHP
- PicoLisp
- Pop11
- PostScript
- Initlib
- PowerShell
- Prolog
- Python
- Q
- R
- Racket
- Raku
- REBOL
- REXX
- Ruby
- Rust
- Scala
- Scheme
- Shen
- Sidef
- Slate
- Smalltalk
- Standard ML
- SuperCollider
- Swift
- Tailspin
- Tcl
- TXR
- Ursala
- VBA
- Verbexx
- Vim Script
- Wart
- ACL2/Omit
- Ada/Omit
- PureBasic/Omit
- TI-89 BASIC/Omit
- Wren
- XQuery
- Yabasic
- Zkl