Y combinator
You are encouraged to solve this task according to the task description, using any language you may know.
In strict functional programming and the lambda calculus, functions (lambda expressions) don't have state and are only allowed to refer to arguments of enclosing functions. This rules out the usual definition of a recursive function wherein a function is associated with the state of a variable and this variable's state is used in the body of the function.
The Y combinator is itself a stateless function that, when applied to another stateless function, returns a recursive version of the function. The Y combinator is the simplest of the class of such functions, called fixed-point combinators.
The task is to define the stateless Y combinator and use it to compute factorials and Fibonacci numbers from other stateless functions or lambda expressions.
- Cf
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>
AppleScript
AppleScript is not terribly "functional" friendly. However, it is capable enough to 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 'funcall' 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>to |Y|(f)
script x to funcall(y) script to funcall(arg) y's funcall(y)'s funcall(arg) end funcall end script f's funcall(result) end funcall end script x's funcall(x)
end |Y|
script
to funcall(f) script to funcall(n) if n = 0 then return 1 n * (f's funcall(n - 1)) end funcall end script end funcall
end script set fact to |Y|(result)
script
to funcall(f) script to funcall(n) if n = 0 then return 0 if n = 1 then return 1 (f's funcall(n - 2)) + (f's funcall(n - 1)) end funcall end script end funcall
end script set fib to |Y|(result)
set facts to {} repeat with i from 0 to 11
set end of facts to fact's funcall(i)
end repeat
set fibs to {} repeat with i from 0 to 20
set end of fibs to fib's funcall(i)
end repeat
{facts:facts, fibs:fibs} (* {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>
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 (*func) (func, func), _; int num;
} func_t;
func new(func(*f)(func, func), func _) {
func x = malloc(sizeof(func_t)); x->func = f; x->_ = _; /* closure, sort of */ x->num = 0; return x;
}
func call(func f, func g) {
return f->func(f, g);
}
func Y(func(*f)(func, func)) {
func _(func x, func y) { return call(x->_, y); } func_t __ = { _ };
func g = call(new(f, 0), &__); g->_ = g; return g;
}
func num(int n) {
func x = new(0, 0); x->num = n; return x;
}
func fac(func f, func _null) {
func _(func self, func n) { int nn = n->num; return nn > 1 ? num(nn * call(self->_, num(nn - 1))->num) : num(1); }
return new(_, f);
}
func fib(func f, func _null) {
func _(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); }
return new(_, f);
}
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#
<lang csharp>using System;
class Program {
delegate Func<int, int> Recursive(Recursive recursive);
static void Main() { Func<Func<Func<int, int>, Func<int, int>>, Func<int, int>> Y = f => ((Recursive)(g => (f(x => g(g)(x)))))((Recursive)(g => f(x => g(g)(x))));
var fac = Y(f => x => x < 2 ? 1 : x * f(x - 1)); var fib = Y(f => x => x < 2 ? x : f(x - 1) + f(x - 2));
Console.WriteLine(fac(6)); Console.WriteLine(fib(6)); }
}</lang> Output:
720 8
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)> fix (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 = fix(almost_fib); auto fac = fix(almost_fac); std::cout << "fib(10) = " << fib(10) << std::endl; std::cout << "fac(10) = " << fac(10) << std::endl; return 0; }</lang>
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>
Sample 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>
Common Lisp
<lang lisp>(defun Y (f)
((lambda (x) (funcall x x)) (lambda (y) (funcall f (lambda (&rest args)
(apply (funcall y y) args))))))
(defun fac (f)
(lambda (n) (if (zerop n)
1 (* n (funcall f (1- n))))))
(defun fib (f)
(lambda (n) (case n (0 0) (1 1) (otherwise (+ (funcall f (- n 1))
(funcall f (- n 2)))))))
? ((mapcar (y #'fac) '(1 2 3 4 5 6 7 8 9 10)) (1 2 6 24 120 720 5040 40320 362880 3628800))
? (mapcar (y #'fib) '(1 2 3 4 5 6 7 8 9 10)) (1 1 2 3 5 8 13 21 34 55)
</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>
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
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
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>
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>
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)
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) // ease syntax colouring confusion with '
let unroll (Roll x) = x //val unroll : 'a mu -> 'a
let fix f = (fun x a -> f (unroll x x) a) (Roll (fun x a -> f (unroll x x) a)) //val fix : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b = <fun>
let fac f = function
0 -> 1 | n -> n * f (n-1)
//val fac : (int -> int) -> int -> int = <fun>
let fib f = function
0 -> 0 | 1 -> 1 | n -> f (n-1) + f (n-2)
//val fib : (int -> int) -> int -> int = <fun>
fix fac 5;; // val it : int = 120
fix fib 8;; // val it : int = 21</lang>
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>
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 := fix(almost_fac) fib := fix(almost_fib) fmt.Println("fac(10) = ", fac(10)) fmt.Println("fib(10) = ", fib(10)) }
func fix(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>
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 in Haskell canot be used 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 = \f -> (\x -> f (unroll x x)) $ Roll (\x -> f (unroll x x))
fac :: Integer -> Integer fac = fix $ \f n -> if (n <= 0) then 1 else n * f (n-1)
fibs :: [Integer] fibs = fix $ \fbs -> 0 : 1 : fix zipP fbs (tail fbs)
where zipP f (x:xs) (y:ys) = x+y : f xs ys
main = do
print $ map fac [1 .. 20] print $ take 20 fibs</lang>
The usual version using recursion, disallowed by the task:
<lang haskell>fix :: (a -> a) -> a fix f = f (fix f)
fac :: Integer -> Integer fac' f n | n <= 0 = 1
| otherwise = n * f (n-1)
fac = fix fac'
-- a simple but wasteful exponential time definition: fib :: Integer -> Integer fib' f 0 = 0 fib' f 1 = 1 fib' f n = f (n-1) + f (n-2) fib = fix fib'
-- Or for far more efficiency, compute a lazy infinite list. This is -- a Y-combinator version of: fibs = 0:1:zipWith (+) fibs (tail fibs) fibs :: [Integer] fibs' a = 0:1:(fix zipP a (tail a))
where zipP f (x:xs) (y:ys) = x+y : f xs ys
fibs = fix fibs'
-- This code shows how the functions can be used: main = do
print $ map fac [1 .. 20] print $ map fib [0 .. 19] print $ take 20 fibs</lang>
J
In J, functions cannot take functions of the same type as arguments. In other words, verbs cannot take verbs and adverbs or conjunctions cannot take adverbs or conjunctions. However, the Y combinator can be implemented indirectly using, for example, the linear representations of verbs. (Y becomes a wrapper which takes a verb as an argument and serializes it, and the underlying self referring system interprets the serialized representation of a verb as the corresponding verb): <lang j>Y=. ((((&>)/)(1 : '(5!:5)<x'))(&([ 128!:2 ,&<)))f.</lang> The factorial and Fibonacci examples: <lang j> u=. [ NB. Function (left)
n=. ] NB. Argument (right) sr=. [ 128!:2 ,&< 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 functions' stateless codings are shown next: <lang j> fac f. Y NB. Showing the stateless recursive factorial function... '1:`(] * [ ([ 128!:2 ,&<) ] - 1:)@.(0: < ])&>/'&([ 128!:2 ,&<)
fac f. NB. Showing the stateless factorial step...
1:`(] * [ ([ 128!:2 ,&<) ] - 1:)@.(0: < ])
Fib f. Y NB. Showing the stateless recursive Fibonacci function...
'(([ ([ 128!:2 ,&<) ] - 2:) + [ ([ 128!:2 ,&<) ] - 1:)^:(1: < ])&>/'&([ 128!:2 ,&<)
Fib f. NB. Showing the stateless Fibonacci step...
(([ ([ 128!:2 ,&<) ] - 2:) + [ ([ 128!:2 ,&<) ] - 1:)^:(1: < ])</lang> A structured derivation of Y follows: <lang j>sr=. [ 128!:2 ,&< NB. Self referring lw=. '(5!:5)<x' (1 :) NB. Linear representation of a word Y=. (&>)/lw(&sr) f. Y=. 'Y'f. NB. Fixing it</lang>
alternate implementation
Another approach uses a J gerund as a "lambda" which can accept a single argument, and `:6 to mark a value which would correspond to the first element of an evaluated list in a lisp-like language.
(Multiple argument lambdas are handled by generating and evaluating an appropriate sequence of these lambdas -- in other words, (lambda (x y z) ...) is implemented as (lambda (x) (lambda (y) (lambda (z) ...))) and that particular example would be used as (((example X) Y) Z)) -- or, using J's syntax, that particular example would be used as: ((example`:6 X)`:6 Y)`:6 Z -- but we can also define a word with the value `:6 for a hypothetical slight increase in clarity.
<lang j>lambda=:3 :0
if. 1=#;:y do. 3 :(y,'=.y',LF,0 :0)` else. (,<#;:y) Defer (3 :(',y,=.y',LF,0 :0))` end.
)
Defer=:2 :0
if. (_1 {:: m) <: #m do. v |. y;_1 }. m else. (y;m) Defer v` end.
)
recursivelY=: lambda 'g recur x'
(g`:6 recur`:6 recur)`:6 x
)
sivelY=: lambda 'g recur'
(recursivelY`:6 g)`:6 recur
)
Y=: lambda 'g'
recur=. sivelY`:6 g recur`:6 recur
)
almost_factorial=: lambda 'f n'
if. 0 >: n do. 1 else. n * f`:6 n-1 end.
)
almost_fibonacci=: lambda 'f n'
if. 2 > n do. n else. (f`:6 n-1) + f`:6 n-2 end.
)
Ev=: `:6</lang>
Example use:
<lang J> (Y Ev almost_factorial)Ev 9 362880
(Y Ev almost_fibonacci)Ev 9
34
(Y Ev almost_fibonacci)Ev"0 i. 10
0 1 1 2 3 5 8 13 21 34</lang>
Note that the names f
and recur
will experience the same value (which will be the value produced by sivelY g
).
Java
Java doesn't (currently) have function types. But we can use a generic function interface in the same way. <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>
<lang java5>import java.util.function.Function;
public class YCombinator {
interface RecursiveFunc<F> extends Function<RecursiveFunc<F>, F> { } public static <A,B> Function<A,B> fix(Function<Function<A,B>, Function<A,B>> f) { RecursiveFunc<Function<A,B>> r = w -> f.apply(x -> w.apply(w).apply(x)); return r.apply(r); }
public static void main(String[] args) { Function<Integer,Integer> fib = fix(f -> n -> {
if (n <= 2) return 1; return f.apply(n - 1) + f.apply(n - 2);
}); Function<Integer,Integer> fac = fix(f -> n -> {
if (n <= 1) return 1; return n * f.apply(n - 1); });
System.out.println("fib(10) = " + fib.apply(10)); System.out.println("fac(10) = " + fac.apply(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
<lang javascript>function Y(f) {
var g = f(function() { 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> 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;
}</lang>
Changing the oder of function application (ie 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.
Joy
<lang joy>DEFINE y == [dup cons] swap concat dup cons i;
fac == [ [pop null] [pop succ] [[dup pred] dip i *] ifte ] y.</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>
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
<lang Mathematica>Y = Function[f, #@# &@Function[x, f[x[x]@# &]]]; factorial = Y@Function[f, If[# < 1, 1, # f[# - 1]] &]; fibonacci = Y@Function[f, If[# < 2, #, f[# - 1] + f[# - 2]] &];</lang>
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 fix (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 {
FuncFunc almost_fac = ^Func(Func f) { return ^(int n) { if (n <= 1) return 1; return n * f(n - 1); }; };
FuncFunc almost_fib = ^Func(Func f) { return ^(int n) { if (n <= 2) return 1; return f(n - 1) + f(n - 2); }; };
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>
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>
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>
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)->($_), 0..9), "\n";
} </lang>
- Output:
1 1 2 6 24 120 720 5040 40320 362880 0 1 1 2 3 5 8 13 21 34
Perl 6
<lang perl6>sub Y ($f) { { .($_) }( -> $y { $f({ $y($y)($^arg) }) } ) } sub fac ($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 Perl 6 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>
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 $factorial(10), "\n"; ?></lang>
with create_function instead of real closures. A little far-fetched, but... <lang php><?php function Y($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>
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>
The 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>
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>
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
<lang rexx>/*REXX program to implement a stateless Y combinator. */ numeric digits 1000 /*allow big 'uns. */
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 fact. */ say ' fact' Y(fact (0 1 2 3 4 5 6 7 8 9 10 11)) /*single fact. */ say ' Dfact' Y(dfact (4 5 6 7 8 9 10 11 12 13)) /*double fact. */ say ' Tfact' Y(tfact (4 5 6 7 8 9 10 11 12 13)) /*triple fact. */ say ' Qfact' Y(qfact (4 5 6 7 8 40)) /*quadruple fact. */ say ' length' Y(length (when for to where whenceforth)) /*lengths of words*/ say 'reverse' Y(reverse (23 678 1007 45 MAS I MA)) /*reverses strings*/ say ' trunc' Y(trunc (-7.0005 12 3.14159 6.4 78.999)) /*truncates numbs.*/ exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────subroutines─────────────────────────*/
Y: lambda=; parse arg Y _; do j=1 for words(_); interpret , 'lambda=lambda' Y'('word(_,j)')'; end; return lambda 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; arg x; !=1; do j=x to 2 by -2;!=!*j; end; return ! tfact: procedure; arg x; !=1; do j=x to 2 by -3;!=!*j; end; return ! qfact: procedure; arg x; !=1; do j=x to 2 by -4;!=!*j; end; return ! fact: procedure; arg x; !=1; do j=2 to x ;!=!*j; end; return !</lang>
output
fib 12586269025 fib 144 89 55 34 21 13 8 5 3 2 1 1 0 fact 8320987112741390144276341183223364380754172606361245952449277696409600000000000000 fact 1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 Dfact 8 15 48 105 384 945 3840 10395 46080 135135 Tfact 4 10 18 28 80 162 280 880 1944 3640 Qfact 4 5 12 21 32 3805072588800 length 4 3 2 5 11 reverse 32 876 7001 54 SAM I AM trunc -7 12 3 6 78
Ruby
Using a lambda:
<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>
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>
Rust
<lang rust>enum Mu<T> { Roll(@fn(Mu<T>) -> T) } fn unroll<T>(Roll(f): Mu<T>) -> @fn(Mu<T>) -> T { f }
type RecFunc<A, B> = @fn(@fn(A) -> B) -> @fn(A) -> B;
fn fix<A, B>(f: RecFunc<A, B>) -> @fn(A) -> B {
let g: @fn(Mu<@fn(A) -> B>) -> @fn(A) -> B = |x| |a| f(unroll(x)(x))(a); g(Roll(g))
}
fn main() {
let fac: RecFunc<uint, uint> = |f| |x| if (x==0) { 1 } else { f(x-1) * x }; let fib : RecFunc<uint, uint> = |f| |x| if (x<2) { 1 } else { f(x-1) + f(x-2) };
let ns = std::vec::from_fn(20, |i| i); println(fmt!("%?", ns.map(|&n| fix(fac)(n)))); println(fmt!("%?", ns.map(|&n| fix(fib)(n))));
}</lang>
Derived from: [1]
Scala
Credit goes to the thread in scala blog <lang scala>def Y[A,B](f: (A=>B)=>(A=>B)) = {
case class W(wf: W=>A=>B) { def apply(w: W) = wf(w) } val g: W=>A=>B = w => f(w(w))(_) g(W(g))
}</lang> Example <lang scala>val fac = Y[Int, Int](f => i => if (i <= 0) 1 else f(i - 1) * i) fac(6) //> res0: Int = 720
val fib = Y[Int, Int](f => i => if (i < 2) i else f(i - 1) + f(i - 2)) fib(6) //> res1: Int = 8</lang>
Scheme
<lang scheme>(define Y
(lambda (f) ((lambda (x) (x x)) (lambda (g) (f (lambda args (apply (g g) args)))))))
(define fac
(Y (lambda (f) (lambda (x) (if (< x 2) 1 (* x (f (- x 1))))))))
(define fib
(Y (lambda (f) (lambda (x) (if (< x 2) x (+ (f (- x 1)) (f (- x 2))))))))
(display (fac 6)) (newline)
(display (fib 6)) (newline)</lang> Output:
720 8
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
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>
Tcl
Y combinator is derived in great detail here.
TXR
This prints out 24, the factorial of 4:
<lang txr>@(do
;; 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 @@
is new in TXR 77. When the currying syntax is nested, code in an inner op/do
can refer to numbered implicit parameters in an outer op/do
. Each additional @
"escapes" out one nesting level.
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.
Wart
<lang python>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>
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