Anonymous recursion
You are encouraged to solve this task according to the task description, using any language you may know.
While implementing a recursive function, it often happens that we must resort to a separate helper function to handle the actual recursion.
This is usually the case when directly calling the current function would waste too many resources (stack space, execution time), causing unwanted side-effects, and/or the function doesn't have the right arguments and/or return values.
So we end up inventing some silly name like foo2 or foo_helper. I have always found it painful to come up with a proper name, and see some disadvantages:
- You have to think up a name, which then pollutes the namespace
- Function is created which is called from nowhere else
- The program flow in the source code is interrupted
Some languages allow you to embed recursion directly in-place. This might work via a label, a local gosub instruction, or some special keyword.
Anonymous recursion can also be accomplished using the Y combinator.
- Task
If possible, demonstrate this by writing the recursive version of the fibonacci function (see Fibonacci sequence) which checks for a negative argument before doing the actual recursion.
11l
<lang 11l>F fib(n)
F f(Int n) -> Int
I n < 2
R n
R @f(n - 1) + @f(n - 2)
R f(n)
L(i) 0..20
print(fib(i), end' ‘ ’)</lang>
- Output:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765
Ada
In Ada you can define functions local to other functions/procedures. This makes it invisible to outside and prevents namespace pollution.
Better would be to use type Natural instead of Integer, which lets Ada do the magic of checking the valid range. <lang Ada> function Fib (X: in Integer) return Integer is
function Actual_Fib (N: in Integer) return Integer is
begin
if N < 2 then
return N;
else
return Actual_Fib (N-1) + Actual_Fib (N-2);
end if;
end Actual_Fib;
begin
if X < 0 then
raise Constraint_Error;
else
return Actual_Fib (X);
end if;
end Fib;</lang>
ALGOL 68
<lang algol68>PROC fibonacci = ( INT x )INT:
IF x < 0
THEN
print( ( "negative parameter to fibonacci", newline ) );
stop
ELSE
PROC actual fibonacci = ( INT n )INT:
IF n < 2
THEN
n
ELSE
actual fibonacci( n - 1 ) + actual fibonacci( n - 2 )
FI;
actual fibonacci( x )
FI;
</lang>
APL
APL has the symbol ∇ which calls the current function recursively.
Functions can be defined and then called in place without ever assigning them a name,
though they are not quite first-class objects (you can't have an array of functions for example).
<lang APL>fib←{ ⍝ Outer function
⍵<0:⎕SIGNAL 11 ⍝ DOMAIN ERROR if argument < 0
{ ⍝ Inner (anonymous) function
⍵<2:⍵
(∇⍵-1)+∇⍵-2 ⍝ ∇ = anonymous recursive call
}⍵ ⍝ Call function in place
}</lang>
AppleScript
<lang applescript>on fibonacci(n) -- "Anonymous recursion" task.
-- For the sake of the task, a needlessly anonymous local script object containing a needlessly recursive handler.
-- The script could easily (and ideally should) be assigned to a local variable.
script
property one : 1
property sequence : {}
on f(n)
if (n < 2) then
set end of my sequence to 0
if (n is 1) then set end of my sequence to one
else
f(n - 1)
set end of my sequence to (item -2 of my sequence) + (end of my sequence)
end if
end f
end script
-- Don't insert any additional code here!
-- Sort out whether the input's positive or negative and tell the object generated above to do the recursive business.
tell result
if (n < 0) then
set its one to -1
set n to -n
end if
f(n)
return its sequence end tell
end fibonacci fibonacci(15) --> {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610} fibonacci(-15) --> {0, -1, -1, -2, -3, -5, -8, -13, -21, -34, -55, -89, -144, -233, -377, -610}</lang>
Or, as the recursion of an anonymous declarative function, enabled by the Y combinator:
<lang applescript>------------ ANONYMOUS RECURSION WITH THE Y-COMBINATOR -------- on run
--------------------- FIBONACCI EXAMPLE -------------------
script
on |λ|(f)
script
on |λ|(n)
if 0 > n then return missing value
if 0 = n then return 0
if 1 = n then return 1
(f's |λ|(n - 2)) + (f's |λ|(n - 1))
end |λ|
end script
end |λ|
end script
unlines(map(showList, chunksOf(12, ¬
map(|Y|(result), enumFromTo(-2, 20)))))
end run
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|
GENERIC FUNCTIONS FOR TEST AND DISPLAY ---------
-- chunksOf :: Int -> [a] -> a on chunksOf(k, xs)
script
on go(ys)
set ab to splitAt(k, ys)
set a to item 1 of ab
if {} ≠ a then
{a} & go(item 2 of ab)
else
a
end if
end go
end script
result's go(xs)
end chunksOf
-- 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
-- intercalate :: String -> [String] -> String
on intercalate(delim, xs)
set {dlm, my text item delimiters} to ¬
{my text item delimiters, delim}
set s to xs as text
set my text item delimiters to dlm
s
end intercalate
-- 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
-- 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
-- showList :: [a] -> String
on showList(xs)
intercalate(", ", map(my str, xs))
end showList
-- splitAt :: Int -> [a] -> ([a], [a])
on splitAt(n, xs)
if n > 0 and n < length of xs then
if class of xs is text then
{items 1 thru n of xs as text, ¬
items (n + 1) thru -1 of xs as text}
else
{items 1 thru n of xs, items (n + 1) thru -1 of xs}
end if
else
if n < 1 then
{{}, xs}
else
{xs, {}}
end if
end if
end splitAt
-- str :: a -> String
on str(x)
x as string
end str
-- unlines :: [String] -> String
on unlines(xs)
-- A single string formed by the intercalation
-- of a list of strings with the newline character.
set {dlm, my text item delimiters} to ¬
{my text item delimiters, linefeed}
set s to xs as text
set my text item delimiters to dlm
s
end unlines</lang>
- Output:
missing value, missing value, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765
AutoHotkey
<lang autohotkey>Fib(n) { nold1 := 1 nold2 := 0 If n < 0 { MsgBox, Positive argument required! Return } Else If n = 0 Return nold2 Else If n = 1 Return nold1 Fib_Label: t := nold2+nold1 If n > 2 { n-- nold2:=nold1 nold1:=t GoSub Fib_Label } Return t }</lang>
AutoIt
<lang autoit> ConsoleWrite(Fibonacci(10) & @CRLF) ; ## USAGE EXAMPLE ConsoleWrite(Fibonacci(20) & @CRLF) ; ## USAGE EXAMPLE ConsoleWrite(Fibonacci(30)) ; ## USAGE EXAMPLE
Func Fibonacci($number)
If $number < 0 Then Return "Invalid argument" ; No negative numbers
If $number < 2 Then ; If $number equals 0 or 1 Return $number ; then return that $number Else ; Else $number equals 2 or more Return Fibonacci($number - 1) + Fibonacci($number - 2) ; FIBONACCI! EndIf
EndFunc </lang>
- Output:
55
6765
832040
Axiom
Using the Aldor compiler in Axiom/Fricas: <lang axiom>#include "axiom" Z ==> Integer; fib(x:Z):Z == { x <= 0 => error "argument outside of range"; f(n:Z,v1:Z,v2:Z):Z == if n<2 then v2 else f(n-1,v2,v1+v2); f(x,1,1); }</lang>The old Axiom compiler has scope issues with calling a local function recursively. One solution is to use the Reference (pointer) domain and initialise the local function with a dummy value: <lang axiom>)abbrev package TESTP TestPackage Z ==> Integer TestPackage : with
fib : Z -> Z
== add
fib x ==
x <= 0 => error "argument outside of range"
f : Reference((Z,Z,Z) -> Z) := ref((n, v1, v2) +-> 0)
f() := (n, v1, v2) +-> if n<2 then v2 else f()(n-1,v2,v1+v2)
f()(x,1,1)</lang>
BaCon
<lang freebasic>
DEF FN fib(x) = FIB(x)
'============================ FUNCTION FIB(int n) TYPE int '============================
IF n < 2 THEN
PRINT n
ELSE
n1 = 0
n2 = 1
FOR i = 1 TO n
sum = n1 + n2
n1 = n2
n2 = sum
NEXT
PRINT n1
END IF
END FUNCTION
'--- less than 2 FIB(0) FIB(1)
'--- greater than or equal to 2 FIB(2) FIB(3) FIB(4) FIB(5) FIB(6) FIB(7) FIB(8) FIB(9)
'--- using an alias 'fib(9) </lang>
BBC BASIC
This works by finding a pointer to the 'anonymous' function and calling it indirectly: <lang bbcbasic> PRINT FNfib(10)
END
DEF FNfib(n%) IF n%<0 THEN ERROR 100, "Must not be negative"
LOCAL P% : P% = !384 + LEN$!384 + 4 : REM Function pointer
(n%) IF n%<2 THEN = n% ELSE = FN(^P%)(n%-1) + FN(^P%)(n%-2)</lang>
- Output:
55
Bracmat
lambda 'light'
The first solution uses macro substitution. In an expression headed by an apostrophe operator with an empty lhs all subexpressions headed by a dollar operator with empty lhs are replaced by the values that the rhs are bound to, without otherwise evaluating the expression. Example: if (x=7) & (y=4) then '($x+3+$y) becomes =7+3+4. Notice that the solution below utilises no other names than arg, the keyword that always denotes a function's argument. The test for negative or non-numeric arguments is outside the recursive part. The function fails if given negative input.
<lang bracmat>( (
=
. !arg:#:~<0
& ( (=.!arg$!arg)
$ (
=
.
' (
. !arg:<2
| (($arg)$($arg))$(!arg+-2)
+ (($arg)$($arg))$(!arg+-1)
)
)
)
$ !arg
)
$ 30 ) </lang> Answer:
832040
pure lambda calculus
(See http://en.wikipedia.org/wiki/Lambda_calculus). The following solution works almost the same way as the previous solution, but uses lambda calculus <lang bracmat>( /(
' ( x
. $x:#:~<0
& ( /('(f.($f)$($f)))
$ /(
' ( r
. /(
' ( n
. $n:<2
| (($r)$($r))$($n+-2)
+ (($r)$($r))$($n+-1)
)
)
)
)
)
$ ($x)
)
)
$ 30 )</lang> Answer:
832040
BQN
𝕊 is a useful symbol in BQN which references the function it is currently in. This can be used to perform anonymous recursion without the need of naming the function block.
The following code calls an anonymous recursive Fibonacci function on each number of the range 0-9. <lang bqn>{
(𝕩<2)◶⟨+´𝕊¨,𝕏⟩𝕩-1‿2
}¨↕10</lang> <lang bqn>⟨ 0 1 1 2 3 5 8 13 21 34 ⟩</lang>
Recursion can also be performed using an internal name defined by a header such as Fact: or Fact 𝕩:. This header is visible inside the block but not outside of it, so from the outside the function is anonymous. The named form allows the outer function to be called within nested blocks, while 𝕊 can only refer to the immediately containing one.
<lang bqn>{Fact 𝕩:
(𝕩<2)◶⟨+´Fact¨,𝕏⟩𝕩-1‿2
}¨↕10</lang>
C
Using scoped function fib_i inside fib, with GCC (required version 3.2 or higher): <lang C>#include <stdio.h>
long fib(long x) {
long fib_i(long n) { return n < 2 ? n : fib_i(n - 2) + fib_i(n - 1); };
if (x < 0) {
printf("Bad argument: fib(%ld)\n", x);
return -1;
}
return fib_i(x);
}
long fib_i(long n) /* just to show the fib_i() inside fib() has no bearing outside it */ {
printf("This is not the fib you are looking for\n");
return -1;
}
int main() {
long x;
for (x = -1; x < 4; x ++)
printf("fib %ld = %ld\n", x, fib(x));
printf("calling fib_i from outside fib:\n");
fib_i(3);
return 0;
}</lang>
- Output:
Bad argument: fib(-1) fib -1 = -1 fib 0 = 0 fib 1 = 1 fib 2 = 1 fib 3 = 2 calling fib_i from outside fib: This is not the fib you are looking for
C#
The inner recursive function (delegate/lambda) has to be named. <lang csharp> static int Fib(int n) {
if (n < 0) throw new ArgumentException("Must be non negativ", "n");
Func<int, int> fib = null; // Must be known, before we can assign recursively to it.
fib = p => p > 1 ? fib(p - 2) + fib(p - 1) : p;
return fib(n);
} </lang>
C++
In C++ (as of the 2003 version of the standard, possibly earlier), we can declare class within a function scope. By giving that class a public static member function, we can create a function whose symbol name is only known to the function in which the class was derived. <lang cpp>double fib(double n) {
if(n < 0)
{
throw "Invalid argument passed to fib";
}
else
{
struct actual_fib
{
static double calc(double n)
{
if(n < 2)
{
return n;
}
else
{
return calc(n-1) + calc(n-2);
}
}
};
return actual_fib::calc(n); }
}</lang>
<lang cpp>#include <functional> using namespace std;
double fib(double n) {
if(n < 0)
throw "Invalid argument";
function<double(double)> actual_fib = [&](double n)
{
if(n < 2) return n;
return actual_fib(n-1) + actual_fib(n-2);
};
return actual_fib(n);
}</lang>
Using a local function object that calls itself using this:
<lang cpp>double fib(double n) {
if(n < 0)
{
throw "Invalid argument passed to fib";
}
else
{
struct
{
double operator()(double n)
{
if(n < 2)
{
return n;
}
else
{
return (*this)(n-1) + (*this)(n-2);
}
}
} actual_fib;
return actual_fib(n); }
}</lang>
Clio
Simple anonymous recursion to print from 9 to 0. <lang clio>10 -> (@eager) fn n:
if n: n - 1 -> print -> recall</lang>
Clojure
The JVM as of now has no Tail call optimization so the default way of looping in Clojure uses anonymous recursion so not to be confusing. <lang clojure> (defn fib [n]
(when (neg? n)
(throw (new IllegalArgumentException "n should be > 0")))
(loop [n n, v1 1, v2 1]
(if (< n 2)
v2
(recur (dec n) v2 (+ v1 v2)))))
</lang> Using an anonymous function
CoffeeScript
<lang coffeescript># This is a rather obscure technique to have an anonymous
- function call itself.
fibonacci = (n) ->
throw "Argument cannot be negative" if n < 0
do (n) ->
return n if n <= 1
arguments.callee(n-2) + arguments.callee(n-1)
- Since it's pretty lightweight to assign an anonymous
- function to a local variable, the idiom below might be
- more preferred.
fibonacci2 = (n) ->
throw "Argument cannot be negative" if n < 0
recurse = (n) ->
return n if n <= 1
recurse(n-2) + recurse(n-1)
recurse(n)
</lang>
Common Lisp
Using Anaphora
This version uses the anaphoric lambda from Paul Graham's On Lisp.
<lang lisp>(defmacro alambda (parms &body body)
`(labels ((self ,parms ,@body))
#'self))</lang>
The Fibonacci function can then be defined as
<lang lisp>(defun fib (n)
(assert (>= n 0) nil "'~a' is a negative number" n)
(funcall
(alambda (n)
(if (>= 1 n)
n (+ (self (- n 1)) (self (- n 2)))))
n))</lang>
Using labels
This puts a function in a local label. The function is not anonymous, but not only is it local, so that its name does not pollute the global namespace, but the name can be chosen to be identical to that of the surrounding function, so it is not a newly invented name
<lang lisp>(defun fib (number)
"Fibonacci sequence function."
(if (< number 0)
(error "Error. The number entered: ~A is negative" number)
(labels ((fib (n a b)
(if (= n 0)
a
(fib (- n 1) b (+ a b)))))
(fib number 0 1))))</lang>
Although name space polution isn't an issue, in recognition of the obvious convenience of anonymous recursive helpers, here is another solution: add the language feature for anonymously recursive blocks: the operator RECURSIVE, with a built-in local operator RECURSE to make recursive calls.
Here is fib rewritten to use RECURSIVE:
<lang lisp>(defun fib (number)
"Fibonacci sequence function."
(if (< number 0)
(error "Error. The number entered: ~A is negative" number)
(recursive ((n number) (a 0) (b 1))
(if (= n 0)
a
(recurse (- n 1) b (+ a b))))))</lang>
Implementation of RECURSIVE: <lang lisp>(defmacro recursive ((&rest parm-init-pairs) &body body)
(let ((hidden-name (gensym "RECURSIVE-")))
`(macrolet ((recurse (&rest args) `(,',hidden-name ,@args)))
(labels ((,hidden-name (,@(mapcar #'first parm-init-pairs)) ,@body))
(,hidden-name ,@(mapcar #'second parm-init-pairs))))))</lang>
RECURSIVE works by generating a local function with LABELS, but with a machine-generated unique name. Furthermore, it provides syntactic sugar so that the initial call to the recursive function takes place implicitly, and the initial values are specified using LET-like syntax. Of course, if RECURSIVE blocks are nested, each RECURSE refers to its own function. There is no way for an inner RECURSIVE to specify recursion to an other RECURSIVE. That is what names are for!
Exercises for reader:
- In the original
fib, the recursive local function can obtain a reference to itself using#'fib. This would allow it to, for instance,(apply #'fib list-of-args). Add a way for RECURSIVE blocks to obtain a reference to themselves. - Add support for &optional and &rest parameters. Optional: also &key and &aux.
- Add LOOPBACK operator whose syntax resembles RECURSE, but which simply assigns the variables and performs a branch back to the top rather than a recursive call.
- Tail recursion optimization is compiler-dependent in Lisp. Modify RECURSIVE so that it walks the expressions and identifies tail-recursive RECURSE calls, rewriting these to use looping code. Be careful that unevaluated literal lists which resemble RECURSE calls are not rewritten, and that RECURSE calls belonging to any nested RECURSIVE invocation are not accidentally treated.
Using the Y combinator
<lang lisp>(setf (symbol-function '!) (symbol-function 'funcall)
(symbol-function '!!) (symbol-function 'apply))
(defmacro ? (args &body body)
`(lambda ,args ,@body))
(defstruct combinator
(name nil :type symbol) (function nil :type function))
(defmethod print-object ((combinator combinator) stream)
(print-unreadable-object (combinator stream :type t) (format stream "~A" (combinator-name combinator))))
(defconstant +y-combinator+
(make-combinator
:name 'y-combinator
:function (? (f) (! (? (g) (! g g))
(? (g) (! f (? (&rest a)
(!! (! g g) a))))))))
(defconstant +z-combinator+
(make-combinator
:name 'z-combinator
:function (? (f) (! (? (g) (! f (? (x) (! (! g g) x))))
(? (g) (! f (? (x) (! (! g g) x))))))))
(defparameter *default-combinator* +y-combinator+)
(defmacro with-y-combinator (&body body)
`(let ((*default-combinator* +y-combinator+))
,@body))
(defmacro with-z-combinator (&body body)
`(let ((*default-combinator* +z-combinator+))
,@body))
(defun x-call (x-function &rest args)
(apply (funcall (combinator-function *default-combinator*) x-function) args))
(defmacro x-function ((name &rest args) &body body)
`(lambda (,name)
(lambda ,args
(macrolet ((,name (&rest args)
`(funcall ,',name ,@args)))
,@body))))
(defmacro x-defun (name args &body body)
`(defun ,name ,args
(x-call (x-function (,name ,@args) ,@body) ,@args)))
- examples
(x-defun factorial (n)
(if (zerop n)
1
(* n (factorial (1- n)))))
(x-defun fib (n)
(case n
(0 0)
(1 1)
(otherwise (+ (fib (- n 1))
(fib (- n 2))))))</lang>
D
<lang d>int fib(in uint arg) pure nothrow @safe @nogc {
assert(arg >= 0);
return function uint(in uint n) pure nothrow @safe @nogc {
static immutable self = &__traits(parent, {});
return (n < 2) ? n : self(n - 1) + self(n - 2);
}(arg);
}
void main() {
import std.stdio;
39.fib.writeln;
}</lang>
- Output:
63245986
With Anonymous Class
In this version anonymous class is created, and by using opCall member function, the anonymous class object can take arguments and act like an anonymous function. The recursion is done by calling opCall inside itself. <lang d>import std.stdio;
int fib(in int n) pure nothrow {
assert(n >= 0);
return (new class {
static int opCall(in int m) pure nothrow {
if (m < 2)
return m;
else
return opCall(m - 1) + opCall(m - 2);
}
})(n);
}
void main() {
writeln(fib(39));
}</lang> The output is the same.
Dylan
This puts a function in a local method binding. The function is not anonymous, but the name fib1 is local and never pollutes the outside namespace.
<lang dylan> define function fib (n)
when (n < 0)
error("Can't take fibonacci of negative integer: %d\n", n)
end;
local method fib1 (n, a, b)
if (n = 0)
a
else
fib1(n - 1, b, a + b)
end
end;
fib1(n, 0, 1)
end </lang>
Déjà Vu
With Y combinator
<lang dejavu>Y f: labda y: labda: f y @y call dup
labda fib n: if <= n 1: 1 else: fib - n 1 fib - n 2 + Y set :fibo
for j range 0 10: !print fibo j</lang>
With recurse
<lang dejavu>fibo-2 n: n 0 1 labda times back-2 back-1: if = times 0: back-2 elseif = times 1: back-1 elseif = times 2: + back-1 back-2 else: recurse -- times back-1 + back-1 back-2 call
for j range 0 10: !print fibo-2 j</lang>
Note that this method starts from 0, while the previous starts from 1.
EchoLisp
A named let provides a local lambda via a label. <lang scheme> (define (fib n) (let _fib ((a 1) (b 1) (n n)) (if (<= n 1) a (_fib b (+ a b) (1- n))))) </lang>
Ela
Using fix-point combinator: <lang ela>fib n | n < 0 = fail "Negative n"
| else = fix (\f n -> if n < 2 then n else f (n - 1) + f (n - 2)) n</lang>
Function 'fix' is defined in standard Prelude as follows: <lang ela>fix f = f (& fix f)</lang>
Elena
ELENA 4.x: <lang elena>import extensions;
fib(n) {
if (n < 0)
{ InvalidArgumentException.raise() };
^ (n)
{
if (n > 1)
{
^ this self(n - 2) + (this self(n - 1))
}
else
{
^ n
}
}(n)
}
public program() {
for (int i := -1, i <= 10, i += 1)
{
console.print("fib(",i,")=");
try
{
console.printLine(fib(i))
}
catch(Exception e)
{
console.printLine:"invalid"
}
};
console.readChar()
}</lang>
- Output:
fib(-1)=invalid 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 fib(10)=55
Elixir
With Y-Combinator: <lang Elixir> fib = fn f -> (
fn x -> if x == 0, do: 0, else: (if x == 1, do: 1, else: f.(x - 1) + f.(x - 2)) end
) end
y = fn x -> (
fn f -> f.(f) end).( fn g -> x.(fn z ->(g.(g)).(z) end) end)
end
IO.inspect y.(&(fib.(&1))).(40) </lang>
- Output:
102334155
Erlang
Two solutions. First fib that use the module to hide its helper. The helper also is called fib so there is no naming problem. Then fib_internal which has the helper function inside itself.
<lang Erlang> -module( anonymous_recursion ). -export( [fib/1, fib_internal/1] ).
fib( N ) when N >= 0 -> fib( N, 1, 0 ).
fib_internal( N ) when N >= 0 -> Fun = fun (_F, 0, _Next, Acc ) -> Acc; (F, N, Next, Acc) -> F( F, N - 1, Acc+Next, Next ) end, Fun( Fun, N, 1, 0 ).
fib( 0, _Next, Acc ) -> Acc;
fib( N, Next, Acc ) -> fib( N - 1, Acc+Next, Next ).
</lang>
F#
Using a nested function:
The function 'fib2' is only visible inside the 'fib' function. <lang fsharp>let fib = function
| n when n < 0 -> None
| n -> let rec fib2 = function
| 0 | 1 -> 1
| n -> fib2 (n-1) + fib2 (n-2)
in Some (fib2 n)</lang>
Using a fixed point combinator: <lang fsharp>let rec fix f x = f (fix f) x
let fib = function
| n when n < 0 -> None | n -> Some (fix (fun f -> (function | 0 | 1 -> 1 | n -> f (n-1) + f (n-2))) n)</lang>
- Output:
Both functions have the same output. <lang fsharp>[-1..5] |> List.map fib |> printfn "%A" [null; Some 1; Some 1; Some 2; Some 3; Some 5; Some 8]</lang>
Factor
One would never use anonymous recursion. The better way defines a private word, like fib2, and recurse by name. This private word would pollute the namespace of one source file.
To achieve anonymous recursion, this solution has a recursive quotation. <lang factor>USING: kernel math ; IN: rosettacode.fibonacci.ar
- fib ( n -- m )
dup 0 < [ "fib of negative" throw ] when
[
! If n < 2, then drop q, else find q(n - 1) + q(n - 2).
[ dup 2 < ] dip swap [ drop ] [
[ [ 1 - ] dip dup call ]
[ [ 2 - ] dip dup call ] 2bi +
] if
] dup call( n q -- m ) ;</lang>
The name q in the stack effect has no significance; call( x x -- x ) would still work.
The recursive quotation has 2 significant disadvantages:
- To enable the recursion, a reference to the quotation stays on the stack. This q impedes access to other things on the stack. This solution must use
dipandswapto move q out of the way. To simplify the code, one might move q to a local variable, but then the recursion would not be anonymous. - Factor cannot infer the stack effect of a recursive quotation. The last line must have
call( n q -- m )instead of plaincall; butcall( n q -- m )defers the stack effect check until runtime. So if the quotation has a wrong stack effect, the compiler would miss the error; only a run offibwould detect the error.
Falcon
Falcon allows a function to refer to itself by use of the fself keyword which is always set to the currently executing function. <lang falcon>function fib(x)
if x < 0
raise ParamError(description|"Negative argument invalid", extra|"Fibbonacci sequence is undefined for negative numbers")
else
return (function(y)
if y == 0
return 0
elif y == 1
return 1
else
return fself(y-1) + fself(y-2)
end
end)(x)
end
end
try
>fib(2)
>fib(3)
>fib(4)
>fib(-1)
catch in e
> e
end</lang>
- Output:
1 2 3 ParamError SS0000 at falcon.core.ParamError._init:(PC:ext.c): Negative argument invalid (Fibbonacci sequence is undefined for negative numbers) Traceback: falcon.core.ParamError._init:0(PC:ext.c) "/home/uDTVwo/prog.fam" prog.fib:3(PC:56) "/home/uDTVwo/prog.fam" prog.__main__:22(PC:132)
FBSL
<lang qbasic>#APPTYPE CONSOLE
FUNCTION Fibonacci(n) IF n < 0 THEN RETURN "Nuts!" ELSE RETURN Fib(n) END IF FUNCTION Fib(m) IF m < 2 THEN Fib = m ELSE Fib = Fib(m - 1) + Fib(m - 2) END IF END FUNCTION END FUNCTION
PRINT Fibonacci(-1.5) PRINT Fibonacci(1.5) PRINT Fibonacci(13.666)
PAUSE</lang> Output:
Nuts! 1.5 484.082 Press any key to continue...
Forth
Recursion is always anonymous in Forth, allowing it to be used in anonymous functions. However, definitions can't be defined during a definition (there are no 'local functions'), and the data stack can't be portably used to get data into a definition being defined.
- and any Forth in which colon-sys consumes zero cells on the data stack.
<lang forth>:noname ( n -- n' )
dup 2 < ?exit 1- dup recurse swap 1- recurse + ; ( xt )
- fib ( +n -- n' )
dup 0< abort" Negative numbers don't exist." [ ( xt from the :NONAME above ) compile, ] ;</lang>
Portability is achieved with a once-off variable (or any temporary-use space with a constant address - i.e., not PAD): <lang forth>( xt from :noname in the previous example ) variable pocket pocket !
- fib ( +n -- n' )
dup 0< abort" Negative numbers don't exist." [ pocket @ compile, ] ;</lang>
Currently, most Forths have started to support embedded definitions (shown here for iForth): <lang forth>: fib ( +n -- ) dup 0< abort" Negative numbers don't exist" [: dup 2 < ?exit 1- dup MYSELF swap 1- MYSELF + ;] execute . ;</lang>
Fortran
Since a hidden named function instead of an anonymous one seems to be ok with implementors, here is the Fortran version: <lang Fortran>integer function fib(n)
integer, intent(in) :: n
if (n < 0 ) then
write (*,*) 'Bad argument: fib(',n,')'
stop
else
fib = purefib(n)
end if
contains
recursive pure integer function purefib(n) result(f)
integer, intent(in) :: n
if (n < 2 ) then
f = n
else
f = purefib(n-1) + purefib(n-2)
end if
end function purefib
end function fib</lang>
FreeBASIC
FreeBASIC does not support nested functions, lambda expressions, functions inside nested types or even (in the default dialect) gosub.
However, for compatibility with old QB code, gosub can be used if one specifies the 'fblite', 'qb' or 'deprecated dialects: <lang freebasic>' FB 1.05.0 Win64
- Lang "fblite"
Option Gosub enables Gosub to be used
' Using gosub to simulate a nested function Function fib(n As UInteger) As UInteger
Gosub nestedFib Exit Function
nestedFib: fib = IIf(n < 2, n, fib(n - 1) + fib(n - 2)) Return
End Function
' This function simulates (rather messily) gosub by using 2 gotos and would therefore work ' even in the default dialect Function fib2(n As UInteger) As UInteger
Goto nestedFib
exitFib: Exit Function
nestedFib: fib2 = IIf(n < 2, n, fib2(n - 1) + fib2(n - 2)) Goto exitFib
End Function
For i As Integer = 1 To 12
Print fib(i); " ";
Next
For j As Integer = 1 To 12
Print fib2(j); " ";
Next
Print Print "Press any key to quit" Sleep</lang>
- Output:
1 1 2 3 5 8 13 21 34 55 89 144 1 1 2 3 5 8 13 21 34 55 89 144
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.
In this page you can see the program(s) related to this task and their results.
Go
Y combinator solution. Go has no special support for anonymous recursion. <lang go>package main
import "fmt"
func main() {
for _, n := range []int{0, 1, 2, 3, 4, 5, 10, 40, -1} {
f, ok := arFib(n)
if ok {
fmt.Printf("fib %d = %d\n", n, f)
} else {
fmt.Println("fib undefined for negative numbers")
}
}
}
func arFib(n int) (int, bool) {
switch {
case n < 0:
return 0, false
case n < 2:
return n, true
}
return yc(func(recurse fn) fn {
return func(left, term1, term2 int) int {
if left == 0 {
return term1+term2
}
return recurse(left-1, term1+term2, term1)
}
})(n-2, 1, 0), true
}
type fn func(int, int, int) int type ff func(fn) fn type fx func(fx) fn
func yc(f ff) fn {
return func(x fx) fn {
return x(x)
}(func(x fx) fn {
return f(func(a1, a2, a3 int) int {
return x(x)(a1, a2, a3)
})
})
}</lang>
- Output:
fib 0 = 0 fib 1 = 1 fib 2 = 1 fib 3 = 2 fib 4 = 3 fib 5 = 5 fib 10 = 55 fib 40 = 102334155 fib undefined for negative numbers
Groovy
Groovy does not explicitly support anonymous recursion. This solution is a kludgy trick that takes advantage of the "owner" scoping variable (reserved word) for closures. <lang groovy>def fib = {
assert it > -1
{i -> i < 2 ? i : {j -> owner.call(j)}(i-1) + {k -> owner.call(k)}(i-2)}(it)
}</lang> Test: <lang groovy>def fib0to20 = (0..20).collect(fib) println fib0to20
try {
println fib(-25)
} catch (Throwable e) {
println "KABOOM!!" println e.message
}</lang>
- Output:
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765]
KABOOM!!
assert it > -1
| |
| false
-25
Haskell
Haskell has two ways to use anonymous recursion. Both methods hide the 'anonymous' function from the containing module, however the first method is actually using a named function.
Named function:
We're defining a function 'real' which is only available from within the fib function. <lang haskell>fib :: Integer -> Maybe Integer fib n
| n < 0 = Nothing
| otherwise = Just $ real n
where real 0 = 1
real 1 = 1
real n = real (n-1) + real (n-2)</lang>
Anonymous function:
This uses the 'fix' function to find the fixed point of the anonymous function. <lang haskell>import Data.Function (fix)
fib :: Integer -> Maybe Integer fib n
| n < 0 = Nothing | otherwise = Just $ fix (\f -> (\n -> if n > 1 then f (n-1) + f (n-2) else 1)) n</lang>
- Output:
Both functions provide the same output when run in GHCI. <lang haskell>ghci> map fib [-4..10] [Nothing,Nothing,Nothing,Nothing,Just 1,Just 1,Just 2,Just 3,Just 5,Just 8,Just 13,Just 21,Just 34,Just 55,Just 89]</lang>
Or, without imports (inlining an anonymous fix)
<lang haskell>fib :: Integer -> Maybe Integer fib n
| n < 0 = Nothing
| otherwise =
Just $
(\f ->
let x = f x
in x)
(\f n ->
if n > 1
then f (n - 1) + f (n - 2)
else 1)
n
-- TEST ---------------------------------------------------------------------- main :: IO () main =
print $
fib <$> [-4 .. 10] >>=
\m ->
case m of
Just x -> [x]
_ -> []</lang>
- Output:
[1,1,2,3,5,8,13,21,34,55,89]
Icon and Unicon
The following solution works in both languages. A cache is used to improve performance.
This example is more a case of can it even be done, and just because we CAN do something - doesn't mean we should do it. The use of co-expressions for this purpose was probably never intended by the language designers and is more than a little bit intensive and definitely NOT recommended.
This example does accomplish the goals of hiding the procedure inside fib so that the type and value checking is outside the recursion. It also does not require an identifier to reference the inner procedure; but, it requires a local variable to remember our return point. Also, each recursion will result in the current co-expression being refreshed, essentially copied, placing a heavy demand on co-expression resources. <lang Icon>procedure main(A)
every write("fib(",a := numeric(!A),")=",fib(a))
end
procedure fib(n)
local source, i
static cache
initial {
cache := table()
cache[0] := 0
cache[1] := 1
}
if type(n) == "integer" & n >= 0 then
return n @ makeProc Template:I := @(source := &source)
end
procedure makeProc(A)
A := if type(A) == "list" then A[1] return (@A, A) # prime and return
end</lang> Some of the code requires some explaining:
- The double curly brace syntax after makeProc serves two different purposes, the outer set is used in the call to create a co-expression. The inner one binds all the expressions together as a single unit.
- At #1 we remember where we came from and receive n from our caller
- At #2 we transmit the new parameters to refreshed copies of the current co-expression setup to act as a normal procedure and cache the result.
- At #3 we transmit the result back to our caller.
- The procedure makeProc consumes the the first transmission to the co-expression which is ignored. Essentially this primes the co-expression to behave like a regular procedure.
For reference, here is the non-cached version: <lang Icon>procedure fib(n)
local source, i
if type(n) == "integer" & n >= 0 then
return n @ makeProc {{
i := @(source := &source)
if i = (0|1) then i@source
((i-1)@makeProc(^¤t) + (i-2)@makeProc(^¤t)) @ source
}}
end</lang> The performance of this second version is 'truly impressive'. And I mean that in a really bad way. By way of example, using default memory settings on a current laptop, a simple recursive non-cached fib out distanced the non-cached fib above by a factor of 20,000. And a simple recursive cached version out distanced the cached version above by a factor of 2,000.
Io
The most natural way to solve this task is to use a nested function whose scope is limited to the helper function. <lang Io>fib := method(x,
if(x < 0, Exception raise("Negative argument not allowed!"))
fib2 := method(n,
if(n < 2, n, fib2(n-1) + fib2(n-2))
)
fib2(x floor)
)</lang>
IS-BASIC
<lang IS-BASIC>100 PROGRAM "Fibonacc.bas" 110 FOR I=0 TO 10 120 PRINT FIB(I); 130 NEXT 140 DEF FIB(K) 150 SELECT CASE K 160 CASE IS<0 170 PRINT "Negative parameter to Fibonacci.":STOP 180 CASE 0 190 LET FIB=0 200 CASE 1 210 LET FIB=1 220 CASE ELSE 230 LET FIB=FIB(K-1)+FIB(K-2) 240 END SELECT 250 END DEF</lang>
J
Copied directly from the fibonacci sequence task, which in turn copied from one of several implementations in an essay on the J Wiki: <lang j> fibN=: (-&2 +&$: -&1)^:(1&<) M."0</lang> Note that this is an identity function for arguments less than 1 (and 1 (and 5)).
Examples: <lang j> fibN 12 144
fibN i.31
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040</lang> (This implementation is doubly recursive except that results are cached across function calls.)
$: is an anonymous reference to the largest containing verb in the sentence.
Note also http://www.jsoftware.com/pipermail/general/2003-August/015571.html which points out that the form
<lang j>basis ` ($: @: g) @. test</lang> which is an anonymous form matches the "tail recursion" pattern is not automatically transformed to satisfy the classic "tail recursion optimization". That optimization would be implemented as transforming this particular example of recursion to the non-recursive <lang j>basis @: (g^:test^:_)</lang>
Of course, that won't work here, because we are adding two recursively obtained results where tail recursion requires that the recursive result is the final result.
Java
Creates an anonymous inner class to do the dirty work. While it does keep the recursive function out of the namespace of the class, it does seem to violate the spirit of the task in that the function is still named.
<lang java>public static long fib(int n) {
if (n < 0)
throw new IllegalArgumentException("n can not be a negative number");
return new Object() {
private long fibInner(int n) {
return (n < 2) ? n : (fibInner(n - 1) + fibInner(n - 2));
}
}.fibInner(n);
}</lang>
Another way is to use the Java Y combinator implementation (the following uses the Java 8 version for better readability). Note that the fib method below is practically the same as that of the version above, with less fibInner.
<lang java5>import java.util.function.Function;
@FunctionalInterface interface SelfApplicable<OUTPUT> {
OUTPUT apply(SelfApplicable<OUTPUT> input);
}
class Utils {
public static <INPUT, OUTPUT> SelfApplicable<Function<Function<Function<INPUT, OUTPUT>, Function<INPUT, OUTPUT>>, Function<INPUT, OUTPUT>>> y() {
return y -> f -> x -> f.apply(y.apply(y).apply(f)).apply(x);
}
public static <INPUT, OUTPUT> Function<Function<Function<INPUT, OUTPUT>, Function<INPUT, OUTPUT>>, Function<INPUT, OUTPUT>> fix() {
return Utils.<INPUT, OUTPUT>y().apply(Utils.<INPUT, OUTPUT>y());
}
public static long fib(int m) {
if (m < 0)
throw new IllegalArgumentException("n can not be a negative number");
return Utils.<Integer, Long>fix().apply(
f -> n -> (n < 2) ? n : (f.apply(n - 1) + f.apply(n - 2))
).apply(m);
}
}</lang>
JavaScript
<lang javascript>function fibo(n) {
if (n < 0) { throw "Argument cannot be negative"; }
return (function(n) {
return (n < 2) ? 1 : arguments.callee(n-1) + arguments.callee(n-2);
})(n);
}</lang>
Note that arguments.callee will not be available in ES5 Strict mode. Instead, you are expected to "name" function (the name is only visible inside function however).
<lang javascript>function fibo(n) {
if (n < 0) { throw "Argument cannot be negative"; }
return (function fib(n) {
return (n < 2) ? 1 : fib(n-1) + fib(n-2);
})(n);
}</lang>
jq
The "recurse" filter supports a type of anonymous recursion, e.g. to generate a stream of integers starting at 0: <lang jq>0 | recurse(. + 1)</lang>
Also, as is the case for example with Julia, jq allows you to define an inner/nested function (in the follow example, aux) that is only defined within the scope of the surrounding function (here fib). It is thus invisible outside the function:
<lang jq>def fib(n):
def aux: if . == 0 then 0
elif . == 1 then 1
else (. - 1 | aux) + (. - 2 | aux)
end;
if n < 0 then error("negative arguments not allowed")
else n | aux
end ;</lang>
Julia
Julia allows you to define an inner/nested function (here, aux) that is only defined within the surrounding function fib scope.
<lang julia>function fib(n)
if n < 0
throw(ArgumentError("negative arguments not allowed"))
end
aux(m) = m < 2 ? one(m) : aux(m-1) + aux(m-2)
aux(n)
end</lang>
K
<lang k>fib: {:[x<0; "Error Negative Number"; {:[x<2;x;_f[x-2]+_f[x-1]]}x]}</lang> Examples: <lang k> fib'!10 0 1 1 2 3 5 8 13 21 34
fib -1
"Error Negative Number"</lang>
Klingphix
<lang>include ..\Utilitys.tlhy
- fib %f !f
%fr
[ %n !n
$n 2 <
( [$n]
[$n 1 - $fr eval $n 2 - $fr eval +] )
if
] !fr
$f 0 <
( ["Error: number is negative"]
[$f true $fr if] )
if
25 fib ?
msec ?
"End " input</lang>
Klong
<lang K> fib::{:[x<0;"error: negative":|x<2;x;.f(x-1)+.f(x-2)]} </lang>
Kotlin
<lang scala>fun fib(n: Int): Int {
require(n >= 0)
fun fib1(k: Int, a: Int, b: Int): Int =
if (k == 0) a else fib1(k - 1, b, a + b)
return fib1(n, 0, 1)
}
fun main(args: Array<String>) {
for (i in 0..20) print("${fib(i)} ")
println()
}</lang>
- Output:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765
Lambdatalk
<lang scheme> 1) defining a tail-recursive function: {def fibo {lambda {:n}
{{lambda {:f :n :a :b} {:f :f :n :a :b}}
{lambda {:f :n :a :b}
{if {< :n 0}
then the number must be positive!
else {if {< :n 1}
then :a
else {:f :f {- :n 1} {+ :a :b} :a}}}} :n 1 0}}}
2) testing: {fibo -1} -> the number must be positive! {fibo 0} -> 1 {fibo 8} -> 34 {fibo 1000} -> 7.0330367711422765e+208 {map fibo {serie 1 20}} -> 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946
We could also avoid any name and write an IIFE
{{lambda {:n}
{{lambda {:f :n :a :b} {:f :f :n :a :b}}
{lambda {:f :n :a :b}
{if {< :n 0}
then the number must be positive!
else {if {< :n 1}
then :a
else {:f :f {- :n 1} {+ :a :b} :a}}}} :n 1 0}}
8}
-> 34
</lang>
Lingo
Lingo does not support anonymous functions. But what comes close: you can create and instantiate an "anonymous class": <lang lingo>on fib (n)
if n<0 then return _player.alert("negative arguments not allowed")
-- create instance of unnamed class in memory only (does not pollute namespace) m = new(#script) r = RETURN m.scriptText = "on fib (me,n)"&r&"if n<2 then return n"&r&"return me.fib(n-1)+me.fib(n-2)"&r&"end" aux = m.script.new() m.erase()
return aux.fib(n)
end</lang>
<lang lingo>put fib(10) -- 55</lang>
LOLCODE
<lang LOLCODE>HAI 1.3
HOW IZ I fib YR x
DIFFRINT x AN BIGGR OF x AN 0, O RLY?
YA RLY, FOUND YR "ERROR"
OIC
HOW IZ I fib_i YR n
DIFFRINT n AN BIGGR OF n AN 2, O RLY?
YA RLY, FOUND YR n
OIC
FOUND YR SUM OF...
I IZ fib_i YR DIFF OF n AN 2 MKAY AN...
I IZ fib_i YR DIFF OF n AN 1 MKAY
IF U SAY SO
FOUND YR I IZ fib_i YR x MKAY
IF U SAY SO
HOW IZ I fib_i YR n
VISIBLE "SRY U CANT HAS FIBS DIS TIEM"
IF U SAY SO
IM IN YR fibber UPPIN YR i TIL BOTH SAEM i AN 5
I HAS A i ITZ DIFF OF i AN 1
VISIBLE "fib(:{i}) = " I IZ fib YR i MKAY
IM OUTTA YR fibber
I IZ fib_i YR 3 MKAY
KTHXBYE</lang>
Lua
Using a Y combinator. <lang lua>local function Y(x) return (function (f) return f(f) end)(function(y) return x(function(z) return y(y)(z) end) end) end
return Y(function(fibs)
return function(n) return n < 2 and 1 or fibs(n - 1) + fibs(n - 2) end
end)</lang> using a metatable (also achieves memoization) <lang lua>return setmetatable({1,1},{__index = function(self, n)
self[n] = self[n-1] + self[n-2] return self[n]
end})</lang>
M2000 Interpreter
We can use a function in string. We can named it so the error say about "Fibonacci" To exclude first check for negative we have to declare a function in anonymous function, which may have a name (a local name) <lang M2000 Interpreter> A$={{ Module "Fibonacci" : Read X :If X<0 then {Error {X<0}} Else Fib=Lambda (x)->if(x>1->fib(x-1)+fib(x-2), x) : =fib(x)}} Try Ok {
Print Function(A$, -12)
} If Error or Not Ok Then Print Error$ Print Function(A$, 12)=144 ' true </lang>
For recursion we can use Lambda() or Lambda$() (for functions which return string) and not name of function so we can use it in a referenced function. Here in k() if we have the fib() we get an error, but with lambda(), interpreter use current function's name.
<lang M2000 Interpreter> Function fib(x) {
If x<0 then Error "argument outside of range"
If x<2 then =x : exit
Def fib1(x)=If(x>1->lambda(x-1)+lambda(x-2), x)
=fib1(x)
} Module CheckIt (&k()) {
Print k(12)
} CheckIt &Fib() Print fib(-2) ' error </lang>
Using lambda function
<lang M2000 Interpreter> fib=lambda -> {
fib1=lambda (x)->If(x>1->lambda(x-1)+lambda(x-2), x)
=lambda fib1 (x) -> {
If x<0 then Error "argument outside of range"
If x<2 then =x : exit
=fib1(x)
}
}() ' using () execute this lambda so fib get the returned lambda Module CheckIt (&k()) {
Print k(12)
} CheckIt &Fib() Try {
Print fib(-2)
} Print Error$ Z=Fib Print Z(12) Dim a(10) a(3)=Z Print a(3)(12)=144 Inventory Alfa = "key1":=Z Print Alfa("key1")(12)=144 </lang>
Using a Group (object in M2000) like a function
A Group may have a name like k (which hold a unique group), or can be unnamed like item in A(4), or can be pointed by a variable (or an array item) (we can use many pointers for the same group)
<lang M2000 Interpreter>
Class Something {
\\ this class is a global function
\\ return a group with a value with one parameter
private:
\\ we can use lambda(), but here we use .fib1() as This.fib1()
fib1=lambda (x)->If(x>1->.fib1(x-1)+.fib1(x-2), x)
public:
Value (x) {
If x<0 then Error "argument outside of range"
If x<2 then =x : exit
=This.fib1(x) \\ we can omit This using .fib1(x)
}
} K=Something() ' K is a static group here Print k(12)=144 Dim a(10) a(4)=Group(K) Print a(4)(12)=144 pk->Something() ' pk is a pointer to group (object in M2000) \\ pointers need Eval to process arguments Print Eval(pk, 12)=144 Inventory Alfa = "Key2":=Group(k), 10*10:=pk Print Alfa("Key2")(12)=144 Print Eval(Alfa("100"),12)=144, Eval(Alfa(100),12)=144 </lang>
Maple
In Maple, the keyword thisproc refers to the currently executing procedure (closure), which need not be named. The following defines a procedure Fib, which uses a recursive, anonymous (unnamed) procedure to implement the Fibonacci sequence. For better efficiency, we use Maple's facility for automatic memoisation ("option remember"). <lang Maple> Fib := proc( n :: nonnegint )
proc( k )
option remember; # automatically memoise
if k = 0 then
0
elif k = 1 then
1
else
# Recurse, anonymously
thisproc( k - 1 ) + thisproc( k - 2 )
end
end( n )
end proc: </lang> For example: <lang Maple> > seq( Fib( i ), i = 0 .. 10 );
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
> Fib( -1 ); Error, invalid input: Fib expects its 1st argument, n, to be of type nonnegint, but received -1 </lang> The check for a negative argument could be put either on the outer Fib procedure, or the anonymous inner procedure (or both). As it wasn't completely clear what was intended, I put it on Fib, which results in a slightly better error message in that it does not reveal how the procedure was actually implemented.
Mathematica / Wolfram Language
An anonymous reference to a function from within itself is named #0, arguments to that function are named #1,#2..#n, n being the position of the argument. The first argument may also be referenced as a # without a following number, the list of all arguments is referenced with ##. Anonymous functions are also known as pure functions in Mathematica. <lang Mathematica>check := #<0& fib := If[check[#],Throw["Negative Argument"],If[#<=1,1,#0[#-2]+#0[#-1]]&[#]]& fib /@ Range[0,10]
{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89}</lang> Making sure that the check is only performed once. <lang Mathematica>check := (Print[#];#<0)& fib /@ Range[0,4] 0 1 2 3 4
{1, 1, 2, 3, 5}</lang>
MATLAB
Not anonymous exactly, but using a nested function solves all the problems stated in the task description.
- does not exist outside of parent function
- does not need a new name, can reuse the parent name
- a nested function can be defined in the place where it is needed
<lang MATLAB> function v = fibonacci(n) assert(n >= 0) v = fibonacci(n,0,1);
% nested function
function a = fibonacci(n,a,b)
if n ~= 0
a = fibonacci(n-1,b,a+b);
end
end
end </lang>
Nemerle
Not anonymous exactly, but using inner function solves all problems stated in task description.
- name is basically the same as outer function and doesn't pollute the namespace
- inner function not expected to be called from anywhere else
- nesting maintains program flow in source code
<lang Nemerle>using System; using System.Console;
module Fib {
Fib(n : long) : long
{
def fib(m : long)
{
|0 => 1
|1 => 1
|_ => fib(m - 1) + fib(m - 2)
}
match(n)
{
|n when (n < 0) => throw ArgumentException("Fib() not defined on negative numbers")
|_ => fib(n)
}
}
Main() : void
{
foreach (i in [-2 .. 10])
{
try {WriteLine("{0}", Fib(i));}
catch {|e is ArgumentException => WriteLine(e.Message)}
}
}
}</lang>
Nim
<lang nim># Using scoped function fibI inside fib proc fib(x: int): int =
proc fibI(n: int): int = if n < 2: n else: fibI(n-2) + fibI(n-1) if x < 0: raise newException(ValueError, "Invalid argument") return fibI(x)
for i in 0..4:
echo fib(i)
- fibI(10) # undeclared identifier 'fibI'</lang>
Output:
0 1 1 2 3
Objective-C
This shows how a method (not regular function) can recursively call itself without explicitly putting its name in the code. <lang objc>#import <Foundation/Foundation.h>
@interface AnonymousRecursion : NSObject { } - (NSNumber *)fibonacci:(NSNumber *)n; @end
@implementation AnonymousRecursion - (NSNumber *)fibonacci:(NSNumber *)n {
int i = [n intValue];
if (i < 0)
@throw [NSException exceptionWithName:NSInvalidArgumentException
reason:@"fibonacci: no negative numbers"
userInfo:nil];
int result;
if (i < 2)
result = 1;
else
result = [[self performSelector:_cmd withObject:@(i-1)] intValue]
+ [[self performSelector:_cmd withObject:@(i-2)] intValue];
return @(result);
} @end
int main (int argc, const char *argv[]) {
@autoreleasepool {
AnonymousRecursion *dummy = [[AnonymousRecursion alloc] init];
NSLog(@"%@", [dummy fibonacci:@8]);
} return 0;
}</lang>
- With internal named recursive block
<lang objc>#import <Foundation/Foundation.h>
int fib(int n) {
if (n < 0)
@throw [NSException exceptionWithName:NSInvalidArgumentException
reason:@"fib: no negative numbers"
userInfo:nil];
int (^f)(int);
__block __weak int (^weak_f)(int); // block cannot capture strong reference to itself
weak_f = f = ^(int n) {
if (n < 2)
return 1;
else
return weak_f(n-1) + weak_f(n-2);
};
return f(n);
}
int main (int argc, const char *argv[]) {
@autoreleasepool {
NSLog(@"%d", fib(8));
} return 0;
}</lang>
When ARC is disabled, the above should be: <lang objc>#import <Foundation/Foundation.h>
int fib(int n) {
if (n < 0)
@throw [NSException exceptionWithName:NSInvalidArgumentException
reason:@"fib: no negative numbers"
userInfo:nil];
__block int (^f)(int);
f = ^(int n) {
if (n < 2)
return 1;
else
return f(n-1) + f(n-2);
};
return f(n);
}
int main (int argc, const char *argv[]) {
@autoreleasepool {
NSLog(@"%d", fib(8));
} return 0;
}</lang>
OCaml
OCaml has two ways to use anonymous recursion. Both methods hide the 'anonymous' function from the containing module, however the first method is actually using a named function.
Named function:
We're defining a function 'real' which is only available from within the fib function. <lang ocaml>let fib n =
let rec real = function
0 -> 1
| 1 -> 1
| n -> real (n-1) + real (n-2)
in
if n < 0 then
None
else
Some (real n)</lang>
Anonymous function:
This uses the 'fix' function to find the fixed point of the anonymous function. <lang ocaml>let rec fix f x = f (fix f) x
let fib n =
if n < 0 then None else Some (fix (fun f -> fun n -> if n <= 1 then 1 else f (n-1) + f (n-2)) n)</lang>
- Output:
# fib 8;; - : int option = Some 34
Ol
This uses named let to create a local function (loop) that only exists inside of function fibonacci. <lang scheme> (define (fibonacci n)
(if (> 0 n)
"error: negative argument."
(let loop ((a 1) (b 0) (count n))
(if (= count 0)
b
(loop (+ a b) a (- count 1))))))
(map fibonacci '(1 2 3 4 5 6 7 8 9 10)))
</lang>
- Output:
'(1 1 2 3 5 8 13 21 34 55)
OxygenBasic
An inner function keeps the name-space clean: <lang oxygenbasic> function fiboRatio() as double
function fibo( double i, j ) as double
if j > 2e12 then return j / i
return fibo j, i + j
end function
return fibo 1, 1
end function
print fiboRatio
</lang>
PARI/GP
This version uses a Y combinator to get a self-reference. <lang parigp>Fib(n)={
my(F=(k,f)->if(k<2,k,f(k-1,f)+f(k-2,f))); if(n<0,(-1)^(n+1),1)*F(abs(n),F)
};</lang>
This version gets a self-reference from self().
<lang parigp>Fib(n)={
my(F=k->my(f=self());if(k<2,k,f(k-1)+f(k-2))); if(n<0,(-1)^(n+1),1)*F(abs(n))
};</lang>
Perl
recur isn't built into Perl, but it's easy to implement.
<lang perl>sub recur (&@) {
my $f = shift; local *recurse = $f; $f->(@_);
}
sub fibo {
my $n = shift;
$n < 0 and die 'Negative argument';
recur {
my $m = shift;
$m < 3 ? 1 : recurse($m - 1) + recurse($m - 2);
} $n;
}</lang> Although for this task, it would be fine to use a lexical variable (closure) to hold an anonymous sub reference, we can also just push it onto the args stack and use it from there: <lang perl>sub fib { my ($n) = @_; die "negative arg $n" if $n < 0; # put anon sub on stack and do a magic goto to it @_ = ($n, sub { my ($n, $f) = @_; # anon sub recurs with the sub ref on stack $n < 2 ? $n : $f->($n - 1, $f) + $f->($n - 2, $f) }); goto $_[1]; }
print(fib($_), " ") for (0 .. 10);</lang>
One can also use caller to get the name of the current subroutine as a string, then call the sub with that string. But this won't work if the current subroutine is anonymous: caller will just return '__ANON__' for the name of the subroutine. Thus, the below program must check the sign of the argument every call, failing the task. Note that under stricture, the line no strict 'refs'; is needed to permit using a string as a subroutine.
<lang perl>sub fibo {
my $n = shift; $n < 0 and die 'Negative argument'; no strict 'refs'; $n < 3 ? 1 : (caller(0))[3]->($n - 1) + (caller(0))[3]->($n - 2);
}</lang>
Perl 5.16 and __SUB__
Perl 5.16 introduced __SUB__ which refers to the current subroutine. <lang Perl>use v5.16; say sub {
my $n = shift; $n < 2 ? $n : __SUB__->($n-2) + __SUB__->($n-1)
}->($_) for 0..10</lang>
Phix
Needs 0.8.1+
using classes
With proof that the private fib_i() does not pollute the outer namespace. <lang Phix>class Fib
private function fib_i(integer n)
return iff(n<2?n:this.fib_i(n-1)+this.fib_i(n-2))
end function
public function fib(integer n)
if n<0 then throw("constraint error") end if
return this.fib_i(n)
end function
end class Fib f = new()
function fib_i(integer i)
return sprintf("this is not the fib_i(%d) you are looking for\n",i)
end function
?f.fib(10) --?f.fib_i(10) -- illegal ?fib_i(10)</lang>
- Output:
55 "this is not the fib_i(10) you are looking for\n"
using a lambda expression
Make of this what you will...
Obviously the inner function does not have to and in fact is not allowed to have a name itself, but it needs to be stored in something with a name before it can be called,
and in being anonymous, in order to effect recursion it must be passed to itself, repeatedly and not really anonymous at all anymore.
<lang Phix>function erm(integer n, f)
return f(n,f)
end function
function fib(integer n)
if n<0 then throw("constraint error") end if
return erm(n,function(integer n,f) return iff(n<2?n:f(n-1,f)+f(n-2,f)) end function)
end function
?fib(10)</lang>
- Output:
55
PHP
In this solution, the function is always called using call_user_func() rather than using function call syntax directly. Inside the function, we get the function itself (without having to refer to the function by name) by relying on the fact that this function must have been passed as the first argument to call_user_func() one call up on the call stack. We can then use debug_backtrace() to get this out.
<lang php><?php function fib($n) {
if ($n < 0)
throw new Exception('Negative numbers not allowed');
$f = function($n) { // This function must be called using call_user_func() only
if ($n < 2)
return 1;
else {
$g = debug_backtrace()[1]['args'][0];
return call_user_func($g, $n-1) + call_user_func($g, $n-2);
}
};
return call_user_func($f, $n);
} echo fib(8), "\n"; ?></lang>
- With internal named recursive function
<lang php><?php function fib($n) {
if ($n < 0)
throw new Exception('Negative numbers not allowed');
$f = function($n) use (&$f) {
if ($n < 2)
return 1;
else
return $f($n-1) + $f($n-2);
};
return $f($n);
} echo fib(8), "\n"; ?></lang>
- With a function object that can call itself using
$this
<lang php><?php class fib_helper {
function __invoke($n) {
if ($n < 2)
return 1;
else
return $this($n-1) + $this($n-2);
}
}
function fib($n) {
if ($n < 0)
throw new Exception('Negative numbers not allowed');
$f = new fib_helper();
return $f($n);
} echo fib(8), "\n"; ?></lang>
PicoLisp
<lang PicoLisp>(de fibo (N)
(if (lt0 N)
(quit "Illegal argument" N) )
(recur (N)
(if (> 2 N)
1
(+ (recurse (dec N)) (recurse (- N 2))) ) ) )</lang>
Explanation: The above uses the 'recur' / 'recurse' function pair, which is defined as a standard language extensions as <lang PicoLisp>(de recur recurse
(run (cdr recurse)) )</lang>
Note how 'recur' dynamically defines the function 'recurse' at runtime, by binding the rest of the expression (i.e. the body of the 'recur' statement) to the symbol 'recurse'.
PostScript
Postscript can make use of the higher order combinators to provide recursion. <lang postscript>% primitive recursion /pfact {
{1} {*} primrec}.
%linear recursion /lfact {
{dup 0 eq}
{pop 1}
{dup pred}
{*}
linrec}.
% general recursion /gfact {
{0 eq}
{succ}
{dup pred}
{i *}
genrec}.
% binary recursion /fib {
{2 lt} {} {pred dup pred} {+} binrec}.</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). It uses the Y combinator. <lang prolog>:- use_module(lambda).
fib(N, _F) :- N < 0, !, write('fib is undefined for negative numbers.'), nl.
fib(N, F) :-
% code of Fibonacci PF = \Nb^R^Rr1^(Nb < 2 ->
R = Nb
;
N1 is Nb - 1, N2 is Nb - 2, call(Rr1,N1,R1,Rr1), call(Rr1,N2,R2,Rr1), R is R1 + R2 ),
% The Y combinator.
Pred = PF +\Nb2^F2^call(PF,Nb2,F2,PF),
call(Pred,N,F).</lang>
Python
<lang python>>>> Y = lambda f: (lambda x: x(x))(lambda y: f(lambda *args: y(y)(*args))) >>> fib = lambda f: lambda n: None if n < 0 else (0 if n == 0 else (1 if n == 1 else f(n-1) + f(n-2))) >>> [ Y(fib)(i) for i in range(-2, 10) ] [None, None, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34]</lang>
Same thing as the above, but modified so that the function is uncurried: <lang python>>>>from functools import partial >>> Y = lambda f: (lambda x: x(x))(lambda y: partial(f, lambda *args: y(y)(*args))) >>> fib = lambda f, n: None if n < 0 else (0 if n == 0 else (1 if n == 1 else f(n-1) + f(n-2))) >>> [ Y(fib)(i) for i in range(-2, 10) ] [None, None, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34]</lang>
A different approach: the function always receives itself as the first argument, and when recursing, makes sure to pass the called function as the first argument also <lang python>>>> from functools import partial >>> Y = lambda f: partial(f, f) >>> fib = lambda f, n: None if n < 0 else (0 if n == 0 else (1 if n == 1 else f(f, n-1) + f(f, n-2))) >>> [ Y(fib)(i) for i in range(-2, 10) ] [None, None, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34]</lang>
An interesting approach using introspection (from http://metapython.blogspot.com/2010/11/recursive-lambda-functions.html) <lang python> >>> from inspect import currentframe >>> from types import FunctionType >>> def myself (*args, **kw): ... caller_frame = currentframe(1) ... code = caller_frame.f_code ... return FunctionType(code, caller_frame.f_globals)(*args, **kw) ... >>> print "factorial(5) =", >>> print (lambda n:1 if n<=1 else n*myself(n-1)) ( 5 ) </lang>
Another way of implementing the "Y" function is given in this post: https://stackoverflow.com/questions/481692/can-a-lambda-function-call-itself-recursively-in-python. The main problem to solve is that the function "fib" can't call itself. Therefore, the function "Y" is used to help "fib" call itself.
<lang python> >>> Y = lambda f: lambda n: f(f,n) >>> fib = lambda f, n: None if n < 0 else (0 if n == 0 else (1 if n == 1 else f(f,n-1) + f(f,n-2))) #same as the first three implementations >>> [ Y(fib)(i) for i in range(-2, 10) ] [None, None, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34] </lang>
All in one line: <lang python> >>> fib_func = (lambda f: lambda n: f(f,n))(lambda f, n: None if n < 0 else (0 if n == 0 else (1 if n == 1 else f(f,n-1) + f(f,n-2)))) >>> [ fib_func(i) for i in range(-2, 10) ] [None, None, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34] </lang>
Qi
Use of anonymous recursive functions is not common in Qi. The philosophy of Qi seems to be that using a "silly name" like "foo2" or "foo_helper" makes the code clearer than using anonymous recursive functions.
However, it can be done, for instance like this:
<lang Qi> (define fib
N -> (let A (/. A N
(if (< N 2)
N
(+ (A A (- N 2))
(A A (- N 1)))))
(A A N)))
</lang>
Quackery
Quackery can do anonymous recursion via the word recurse.
[ dup 0 < iff
$ "negative argument passed to fibonacci"
fail
[ dup 2 < if done
dup 1 - recurse
swap 2 - recurse + ] ] is fibonacci ( n --> n )
recurse causes the current nest (i.e. the on that starts [ dup and ends + ] to be evaluated recursively. This means that if, for example, the first recursive call was inside one or more nested nests, it would not work as desired. So the first instance of recurse in:
( *** faulty code *** )
[ dup 0 < iff
$ "negative argument passed to fibonacci"
fail
[ dup 2 < if done
[ [ [ [ [ [
[ dup 1 - recurse ]
] ] ] ] ] ]
swap 2 - recurse + ] ] is fibonacci ( n --> n )
would cause the nest [ dup 1 - recurse ]to be evaluated, and the program would go into a tailspin, recursively evaluating that nest until the return stack overflowed.
This limitation can be overcome with the understanding that recursion can be factored out into two ideas, i.e. self-reference and evaluation. The self-reference word in Quackery is this, which puts a pointer to the current nest on the data stack (usually just called "the stack") and the evaluation word is do, which takes an item from the stack and evaluates it. So this do is equivalent to recurse.
The final example fixes the previous example by using this and do to put the pointer to the current nest on the stack at the correct level of nesting and evaluate it within the nested nests. See also Even or Odd#Quackery: With Anonymous Mutual recursion.
[ dup 0 < iff
$ "negative argument passed to fibonacci"
fail
[ dup 2 < if done
this
[ [ [ [ [ [
[ dip [ dup 1 - ] do ]
] ] ] ] ] ]
swap 2 - recurse + ] ] is fibonacci ( n --> n )
Quackery also provides named recursion with forward and resolves, so this is usually not necessary but can be useful in unusual circumstances.
R
R provides Recall() as a wrapper which finds the calling function, with limitations; Recall will not work if passed to another function as an argument. <lang R>fib2 <- function(n) {
(n >= 0) || stop("bad argument")
( function(n) if (n <= 1) 1 else Recall(n-1)+Recall(n-2) )(n)
}</lang>
Racket
In Racket, local helper function definitions inside of a function are only visible locally and do not pollute the module or global scope.
<lang racket>
- lang racket
- Natural -> Natural
- Calculate factorial
(define (fact n)
(define (fact-helper n acc)
(if (= n 0)
acc
(fact-helper (sub1 n) (* n acc))))
(unless (exact-nonnegative-integer? n)
(raise-argument-error 'fact "natural" n))
(fact-helper n 1))
- Unit tests, works in v5.3 and newer
(module+ test
(require rackunit) (check-equal? (fact 0) 1) (check-equal? (fact 5) 120))
</lang>
This calculates the slightly more complex Fibonacci funciton: <lang racket>
- lang racket
- Natural -> Natural
- Calculate fibonacci
(define (fibb n)
(define (fibb-helper n fibb_n-1 fibb_n-2)
(if (= 1 n)
fibb_n-1
(fibb-helper (sub1 n) (+ fibb_n-1 fibb_n-2) fibb_n-1)))
(unless (exact-nonnegative-integer? n)
(raise-argument-error 'fibb "natural" n))
(if (zero? n) 0 (fibb-helper n 1 0)))
- Unit tests, works in v5.3 and newer
(module+ test
(require rackunit)
(check-exn exn:fail? (lambda () (fibb -2)))
(check-equal?
(for/list ([i (in-range 21)]) (fibb i))
'(0 1 1 2 3 5 8 13 21 34 55 89 144 233
377 610 987 1597 2584 4181 6765)))
</lang>
Also with the help of first-class functions in Racket, anonymous recursion can be implemented using fixed-points operators:
<lang racket>
- lang racket
- We use Z combinator (applicative order fixed-point operator)
(define Z
(λ (f)
((λ (x) (f (λ (g) ((x x) g))))
(λ (x) (f (λ (g) ((x x) g)))))))
(define fibonacci
(Z (λ (fibo)
(λ (n)
(if (<= n 2)
1
(+ (fibo (- n 1))
(fibo (- n 2))))))))
</lang>
> (fibonacci -2) 1 > (fibonacci 5) 5 > (fibonacci 10) 55
Raku
(formerly Perl 6)
In addition to the methods in the Perl entry above, and the Y-combinator described in Y_combinator, you may also refer to an anonymous block or function from the inside: <lang perl6>sub fib($n) {
die "Naughty fib" if $n < 0;
return {
$_ < 2
?? $_
!! &?BLOCK($_-1) + &?BLOCK($_-2);
}($n);
}
say fib(10);</lang> However, using any of these methods is insane, when Raku provides a sort of inside-out combinator that lets you define lazy infinite constants, where the demand for a particular value is divorced from dependencies on more primitive values. This operator, known as the sequence operator, does in a sense provide anonymous recursion to a closure that refers to more primitive values. <lang perl6>constant @fib = 0, 1, *+* ... *; say @fib[10];</lang> Here the closure, *+*, is just a quick way to write a lambda, -> $a, $b { $a + $b }. The sequence operator implicitly maps the two arguments to the -2nd and -1st elements of the sequence. So the sequence operator certainly applies an anonymous lambda, but whether it's recursion or not depends on whether you view a sequence as iteration or as simply a convenient way of memoizing a recursion. Either view is justifiable.
At this point someone may complain that the solution is doesn't fit the specified task because the sequence operator doesn't do the check for negative. True, but the sequence operator is not the whole of the solution; this check is supplied by the subscripting operator itself when you ask for @fib[-1]. Instead of scattering all kinds of arbitrary boundary conditions throughout your functions, the sequence operator maps them quite naturally to the boundary of definedness at the start of a list.
REBOL
<lang rebol> fib: func [n /f][ do f: func [m] [ either m < 2 [m][(f m - 1) + f m - 2]] n] </lang>
REXX
[Modeled after the Fortran example.]
Since a hidden named function (instead of an anonymous function) seems
to be OK with the implementers, here are the REXX versions.
simplistic
<lang rexx>/*REXX program to show anonymous recursion (of a function or subroutine). */ numeric digits 1e6 /*in case the user goes ka-razy with X.*/ parse arg x . /*obtain the optional argument from CL.*/ if x== | x=="," then x= 12 /*Not specified? Then use the default.*/ w= length(x) /*W: used for formatting the output. */
do j=0 for x+1 /*use the argument as an upper limit.*/
say 'fibonacci('right(j, w)") =" fib(j)
end /*j*/ /* [↑] show Fibonacci sequence: 0 ──► X*/
exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ fib: procedure; parse arg z; if z>=0 then return .(z)
say "***error*** argument can't be negative."; exit
.: procedure; parse arg #; if #<2 then return #; return .(#-1) + .(#-2)</lang>
- output when using the input of: 12
fibonacci( 0) = 0 fibonacci( 1) = 1 fibonacci( 2) = 1 fibonacci( 3) = 2 fibonacci( 4) = 3 fibonacci( 5) = 5 fibonacci( 6) = 8 fibonacci( 7) = 13 fibonacci( 8) = 21 fibonacci( 9) = 34 fibonacci(10) = 55 fibonacci(11) = 89 fibonacci(12) = 144
memoization
Since the above REXX version is very slow for larger numbers, the following version was added that incorporates memoization.
It's many orders of magnitude faster for larger values.
<lang rexx>/*REXX program to show anonymous recursion of a function or subroutine with memoization.*/
numeric digits 1e6 /*in case the user goes ka-razy with X.*/
parse arg x . /*obtain the optional argument from CL.*/
if x== | x=="," then x= 12 /*Not specified? Then use the default.*/
@.= .; @.0= 0; @.1= 1 /*used to implement memoization for FIB*/
w= length(x) /*W: used for formatting the output. */
do j=0 for x+1 /*use the argument as an upper limit.*/
say 'fibonacci('right(j, w)") =" fib(j)
end /*j*/ /* [↑] show Fibonacci sequence: 0 ──► X*/
exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ fib: procedure expose @.; arg z; if z>=0 then return .(z)
say "***error*** argument can't be negative."; exit
.: procedure expose @.; arg #; if @.#\==. then return @.#; @.#=.(#-1)+.(#-2); return @.#</lang>
- output is identical to the 1st REXX version.
Ring
<lang ring>
- Project : Anonymous recursion
t=0 for x = -2 to 12
n = x
recursion()
if x > -1
see t + nl
ok
next
func recursion()
nold1=1
nold2=0
if n < 0
see "positive argument required!" + nl
return
ok
if n=0
t=nold2
return t
ok
if n=1
t=nold1
return t
ok
while n
t=nold2+nold1
if n>2
n=n-1
nold2=nold1
nold1=t
loop
ok
return t
end
return t
</lang> Output:
positive argument required! positive argument required! 0 1 1 2 3 5 8 13 21 34 55 89 144
Ruby
Ruby has no keyword for anonymous recursion.
We can recurse a block of code, but we must provide the block with a reference to itself. The easiest solution is to use a local variable.
Ruby with local variable
<lang ruby>def fib(n)
raise RangeError, "fib of negative" if n < 0
(fib2 = proc { |m| m < 2 ? m : fib2[m - 1] + fib2[m - 2] })[n]
end</lang>
<lang ruby>(-2..12).map { |i| fib i rescue :error } => [:error, :error, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144]</lang>
Here 'fib2' is a local variable of the fib() method. Only the fib() method, or a block inside the fib() method, can call this 'fib2'. The rest of this program cannot call this 'fib2', but it can use the name 'fib2' for other things.
- The fib() method has two local variables 'fib2' and 'n'.
- The block has a local variable 'm' and closes on both 'fib2' and 'n'.
Caution! The recursive block has a difference from Ruby 1.8 to Ruby 1.9. Here is the same method, except changing the block parameter from 'm' to 'n', so that block 'n' and method 'n' have the same name. <lang ruby>def fib(n)
raise RangeError, "fib of negative" if n < 0
(fib2 = proc { |n| n < 2 ? n : fib2[n - 1] + fib2[n - 2] })[n]
end</lang> <lang ruby># Ruby 1.9 (-2..12).map { |i| fib i rescue :error } => [:error, :error, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144]
- Ruby 1.8
(-2..12).map { |i| fib i rescue :error } => [:error, :error, 0, 1, 0, -3, -8, -15, -24, -35, -48, -63, -80, -99, -120]</lang> Ruby 1.9 still shows the correct answer, but Ruby 1.8 shows the wrong answer. With Ruby 1.9, 'n' is still a local variable of the block. With Ruby 1.8, 'n' of the block closes on 'n' of the fib() method. All calls to the block share the 'n' of one call to the method. So fib2[n - 1] changes the value of 'n', and fib2[n - 2] uses the wrong value of 'n', thus the wrong answer.
Ruby with Hash
<lang ruby>def fib(n)
raise RangeError, "fib of negative" if n < 0
Hash.new { |fib2, m|
fib2[m] = (m < 2 ? m : fib2[m - 1] + fib2[m - 2]) }[n]
end</lang> This uses a Hash to memoize the recursion. After fib2[m - 1] returns, fib2[m - 2] uses the value in the Hash, without redoing the calculations.
- The fib() method has one local variable 'n'.
- The block has two local variables 'fib2' and 'm', and closes on 'n'.
Ruby with recur/recurse
<lang ruby>require 'continuation' unless defined? Continuation
module Kernel
module_function
def recur(*args, &block)
cont = catch(:recur) { return block[*args] }
cont[block]
end
def recurse(*args)
block = callcc { |cont| throw(:recur, cont) }
block[*args]
end
end
def fib(n)
raise RangeError, "fib of negative" if n < 0
recur(n) { |m| m < 2 ? m : (recurse m - 1) + (recurse m - 2) }
end</lang>
Our recursive block now lives in the 'block' variable of the Kernel#recur method.
To start, Kernel#recur calls the block once. From inside the block, Kernel#recurse calls the block again. To find the block, recurse() plays a trick. First, Kernel#callcc creates a Continuation. Second, throw(:recur, cont) unwinds the call stack until it finds a matching Kernel#catch(:recur), which returns our Continuation. Third, Kernel#recur uses our Continuation to continue the matching Kernel#callcc, which returns our recursive block.
Ruby with arguments.callee
<lang ruby>require 'continuation' unless defined? Continuation
module Kernel
module_function
def function(&block)
f = (proc do |*args|
(class << args; self; end).class_eval do
define_method(:callee) { f }
end
ret = nil
cont = catch(:function) { ret = block.call(*args); nil }
cont[args] if cont
ret
end)
end
def arguments
callcc { |cont| throw(:function, cont) }
end
end
def fib(n)
raise RangeError, "fib of negative" if n < 0
function { |m|
if m < 2
m
else
arguments.callee[m - 1] + arguments.callee[m - 2]
end
}[n]
end</lang> Our recursive block now lives in the 'block' variable of the Kernel#function method. Another block 'f' wraps our original block and sets up the 'arguments' array. Kernel#function returns this wrapper block. Kernel#arguments plays a trick to get the array of arguments from 'f'; this array has an extra singleton method #callee which returns 'f'.
Rust
<lang rust>fn fib(n: i64) -> Option<i64> {
// A function declared inside another function does not pollute the outer namespace.
fn actual_fib(n: i64) -> i64 {
if n < 2 {
n
} else {
actual_fib(n - 1) + actual_fib(n - 2)
}
}
if n < 0 {
None
} else {
Some(actual_fib(n))
}
}
fn main() {
println!("Fib(-1) = {:?}", fib(-1));
println!("Fib(0) = {:?}", fib(0));
println!("Fib(1) = {:?}", fib(1));
println!("Fib(2) = {:?}", fib(2));
println!("Fib(3) = {:?}", fib(3));
println!("Fib(4) = {:?}", fib(4));
println!("Fib(5) = {:?}", fib(5));
println!("Fib(10) = {:?}", fib(10));
}
- [test]
fn test_fib() {
assert_eq!(fib(0).unwrap(), 0); assert_eq!(fib(1).unwrap(), 1); assert_eq!(fib(2).unwrap(), 1); assert_eq!(fib(3).unwrap(), 2); assert_eq!(fib(4).unwrap(), 3); assert_eq!(fib(5).unwrap(), 5); assert_eq!(fib(10).unwrap(), 55);
}
- [test]
fn test_invalid_argument() {
assert_eq!(fib(-1), None);
}</lang>
Scala
Using a Y-combinator: <lang scala>def Y[A, B](f: (A ⇒ B) ⇒ (A ⇒ B)): A ⇒ B = f(Y(f))(_)
def fib(n: Int): Option[Int] =
if (n < 0) None
else Some(Y[Int, Int](f ⇒ i ⇒
if (i < 2) 1
else f(i - 1) + f(i - 2))(n))
-2 to 5 map (n ⇒ (n, fib(n))) foreach println</lang>
- Output:
(-2,None) (-1,None) (0,Some(1)) (1,Some(1)) (2,Some(2)) (3,Some(3)) (4,Some(5)) (5,Some(8))
Scheme
This uses named let to create a function (aux) that only exists inside of fibonacci: <lang scheme>(define (fibonacci n)
(if (> 0 n)
"Error: argument must not be negative."
(let aux ((a 1) (b 0) (count n))
(if (= count 0)
b
(aux (+ a b) a (- count 1))))))
(map fibonacci '(1 2 3 4 5 6 7 8 9 10))</lang>
- Output:
'(1 1 2 3 5 8 13 21 34 55)
Seed7
Uses a local function to do the dirty work. The local function has a name, but it is not in the global namespace. <lang seed7>$ include "seed7_05.s7i";
const func integer: fib (in integer: x) is func
result
var integer: fib is 0;
local
const func integer: fib1 (in integer: n) is func
result
var integer: fib1 is 0;
begin
if n < 2 then
fib1 := n;
else
fib1 := fib1(n-2) + fib1(n-1);
end if;
end func;
begin
if x < 0 then
raise RANGE_ERROR;
else
fib := fib1(x);
end if;
end func;
const proc: main is func
local
var integer: i is 0;
begin
for i range 0 to 4 do
writeln(fib(i));
end for;
end func;</lang>
- Output:
0 1 1 2 3
Sidef
__FUNC__ refers to the current function. <lang ruby>func fib(n) {
return NaN if (n < 0)
func (n) {
n < 2 ? n
: (__FUNC__(n-1) + __FUNC__(n-2))
}(n)
}</lang>
__BLOCK__ refers to the current block.
<lang ruby>func fib(n) {
return NaN if (n < 0)n < 2 ? n : (__BLOCK__(n-1) + __BLOCK__(n-2)) }(n) }</lang>
Smalltalk
In Smalltalk, things are referred to by a name, so the following may not fully qualify as solutions.
Also: doing things like this is not considered "good style" (give things a good name to make the meaning obvious).
Notice, that none of the two solutions below pollutes any namespace; in (a), the variable introduced is strictly local to the method, in (b) not even a method-local is introduced (so maybe (b) does qualify, after all).
a) Use a funny local name (in this case: "_"). Here the closure is defined as "_", and then evaluated (by sending it a value: message). <lang smalltalk> myMethodComputingFib:arg
^ (_ := [:n | n <= 1
ifTrue:[n]
ifFalse:[(_ value:(n - 1))+(_ value:(n - 2))]]
) value:arg.</lang>
b) Define it in a local scope to not infect the outer scopes.
^ [ |
[:n | n <= 1
ifTrue:[1]
ifFalse:[(fib value:(n - 1))+(fun value:(n - 2))]] value:arg.
] value.</lang>
To completely make it anonymous, we could use reflection to get at the current executed block (via thisContext), but that is too ugly and obfuscating to be shown here. SparklingAs a function expression: <lang Sparkling>function(n, f) { return f(n, f); }(10, function(n, f) { return n < 2 ? 1 : f(n - 1, f) + f(n - 2, f); }) </lang> When typed into the REPL: <lang Sparkling>spn:1> function(n, f) { return f(n, f); }(10, function(n, f) { return n < 2 ? 1 : f(n - 1, f) + f(n - 2, f); }) = 89</lang> Standard MLML does not have a built-in construct for anonymous recursion, but you can easily write your own fix-point combinator: <lang sml>fun fix f x = f (fix f) x fun fib n = if n < 0 then raise Fail "Negative"
else
fix (fn fib =>
(fn 0 => 0
|
1 => 1 | n => fib (n-1) + fib (n-2))) n</lang>
Instead of using a fix-point, the more common approach is to locally define a recursive function and call it once: <lang sml>fun fib n = let
fun fib 0 = 0
|
fib 1 = 1 | fib n = fib (n-1) + fib (n-2)
in
if n < 0 then
raise Fail "Negative"
else
fib n
end</lang>
In this example the local function has the same name as the outer function. This is fine: the local definition shadows the outer definition, so the line "fib n" will refer to our helper function. Another variation is possible. Instead, we could define the recursive "fib" at top-level, then shadow it with a non-recursive wrapper. To force the wrapper to be non-recursive, we use the "val" syntax instead of "fun": <lang sml>fun fib 0 = 0 |
fib 1 = 1 | fib n = fib (n-1) + fib (n-2)
val fib = fn n => if n < 0 then raise Fail "Negative" else fib n</lang> SuperColliderSuperCollider has a keyword "thisFunction", which refers to the current function context. The example below uses this for anonymous recursion. One may think that "thisFunction" would refer to the second branch of the if statement, but because if statements are inlined, the function is the outer one. <lang SuperCollider> ( f = { |n| if(n >= 0) { if(n < 2) { n } { thisFunction.(n-1) + thisFunction.(n-2) } } }; (0..20).collect(f) ) </lang> Swift
<lang swift>let fib: Int -> Int = { func f(n: Int) -> Int {
assert(n >= 0, "fib: no negative numbers")
return n < 2 ? 1 : f(n-1) + f(n-2)
}
return f
}() print(fib(8))</lang> <lang swift>let fib: Int -> Int = { var f: (Int -> Int)!
f = { n in
assert(n >= 0, "fib: no negative numbers")
return n < 2 ? 1 : f(n-1) + f(n-2)
}
return f
}() println(fib(8))</lang>
<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)($0) } }
return r.o(r)
} func fib(n: Int) -> Int { assert(n >= 0, "fib: no negative numbers")
return y {f in {n in n < 2 ? 1 : f(n-1) + f(n-2)}} (n)
} println(fib(8))</lang> TailspinSee the actual fibonacci solution Fibonacci_sequence#State_machine TclThis solution uses Tcl 8.5's lambda terms, extracting the current term from the call stack using introspection (storing it in a local variable only for convenience, with that not in any way being the name of the lambda term; just what it is stored in, and only as a convenience that keeps the code shorter). The lambda terms are applied with the # sanity checks
if {[incr n 0] < 0} {error "argument may not be negative"}
apply {x {
if {$x < 2} {return $x} # Extract the lambda term from the stack introspector for brevity set f [lindex [info level 0] 1] expr {[apply $f [incr x -1]] + [apply $f [incr x -1]]} }} $n }</lang> Demonstrating: <lang tcl>puts [fib 12]</lang>
144 The code above can be written without even using a local variable to hold the lambda term, though this is generally less idiomatic because the code ends up longer and clumsier: <lang tcl>proc fib n { if {[incr n 0] < 0} {error "argument may not be negative"}
apply {x {expr {
$x < 2
? $x
: [apply [lindex [info level 0] 1] [incr x -1]]
+ [apply [lindex [info level 0] 1] [incr x -1]]
}}} $n
}</lang>
However, we can create a if {[incr n 0] < 0} {error "argument may not be negative"}
apply {x {expr {
$x < 2 ? $x : recurse([incr x -1]) + recurse([incr x -1])
}}} $n
}</lang> TXRFor the Y combinator approach in TXR, see the Y combinator task. The following easy transliteration of one of the Common Lisp solutions shows the conceptual and cultural compatibility between TXR Lisp macros and CL macros: <lang txrlisp>(defmacro recursive ((. parm-init-pairs) . body) (let ((hidden-name (gensym "RECURSIVE-")))
^(macrolet ((recurse (. args) ^(,',hidden-name ,*args)))
(labels ((,hidden-name (,*[mapcar first parm-init-pairs]) ,*body))
(,hidden-name ,*[mapcar second parm-init-pairs])))))
(defun fib (number) (if (< number 0)
(error "Error. The number entered: ~a is negative" number)
(recursive ((n number) (a 0) (b 1))
(if (= n 0)
a
(recurse (- n 1) b (+ a b))))))
(put-line `fib(10) = @(fib 10)`) (put-line `fib(-1) = @(fib -1)`))</lang>
$ txr anonymous-recursion.txr fib(10) = 55 txr: unhandled exception of type error: txr: possibly triggered by anonymous-recursion.txr:9 txr: Error. The number entered: -1 is negative Aborted (core dumped) UNIX ShellThe shell does not have anonymous functions. Every function must have a name. However, one can create a subshell such that some function, which has a name in the subshell, is effectively anonymous to the parent shell. <lang bash>fib() { if test 0 -gt "$1"; then
echo "fib: fib of negative" 1>&2
return 1
else
(
fib2() {
if test 2 -gt "$1"; then
echo "$1"
else
echo $(( $(fib2 $(($1 - 1)) ) + $(fib2 $(($1 - 2)) ) ))
fi
}
fib2 "$1"
)
fi
}</lang> <lang bash>$ for i in -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12; do > fib $i > done fib: fib of negative fib: fib of negative 0 1 1 2 3 5 8 13 21 34 55 89 144</lang> Ursala<lang Ursala>#import nat fib = ~&izZB?( # test the sign bit of the argument <'fib of negative'>!%, # throw an exception if it's negative
{0,1}^?<a( # test the argument to a recursively defined function
~&a, # if the argument was a member of {0,1}, return it
sum^|W( # otherwise return the sum of two recursive calls
~&, # to the function thus defined
predecessor^~( # with the respective predecessors of
~&, # the given argument
predecessor)))) # and the predecessor thereof</lang>
Anonymous recursion is often achieved using the recursive conditional operator, The recursive conditional operator UTFoolSolution with anonymous class <lang UTFool> ··· http://rosettacode.org/wiki/Anonymous_recursion ··· ⟦import java.util.function.UnaryOperator;⟧ ■ AnonymousRecursion § static
▶ main
• args⦂ String[]
if 0 > Integer.valueOf args[0]
System.out.println "negative argument"
else
System.out.println *UnaryOperator⟨Integer⟩° ■
▶ apply⦂ Integer
• n⦂ Integer
⏎ n ≤ 1 ? n ! (apply n - 1) + (apply n - 2)
°.apply Integer.valueOf args[0]
</lang> VBA<lang vb> Sub Main() Debug.Print F(-10) Debug.Print F(10) End Sub Private Function F(N As Long) As Variant If N < 0 Then
F = "Error. Negative argument"
ElseIf N <= 1 Then
F = N
Else
F = F(N - 1) + F(N - 2)
End If
End Function</lang>
Error. Negative argument 55 Wart<lang wart>def (fib n) if (n >= 0)
(transform n :thru (afn (n)
(if (n < 2)
n
(+ (self n-1)
(self n-2)))))</lang>
WDTE<lang WDTE>let str => 'strings'; let fib n => switch n { < 0 => str.format 'Bad argument: {q}' n;
default => n -> (@ memo s n => switch n {
== 0 => 0; == 1 => 1;
default => + (s (- n 1)) (s (- n 2));
});
};</lang> In WDTE, a lambda, defined in a block delineated by Wren<lang ecmascript>class Fibonacci { static compute(n) {
var fib
fib = Fn.new {|n|
if (n < 2) return n
return fib.call(n - 1) + fib.call(n - 2)
}
if (n < 0) return null
return fib.call(n)
}
} System.print(Fibonacci.compute(36))</lang>
14930352 x86 Assembly32 bit Fakes the actual first call to the function that generates Fibonacci numbers. The address of the instructions after the function get put on the stack and then execution continues into the actual function. When the recursion is complete, instead of returning to the location of the call it goes to the end of the loop. <lang asm>
global _start section .text _start: mov ecx, 48 ; Initialize loop counter .loop: mov ebx, 48 ; Calculate which Fn will be computed
sub ebx, ecx ; Takes into account the reversed nature
push ebx ; Pass the parameter in on the stack
push .done ; Emulate a call but "return" to end of loop
; The return adress is manually set on the stack
.fib: push ebp ; Setup stack frame mov ebp, esp push ebx ; Save needed registers xor eax, eax mov ebx, [ebp + 8] ; Get the parameter cmp ebx, 1 ; Test for base cases jl .return mov eax, 1 je .return dec ebx ; Calculate fib(n-1) push ebx call .fib mov [esp], eax ; Save result on top of parameter in stack dec ebx ; Calculate fib(n-2) push ebx call .fib add eax, [esp + 4] ; Add the first to the second add esp, 8 ; Reset local stack .return: pop ebx ; Restore modified registers mov esp, ebp ; Tear down stack frame and return pop ebp ret .done: mov [esp], ecx ; Save the counter between calls push eax ; Print the number call print_num add esp, 4 pop ecx ; Restore the loop counter loop .loop ; Loop until 0 mov eax, 0x01 ; sys_exit(int error) xor ebx, ebx ; error = 0 (success) int 0x80 ; syscall
print_num: push ebp mov ebp, esp sub esp, 11 ; Save space for digits and newline lea ecx, [ebp - 1] ; Save a pointer to after the buffer mov BYTE [ecx], 0x0A ; Set the newline at the end mov eax, [ebp + 8] ; Get the parameter mov ebx, DWORD 10 ; Divisor .loop: dec ecx ; Move pointer to next digit xor edx, edx div ebx ; Extract one digit, quot in eax, rem in edx add dl, 0x30 ; Convert remainder to ascii mov [ecx], dl ; Save the ascii form cmp eax, 0 ; Loop until all digits have been converted jg .loop mov eax, 0x04 ; sys_write(int fd, char **buf, int len) mov ebx, 1 ; stdout mov edx, ebp ; Calculate the length sub edx, ecx ; address after newline - address of first digit int 0x80 mov esp, ebp pop ebp ret </lang> XPL0In XPL0 you can nest functions/procedures inside other functions/procedures up to eight levels deep. This makes those nested functions invisible to the outside, thus preventing namespace pollution. <lang XPL0>include c:\cxpl\codes; func Fib(X); int X; func ActualFib(N);
int N;
[if N<2 then return N
else return ActualFib(N-1) + ActualFib(N-2);
]; \ActualFib;
[if X<0 then [Text(0, "Error "); return 0] else return ActualFib(X); ]; \Fib; [IntOut(0, Fib(8)); CrLf(0); IntOut(0, Fib(-2)); CrLf(0); ]</lang>
21 Error 0 Yabasic<lang Yabasic>print Fibonacci(-10) print Fibonacci(10)
If number < 0 print "Invalid argument: "; : return number
If number < 2 Then
Return number
Else
Return Fibonacci(number - 1) + Fibonacci(number - 2)
EndIf
end sub</lang> zkl<lang zkl>fcn fib(n){ if (n<0) throw(Exception.ValueError);
fcn(n){
if (n < 2) return(1);
else return(self.fcn(n-1) + self.fcn(n-2));
}(n);
} fib(8) .println(); fib(-8).println(); </lang>
34 ValueError thrown ZX Spectrum Basic<lang zxbasic>10 INPUT "Enter a number: ";n 20 LET t=0 30 GO SUB 60 40 PRINT t 50 STOP 60 LET nold1=1: LET nold2=0 70 IF n<0 THEN PRINT "Positive argument required!": RETURN 80 IF n=0 THEN LET t=nold2: RETURN 90 IF n=1 THEN LET t=nold1: RETURN 100 LET t=nold2+nold1 110 IF n>2 THEN LET n=n-1: LET nold2=nold1: LET nold1=t: GO SUB 100 120 RETURN </lang> |
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