Mutual recursion: Difference between revisions

Content added Content deleted

Inline

Revision as of 17:33, 25 June 2009

Two functions are said to be mutually recursive if the first calls the second, and in turn the second calls the first.

Write two mutually recursive functions that compute members of the Hofstadter Female and Male sequences defined as:

{\begin{aligned}F(0)&=1\ ;\ M(0)=0\\F(n)&=n-M(F(n-1)),\quad n>0\\M(n)&=n-F(M(n-1)),\quad n>0.\end{aligned}}

(If a language does not allow for a solution using mutually recursive functions then state this rather than give a solution by other means).

ALGOL 68

Translation of: C

Works with: ALGOL 68 version Standard - no extensions to language used

Works with: ALGOL 68G version Any - tested with release mk15-0.8b.fc9.i386

Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8.8d.fc9.i386

<lang algol>PROC (INT)INT m; # ONLY required for ELLA ALGOL 68RS - an official subset OF full ALGOL 68 #

PROC f = (INT n)INT:

 IF n = 0 THEN 1
 ELSE n - m(f(n-1)) FI;

m := (INT n)INT:

 IF n = 0 THEN 0
 ELSE n - f(m(n-1)) FI;

main: (

 FOR i FROM 0 TO 19 DO
   print(whole(f(i),-3))
 OD;
 new line(stand out);
 FOR i FROM 0 TO 19 DO
   print(whole(m(i),-3))
 OD;
 new line(stand out)

)</lang> Output:

  1  1  2  2  3  3  4  5  5  6  6  7  8  8  9  9 10 11 11 12
  0  0  1  2  2  3  4  4  5  6  6  7  7  8  9  9 10 11 11 12

To let C see functions that will be used, it is enough to declare them. Normally this is done in a header file; in this example we do it directly in the code. If we do not declare them explicity, they get an implicit declaration (if implicit declaration matches the use, everything's fine; but it is better however to write an explicit declaration)

<lang c>#include <stdio.h>

/* let us declare our functions; indeed here we need

  really only M declaration, so that F can "see" it
  and the compiler won't complain with a warning */

int F(int n); int M(int n);

int F(int n) {

 if ( n==0 ) return 1;
 return n - M(F(n-1));

}

int M(int n) {

 if ( n == 0 ) return 0;
 return n - F(M(n-1));

}

int main() {

 int i;

 for(i=0; i < 20; i++) {
   printf("%2d ", F(i));
 }
 printf("\n");
 for(i=0; i < 20; i++) {
   printf("%2d ", M(i));
 }
 printf("\n");
 return 0;

}</lang>

C++

C++ has prior declaration rules similar to those stated above for C, if we would use two functions. Instead here we define M and F as static (class) methods of a class, and specify the bodies inline in the declaration of the class. Inlined methods in the class can still call other methods or access fields in the class, no matter what order they are declared in, without any additional pre-declaration. This is possible because all the possible methods and fields are declared somewhere in the class declaration, which is known the first time the class declaration is parsed. <lang cpp>#include <iostream>

include <vector>
include <iterator>

class Hofstadter { public:

 static int F(int n) {
   if ( n == 0 ) return 1;
   return n - M(F(n-1));
 }
 static int M(int n) {
   if ( n == 0 ) return 0;
   return n - F(M(n-1));
 }

};

using namespace std;

int main() {

 int i;
 vector<int> ra, rb;

 for(i=0; i < 20; i++) {
   ra.push_back(Hofstadter::F(i));
   rb.push_back(Hofstadter::M(i));
 }
 copy(ra.begin(), ra.end(),
      ostream_iterator<int>(cout, " "));
 cout << endl;
 copy(rb.begin(), rb.end(),
      ostream_iterator<int>(cout, " "));
 cout << endl;
 return 0;

}</lang>

The following version shows better what's going on and why we seemingly didn't need pre-declaration (like C) when "encapsulating" the functions as static (class) methods.

This version is equivalent to the above but does not inline the definition of the methods into the definition of the class. Here the method declarations in the class definition serves as the "pre-declaration" for the methods, as in C.

<lang cpp>class Hofstadter { public:

 static int F(int n);
 static int M(int n);

};

int Hofstadter::F(int n) {

 if ( n == 0 ) return 1;
 return n - M(F(n-1));

}

int Hofstadter::M(int n) {

 if ( n == 0 ) return 0;
 return n - F(M(n-1));

}</lang>

Common Lisp

<lang lisp>(defun m (n)

   (if (zerop n)
       0
       (- n (f (m (- n 1))))))

(defun f (n)

   (if (zerop n)
       1
       (- n (m (f (- n 1))))))

</lang>

D

<lang d> long male (long n) { if (n) return n - female (male (n - 1)); else return 0; }

long female (long n) { if (n) return n - male (female (n - 1)); else return 1; } </lang>

E

In E, nouns (variable names) always refer to preceding definitions, so to have mutual recursion, either one must be forward-declared or we must use a recursive def construct. Either one of these is syntactic sugar for first binding the noun to an E promise (a reference with an undetermined target), then resolving the promise to the value.

Recursive def:

<lang e> def [F, M] := [

 fn n { if (n <=> 0) { 1 } else { n - M(F(n - 1)) } },
 fn n { if (n <=> 0) { 0 } else { n - F(M(n - 1)) } },

] </lang>

Forward declaration:

<lang e> def M def F(n) { return if (n <=> 0) { 1 } else { n - M(F(n - 1)) } } bind M(n) { return if (n <=> 0) { 0 } else { n - F(M(n - 1)) } } </lang>

def M binds M to a promise, and stashes the resolver for that promise where bind can get to it. When def F... is executed, the function F closes over the promise which is the value of M. bind M... uses the resolver to resolve M to the provided definition. The recursive def operates similarly, except that it constructs promises for every variable on the left side ([F, M]), executes the right side ([fn ..., fn ...]) and collects the values, then resolves each promise to its corresponding value.

But you don't have to worry about that to use it.

Erlang

<lang erlang>-module(mutrec). -export([mutrec/0, f/1, m/1]).

f(0) -> 1; f(N) -> N - m(f(N-1)).

m(0) -> 0; m(N) -> N - f(m(N-1)).

mutrec() -> lists:map(fun(X) -> io:format("~w ", [f(X)]) end, lists:seq(0,19)), io:format("~n", []), lists:map(fun(X) -> io:format("~w ", [m(X)]) end, lists:seq(0,19)), io:format("~n", []).</lang>

FALSE

[$[$1-f;!m;!-1-]?1+]f:
[$[$1-m;!f;!-  ]?  ]m:
[0[$20\>][\$@$@!." "1+]#%%]t:
 f; t;!"
"m; t;!

Forth

Forward references required for mutual recursion may be set up using DEFER. <lang forth> defer m

f ( n -- n )

 dup 0= if 1+ exit then
 dup 1- recurse m - ;

noname ( n -- n )

 dup 0= if exit then
 dup 1- recurse f - ;

is m

test ( xt n -- ) cr 0 do i over execute . loop drop ;

' m 20 test \ 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 ' f 20 test \ 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 </lang>

Fortran

As long as the code of the two functions is inside the same "block" (module or program) we don't need special care. Otherwise, we should "load" at least the interface of the other function (each module will load mutually the other; of course the compiler won't enter in a infinite loop), e.g. by using a "use" (we do that if M and F function are inside different modules)

Works with: Fortran version 95 and later

<lang fortran>module MutualRec

 implicit none

contains

 pure recursive function m(n) result(r)
   integer :: r
   integer, intent(in) :: n
   if ( n == 0 ) then
      r = 0
      return
   end if
   r = n - f(m(n-1))
 end function m
 
 pure recursive function f(n) result(r)
   integer :: r
   integer, intent(in) :: n
   if ( n == 0 ) then
      r = 1
      return
   end if
   r = n - m(f(n-1))
 end function f

end module</lang>

I've added the attribute pure so that we can use them in a forall statement.

<lang fortran>program testmutrec

 use MutualRec
 implicit none

 integer :: i
 integer, dimension(20) :: a = (/ (i, i=0,19) /), b = (/ (i, i=0,19) /)
 integer, dimension(20) :: ra, rb
 
 forall(i=1:20) 
    ra(i) = m(a(i))
    rb(i) = f(b(i))
 end forall

 write(*,'(20I3)') rb
 write(*,'(20I3)') ra

end program testmutrec</lang>

Groovy

Solution: <lang groovy>def f, m // recursive closures must be declared before they are defined f = { n -> n == 0 ? 1 : n - m(f(n-1)) } m = { n -> n == 0 ? 0 : n - f(m(n-1)) }</lang>

Test program: <lang groovy>println 'f(0..20): ' + (0..20).collect { f(it) } println 'm(0..20): ' + (0..20).collect { m(it) }</lang>

Output:

f(0..20): [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13]
m(0..20): [0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12]

Haskell

Haskell's definitions constructs (at the top level, or inside a let or where construct) are always mutually-recursive: <lang haskell>f 0 = 1 f n | n > 0 = n - m (f $ n-1)

m 0 = 0 m n | n > 0 = n - f (m $ n-1)

main = do

      print $ map f [0..19]
      print $ map m [0..19]</lang>

J

   F =: 1:`(-  M @ $: @ <:) @.*
   M =: 0:`(-  F @ $: @ <:) @.*

Java

Translation of: C

<lang java5>public static int f(int n) {

 if ( n == 0 ) return 1;
 return n - m(f(n - 1));

}

public static int m(int n) {

 if ( n == 0 ) return 0;
 return n - f(m(n - 1));

}

public static void main(String args[]){

  for(int i=0; i < 20; i++) {
     System.out.println(f(i));
  }
  System.out.println();
  for(i=0; i < 20; i++) {
     System.out.println(m(i));
  }

}</lang>

Logo

Like Lisp, symbols in Logo are late-bound so no special syntax is required for forward references.

<lang logo> to m :n

 if 0 = :n [output 0]
 output :n - f m :n-1

end to f :n

 if 0 = :n [output 1]
 output :n - m f :n-1

end

show cascade 20 [lput m #-1 ?] [] [1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12] show cascade 20 [lput f #-1 ?] [] [0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12] </lang>

M4

<lang m4>define(`female',`ifelse(0,$1,1,`eval($1 - male(female(decr($1))))')')dnl define(`male',`ifelse(0,$1,0,`eval($1 - female(male(decr($1))))')')dnl define(`loop',`ifelse($1,$2,,`$3($1) loop(incr($1),$2,`$3')')')dnl loop(0,20,`female') loop(0,20,`male')</lang>

Mathematica

Without caching: <lang Mathematica>

f[0]:=1
m[0]:=0
f[n_]:=n-m[f[n-1]]
m[n_]:=n-f[m[n-1]]

</lang> With caching: <lang Mathematica>

f[0]:=1
m[0]:=0
f[n_]:=f[n]=n-m[f[n-1]]
m[n_]:=m[n]=n-f[m[n-1]]

</lang> Example finding f(1) to f(30) and m(1) to m(30): <lang Mathematica>

m /@ Range[30] 
f /@ Range[30]

</lang> gives back: <lang Mathematica>

{0,1,2,2,3,4,4,5,6,6,7,7,8,9,9,10,11,11,12,12,13,14,14,15,16,16,17,17,18,19}
{1,2,2,3,3,4,5,5,6,6,7,8,8,9,9,10,11,11,12,13,13,14,14,15,16,16,17,17,18,19}

</lang>

Objective-C

Objective-C has prior declaration rules similar to those stated above for C, for C-like types. In this example we show the use of a two class method; this works since we need an interface block that is like declaration of functions in C code.

<lang objc>#import <objc/Object.h>

@interface Hofstadter : Object + (int)M: (int)n; + (int)F: (int)n; @end

@implementation Hofstadter + (int)M: (int)n {

 if ( n == 0 ) return 0;
 return n - [self F: [self M: (n-1)]];

} + (int)F: (int)n {

 if ( n == 0 ) return 1;
 return n - [self M: [self F: (n-1)]];

} @end

int main() {

 int i;

 for(i=0; i < 20; i++) {
   printf("%3d ", [Hofstadter F: i]);
 }
 printf("\n");
 for(i=0; i < 20; i++) {
   printf("%3d ", [Hofstadter M: i]);
 }
 printf("\n");
 return 0;

}</lang>

OCaml

<lang ocaml>let rec f = function

 | 0 -> 1
 | n -> n - m(f(n-1))

and m = function

 | 0 -> 0
 | n -> n - f(m(n-1))

</lang>

The let rec f ... and m ... construct indicates that the functions call themselves (rec) and each other (and).

Octave

We don't need to pre-declare or specify in some other way a function that will be defined later; but both must be declared before their use.
(The code is written to handle vectors, as the testing part shows)

<lang octave>function r = F(n)

 for i = 1:length(n)
   if (n(i) == 0)
     r(i) = 1;
   else
     r(i) = n(i) - M(F(n(i)-1));
   endif
 endfor

endfunction

function r = M(n)

 for i = 1:length(n)
   if (n(i) == 0)
     r(i) = 0;
   else
     r(i) = n(i) - F(M(n(i)-1));
   endif
 endfor

endfunction</lang>

<lang octave># testing ra = F([0:19]); rb = M([0:19]); disp(ra); disp(rb);</lang>

Pascal

In Pascal we need to pre-declare functions/procedures; to do so, the forward statement is used.

<lang pascal>Program MutualRecursion;

{M definition comes after F which uses it} function M(n : Integer) : Integer; forward;

function F(n : Integer) : Integer; begin

  if n = 0 then
     F := 1
  else
     F := n - M(F(n-1));

end;

function M(n : Integer) : Integer; begin

  if n = 0 then
     M := 0
  else
     M := n - F(M(n-1));

end;

var

  i : Integer;

begin

  for i := 0 to 19 do begin
     write(F(i) : 4)
  end;
  writeln;
  for i := 0 to 19 do begin
     write(M(i) : 4)
  end;
  writeln;

end.</lang>

Perl

<lang perl>use strict;

sub F {

   my $n = shift;
   return 1 if $n==0;
   return $n - M(F($n-1));

}

sub M {

   my $n = shift;
   return 0 if $n==0;
   return $n - F(M($n-1));

}

my @ra = (); my @rb = (); for(my $i=0; $i < 20; $i++) {

   push @ra, F($i);
   push @rb, M($i);

} print join(" ", @ra) . "\n"; print join(" ", @rb) . "\n";</lang>

PHP

<lang php><?php function F($n) {

 if ( $n == 0 ) return 1;
 return $n - M(F($n-1));

}

function M($n) {

 if ( $n == 0) return 0;
 return $n - F(M($n-1));

}

$ra = array(); $rb = array(); for($i=0; $i < 20; $i++) {

 array_push($ra, F($i));
 array_push($rb, M($i));

} echo implode(" ", $ra) . "\n"; echo implode(" ", $rb) . "\n"; ?></lang>

Prolog

<lang prolog>female(0,1). female(N,F) :- N>0, N1 is N-1, female(N1,R), male(R, R1), F is N-R1.

male(0,0). male(N,F) :- N>0, N1 is N-1, male(N1,R), female(R, R1), F is N-R1.</lang>

Works with: GNU Prolog

<lang prolog>flist(S) :- for(X, 0, S), female(X, R), format('~d ', [R]), fail. mlist(S) :- for(X, 0, S), male(X, R), format('~d ', [R]), fail.</lang>

Testing

| ?- flist(19).
1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 

no
| ?- mlist(19).
0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12

Python

Works with: Python version 3.0

.

Works with: Python version 2.6

<lang python>def F(n): return 1 if n == 0 else n - M(F(n-1)) def M(n): return 0 if n == 0 else n - F(M(n-1))

print ([ F(n) for n in range(20) ]) print ([ M(n) for n in range(20) ])</lang>

Output:

[1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12]
[0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12]

In python there is no need to pre-declare M for it to be used in the definition of F. (However M must be defined before F calls it).

Ruby

<lang ruby>def F(n)

 n == 0 ? 1 : n - M(F(n-1))

end def M(n)

 n == 0 ? 0 : n - F(M(n-1))

end

p (Array.new(20) {|n| F(n) }) p (Array.new(20) {|n| M(n) })</lang>

Output:

[1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12]
[0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12]

In ruby there is no need to pre-declare M for it to be used in the definition of F. (However M must be defined before F calls it).

Scheme

define declarations are automatically mutually recursive: <lang scheme> (define (F n)

 (if (= n 0) 1
     (- n (M (F (- n 1))))))

(define (M n)

 (if (= n 0) 0
     (- n (F (M (- n 1))))))

</lang>

If you wanted to use a let-like construct to create local bindings, you would do the following. The define construct above is just a syntactic sugar for the following where the entire rest of the scope is used as the body. <lang scheme> (letrec ((F (lambda (n)

             (if (= n 0) 1
                 (- n (M (F (- n 1)))))))
        (M (lambda (n)
             (if (= n 0) 0
                 (- n (F (M (- n 1))))))))
 (F 19)) # evaluates to 12

</lang>

The letrec indicates that the definitions can be recursive, and fact that we placed these two in the same letrec block makes them mutually recursive.

Smalltalk

Using block closure.

<lang smalltalk>|F M ra rb|

F := [ :n |

 (n == 0)
   ifTrue: [ 1 ]
   ifFalse: [ n - (M value: (F value: (n-1))) ]

].

M := [ :n |

 (n == 0)
   ifTrue: [ 0 ]
   ifFalse: [ n - (F value: (M value: (n-1))) ]

].

ra := OrderedCollection new. rb := OrderedCollection new. 0 to: 19 do: [ :i |

 ra add: (F value: i).
 rb add: (M value: i)

].

ra displayNl. rb displayNl.</lang>

SNUSP

The program shown calculates F(3) and demonstrates simple and mutual recursion. <lang SNUSP>

      /======\

F==!/=!\?\+# | />-<-\

   |    \@\-@/@\===?/<#
   |      |    |

$+++/======|====/

   !    /=/       /+<<-\
   |    \!/======?\>>=?/<#  dup
   |      \<<+>+>-/
   !      !
   \======|====\
   |      |    |
   |  /===|==\ |

M==!\=!\?\#| | |

        \@/-@/@/===?\<#
       ^       \>-<-/
       | ^  ^ ^ ^
       | |  | | subtract from n
       | |  | mutual recursion
       | |  recursion
       | n-1
       check for zero

</lang>

Standard ML

<lang sml>fun f 0 = 1

 | f n = n - m (f (n-1))

and m 0 = 0

 | m n = n - f (m (n-1))

</lang>

The fun construct creates recursive functions, and the and allows a group of functions to call each other. The above is just a shortcut for the following:

<lang sml>val rec f = fn 0 => 1

            | n => n - m (f (n-1))

and m = fn 0 => 0

        | n => n - f (m (n-1))

</lang>

which indicates that the functions call themselves (rec) and each other (and).

Tcl

<lang tcl>proc m {n} {

   if { $n == 0 } { expr 0; } else {

expr {$n - [f [m [expr {$n-1}] ]]};

} proc f {n} {

   if { $n == 0 } { expr 1; } else {

expr {$n - [m [f [expr {$n-1}] ]]};

}

for {set i 0} {$i < 20} {incr i} {

   puts -nonewline [f $i];
   puts -nonewline " ";

} puts "" for {set i 0} {$i < 20} {incr i} {

   puts -nonewline [m $i];
   puts -nonewline " ";

} puts ""</lang>

UNIX Shell

Works with: Bourne Again SHell

<lang bash>M() {

   local n
   n=$1
   if $n -eq 0 ; then

echo -n 0

   else

echo -n $(( n - $(F $(M $((n-1)) ) ) ))

fi

}

F() {

   local n
   n=$1
   if $n -eq 0 ; then

echo -n 1

   else

echo -n $(( n - $(M $(F $((n-1)) ) ) ))

fi

}

for((i=0; i < 20; i++)); do

   F $i
   echo -n " "

done echo for((i=0; i < 20; i++)); do

   M $i
   echo -n " "

done echo</lang>

Ursala

Forward declarations are not an issue in Ursala, which allows any definition to depend on any symbol declared in within the same scope. However, cyclic dependences are not allowed unless the programmer is prepared to explain what they mean. If the recurrence can be solved using a fixed point combinator, the compiler can be directed to use one by the #fix directive. A fixed point combinator appropriate for first order functions is available as a library function, as shown below. User defined fixed point combinators allowing recursive definitions of things other than functions are also possible. Note that this feature is rarely needed for defining functions because the language provides other more convenient mechanisms.

import std
import nat
import sol

fix extensional_function_fixer

F = ~&?\1! difference^/~& M+ F+ predecessor M = ~&?\0! difference^/~& F+ M+ predecessor</lang> This test program applies both functions to the first twenty natural numbers. <lang Ursala>#cast %nLW

test = ^(F*,M*) iota 20</lang> output:

(
   <1,1,2,2,3,3,4,5,5,6,6,7,8,8,9,9,10,11,11,12>,
   <0,0,1,2,2,3,4,4,5,6,6,7,7,8,9,9,10,11,11,12>)

x86 assembly

Works with: nasm

Since all "labels" (symbols), if not local, can be seen by the whole code in the same source unit, we don't need special care to let the subroutine func_f call func_m. If the function would have been in another source unit, we should have declared it extern (the linker will resolve the symbol), as done for printf.
(It must be linked with the C standard library libc or similar and a startup code; lazyly a gcc mutrec.o works, being mutrec.o produced by e.g. nasm -f elf mutrec.asm) <lang asm> global main extern printf

section .text

func_f mov eax, [esp+4] cmp eax, 0 jz f_ret dec eax push eax call func_f mov [esp+0], eax call func_m add esp, 4 mov ebx, [esp+4] sub ebx, eax mov eax, ebx ret f_ret mov eax, 1 ret

func_m mov eax, [esp+4] cmp eax, 0 jz m_ret dec eax push eax call func_m mov [esp+0], eax call func_f add esp, 4 mov ebx, [esp+4] sub ebx, eax mov eax, ebx ret m_ret xor eax, eax ret

main mov edx, func_f call output_res mov edx, func_m call output_res ret

output_res xor ecx, ecx loop0 push ecx call edx

       push    edx

push eax push form call printf add esp, 8

pop edx

       pop     ecx

inc ecx cmp ecx, 20 jnz loop0

push newline call printf add esp, 4

ret

section .rodata form db '%d ',0 newline db 10,0

end</lang>