Mutual recursion
Two functions are said to be mutually recursive if the first calls the second, and in turn the second calls the first.
You are encouraged to solve this task according to the task description, using any language you may know.
Write two mutually recursive functions that compute members of the Hofstadter Female and Male sequences defined as:
(If a language does not allow for a solution using mutually recursive functions
then state this rather than give a solution by other means).
8080 Assembly
The 8080 processor has built-in support for recursion, at the instruction level.
The processor keeps a stack pointer, called SP,
which is a 16-bit register that can be set by the program to point anywhere in the address space.
The stack pointer points to the topmost word on the stack. The stack grows downward into memory:
when a word is pushed onto the stack, the SP is decremented by 2, and the word written
at the new location. When a word is popped from the stack, it is read from the location the
SP is pointing to, and afterwards the SP is incremented by 2.
The instruction set includes a call instruction, which pushes the location of the
next instruction onto the stack, and then jumps to the given location. Its counterpart is the
ret instruction, which pops a location from the stack and jumps there.
There are also push and pop instructions, to push and
pop the values of register
pairs on and off the stack directly. This can be used, among other things, to save 'local variables'
in a recursive routine, as the code below does.
<lang 8080asm> org 100h jmp test ;;; Implementation of F(A). F: ana a ; Zero? jz one ; Then set A=1 mov b,a ; Otherwise, set B=A, push b ; And put it on the stack dcr a ; Set A=A-1 call F ; Set A=F(A-1) call M ; Set A=M(F(A-1)) pop b ; Retrieve input value cma ; (-A)+B is actually one cycle faster inr a ; than C=A;A=B;A-=B, and equivalent add b ret one: mvi a,1 ; Set A to 1, ret ; and return. ;;; Implementation of M(A). M: ana a ; Zero? rz ; Then keep it that way and return. mov b,a push b ; Otherwise, same deal as in F, dcr a ; but M and F are called in opposite call M ; order. call F pop b cma inr a add b ret ;;; Demonstration code. test: lhld 6 ; Set stack pointer to highest usable sphl ; memory. ;;; Print F([0..15]) lxi d,fpfx ; Print "F: " mvi c,9 call 5 xra a ; Start with N=0 floop: push psw ; Keep N call F ; Get value for F(N) call pdgt ; Print it pop psw ; Restore N inr a ; Next N cpi 16 ; Done yet? jnz floop ;;; Print M([0..15]) lxi d,mpfx ; Print "\r\nM: " mvi c,9 call 5 xra a ; Start with N=0 mloop: push psw ; same deal as above call M call pdgt pop psw ; Restore N inr a cpi 16 jnz mloop rst 0 ; Explicit exit, we got rid of system stack ;;; Print digit and space pdgt: adi '0' ; ASCII mov e,a mvi c,2 call 5 mvi e,' ' ; Space mvi c,2 jmp 5 ; Tail call optimization fpfx: db 'F: $' mpfx: db 13,10,'M: $'</lang>
- Output:
F: 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 M: 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9
ABAP
This works for ABAP Version 7.40 and can be implemented in procedural ABAP as well, but with classes it is much more readable. As this allows a method with a returning value to be an input for a subsequent method call.
<lang ABAP> report z_mutual_recursion.
class hoffstadter_sequences definition.
public section.
class-methods:
f
importing
n type int4
returning
value(result) type int4,
m
importing
n type int4
returning
value(result) type int4.
endclass.
class hoffstadter_sequences implementation.
method f.
result = cond int4(
when n eq 0
then 1
else n - m( f( n - 1 ) ) ).
endmethod.
method m.
result = cond int4(
when n eq 0
then 0
else n - f( m( n - 1 ) ) ).
endmethod.
endclass.
start-of-selection.
write: |{ reduce string(
init results = |f(0 - 19): { hoffstadter_sequences=>f( 0 ) }|
for i = 1 while i < 20
next results = |{ results }, { hoffstadter_sequences=>f( i ) }| ) }|, /.
write: |{ reduce string(
init results = |m(0 - 19): { hoffstadter_sequences=>m( 0 ) }|
for i = 1 while i < 20
next results = |{ results }, { hoffstadter_sequences=>m( i ) }| ) }|, /.
</lang>
- Output:
f(0 - 19): 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12 m(0 - 19): 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12
ACL2
<lang lisp>(mutual-recursion
(defun f (n)
(declare (xargs :mode :program))
(if (zp n)
1
(- n (m (f (1- n))))))
(defun m (n)
(declare (xargs :mode :program))
(if (zp n)
0
(- n (f (m (1- n)))))))</lang>
Ada
<lang Ada>with Ada.Text_Io; use Ada.Text_Io; procedure Mutual_Recursion is
function M(N : Integer) return Integer;
function F(N : Integer) return Integer is
begin
if N = 0 then
return 1;
else
return N - M(F(N - 1));
end if;
end F;
function M(N : Integer) return Integer is
begin
if N = 0 then
return 0;
else
return N - F(M(N-1));
end if;
end M;
begin
for I in 0..19 loop
Put_Line(Integer'Image(F(I)));
end loop;
New_Line;
for I in 0..19 loop
Put_Line(Integer'Image(M(I)));
end loop;
end Mutual_recursion;</lang>
<lang ada>with Ada.Text_Io; use Ada.Text_Io; procedure Mutual_Recursion is
function M(N: Natural) return Natural;
function F(N: Natural) return Natural;
function M(N: Natural) return Natural is
(if N = 0 then 0 else N – F(M(N–1)));
function F(N: Natural) return Natural is
(if N =0 then 1 else N – M(F(N–1)));
begin
for I in 0..19 loop
Put_Line(Integer'Image(F(I)));
end loop;
New_Line;
for I in 0..19 loop
Put_Line(Integer'Image(M(I)));
end loop;
end Mutual_recursion;</lang>
Aime
<lang aime>integer F(integer n); integer M(integer n);
integer F(integer n) {
integer r;
if (n) {
r = n - M(F(n - 1));
} else {
r = 1;
} return r;
}
integer M(integer n) {
integer r;
if (n) {
r = n - F(M(n - 1));
} else {
r = 0;
} return r;
}
integer main(void) {
integer i;
i = 0;
while (i < 20) {
o_winteger(3, F(i)); i += 1;
}
o_byte('\n');
i = 0;
while (i < 20) {
o_winteger(3, M(i)); i += 1;
}
o_byte('\n');
return 0;
}</lang>
ALGOL 68
<lang algol68>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
ALGOL W
<lang algolw>begin
% define mutually recursive funtions F and M that compute the elements % % of the Hofstadter Female and Male sequences %
integer procedure F ( integer value n ) ;
if n = 0 then 1 else n - M( F( n - 1 ) );
integer procedure M ( integer value n ) ;
if n = 0 then 0 else n - F( M( n - 1 ) );
% print the first few elements of the sequences % i_w := 2; s_w := 1; % set I/O formatting % write( "F: " ); for i := 0 until 20 do writeon( F( i ) ); write( "M: " ); for i := 0 until 20 do writeon( M( i ) );
end.</lang>
APL
<lang apl>f ← {⍵=0:1 ⋄ ⍵-m∇⍵-1} m ← {⍵=0:0 ⋄ ⍵-f∇⍵-1} ⍉'nFM'⍪↑(⊢,f,m)¨0,⍳20</lang>
- Output:
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 F 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 M 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12
AppleScript
<lang AppleScript>-- f :: Int -> Int on f(x)
if x = 0 then
1
else
x - m(f(x - 1))
end if
end f
-- m :: Int -> Int on m(x)
if x = 0 then
0
else
x - f(m(x - 1))
end if
end m
-- TEST
on run
set xs to range(0, 19)
{map(f, xs), map(m, xs)}
end run
-- GENERIC FUNCTIONS
-- 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 lambda(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 lambda : f
end script
end if
end mReturn
-- range :: Int -> Int -> [Int] on range(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 range</lang>
- Output:
<lang AppleScript>{{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}}</lang>
ARM Assembly
Unlike on the x86 family of processors, the ARM instruction set does not include specialized
call and ret instructions. However, the program counter is a visible
register (r15, also called pc), so it can be loaded and saved
as any other. Nor is there a specialized stack pointer, though the load and store instructions offer
pre- and postincrement as well as pre- and postdecrement on the register used as a pointer, making
any register usable as a stack pointer.
By convention, r13 is used as the system stack pointer and is therefore also
called sp, and r14 is used to store the return address for
a function, and is therefore also called the *link register* or lr.
The assembler recognizes push {x} and pop {x} instructions, though these
are really pseudoinstructions, that generate the exact same machine code as
ldmia r13!,{x} and stmdb r13!,{x},
these being, respectively, load with postincrement and store with predecrement on r13.
The link register is slightly special in that there is a family of branch-and-link instructions
(bl). These are the same as mov r14,pc ; mov/ldr pc,<destination>, but in
one machine instruction instead of two. This is the general way of calling subroutines,
meaning no stack access is necessary unless the subroutine wants to call others in turn, in which case
the link register must be saved by hand (as the code below shows several ways of doing).
<lang>.text .global _start @@@ Implementation of F(n), n in R0. n is considered unsigned. F: tst r0,r0 @ n = 0? moveq r0,#1 @ In that case, the result is 1 bxeq lr @ And we can return to the caller push {r0,lr} @ Save link register and argument to stack sub r0,r0,#1 @ r0 -= 1 = n-1 bl F @ r0 = F(r0) = F(n-1) bl M @ r0 = M(r0) = M(F(n-1)) pop {r1,lr} @ Restore link register and argument in r1 sub r0,r1,r0 @ Result is n-F(M(n-1)) bx lr @ Return to caller.
@@@ Implementation of M(n), n in R0. n is considered unsigned. M: tst r0,r0 @ n = 0? bxeq lr @ In that case the result is also 0; return. push {r0,lr} @ Save link register and argument to stack sub r0,r0,#1 @ r0 -= 1 = n-1 bl M @ r0 = M(r0) = M(n-1) bl F @ r0 = M(r0) = F(M(n-1)) pop {r1,lr} @ Restore link register and argument in r1 sub r0,r1,r0 @ Result is n-M(F(n-1)) bx lr @ Return to caller
@@@ Print F(0..15) and M(0..15) _start: ldr r1,=fmsg @ Print values for F ldr r4,=F bl prfn ldr r1,=mmsg @ Print values for M ldr r4,=M bl prfn mov r7,#1 @ Exit process swi #0
@@@ Helper function for output: print [r1], then [r4](0..15) @@@ This assumes [r4] preserves r3 and r4; M and F do. prfn: push {lr} @ Keep link register bl pstr @ Print the string mov r3,#0 @ Start at 0 1: mov r0,r3 @ Call the function in r4 with current number blx r4 add r0,r0,#'0 @ Make ASCII digit ldr r1,=dgt @ Store in digit string strb r0,[r1] ldr r1,=dstr @ Print result bl pstr add r3,r3,#1 @ Next number cmp r3,#15 @ Keep going up to and including 15 bls 1b ldr r1,=nl @ Print newline afterwards bl pstr pop {pc} @ Return to address on stack @@@ Print length-prefixed string r1 to stdout pstr: push {lr} @ Keep link register mov r0,#1 @ stdout = 1 ldrb r2,[r1],#1 @ r2 = length prefix mov r7,#4 @ 4 = write syscall swi #0 pop {pc} @ Return to address on stack .data fmsg: .ascii "\3F: " mmsg: .ascii "\3M: " dstr: .ascii "\2" dgt: .ascii "* " nl: .ascii "\1\n"</lang>
- Output:
F: 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 M: 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9
Arturo
<lang rebol>f: $[n][ if? n=0 -> 1 else -> n-m f n-1 ] m: $[n][ if? n=0 -> 0 else -> n-f m n-1 ]
loop 0..20 'i [ print ["f(" i ")=" f i] print ["m(" i ")=" m i] print "" ]</lang>
- Output:
f( 0 )= 1 m( 0 )= 0 f( 1 )= 1 m( 1 )= 0 f( 2 )= 2 m( 2 )= 1 f( 3 )= 2 m( 3 )= 2 f( 4 )= 3 m( 4 )= 2 f( 5 )= 3 m( 5 )= 3 f( 6 )= 4 m( 6 )= 4 f( 7 )= 5 m( 7 )= 4 f( 8 )= 5 m( 8 )= 5 f( 9 )= 6 m( 9 )= 6 f( 10 )= 6 m( 10 )= 6 f( 11 )= 7 m( 11 )= 7 f( 12 )= 8 m( 12 )= 7 f( 13 )= 8 m( 13 )= 8 f( 14 )= 9 m( 14 )= 9 f( 15 )= 9 m( 15 )= 9 f( 16 )= 10 m( 16 )= 10 f( 17 )= 11 m( 17 )= 11 f( 18 )= 11 m( 18 )= 11 f( 19 )= 12 m( 19 )= 12 f( 20 )= 13 m( 20 )= 12
AutoHotkey
<lang AutoHotkey>Loop 20
i := A_Index-1, t .= "`n" i "`t " M(i) "`t " F(i)
MsgBox x`tmale`tfemale`n%t%
F(n) {
Return n ? n - M(F(n-1)) : 1
}
M(n) {
Return n ? n - F(M(n-1)) : 0
}</lang>
This one is an alternative to the above.
<lang AutoHotkey>main() Return
F(n) {
If (n == 0) Return 1 Else Return n - M(F(n-1))
}
M(n) {
If (n == 0) Return 0 Else Return n - F(M(n-1)) ;
}
main() {
i = 0
While, i < 20
{
male .= M(i) . "`n"
female .= F(i) . "`n"
i++
}
MsgBox % "male:`n" . male
MsgBox % "female:`n" . female
}</lang>
AWK
In AWK it is enough that both functions are defined somewhere. It matters not whether the BEGIN block is before or after the function definitions.
<lang awk>cat mutual_recursion.awk:
- !/usr/local/bin/gawk -f
- User defined functions
function F(n) { return n == 0 ? 1 : n - M(F(n-1)) }
function M(n) { return n == 0 ? 0 : n - F(M(n-1)) }
BEGIN {
for(i=0; i <= 20; i++) {
printf "%3d ", F(i)
}
print ""
for(i=0; i <= 20; i++) {
printf "%3d ", M(i)
}
print ""
}</lang>
- Output:
$ awk -f mutual_recursion.awk 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12
BaCon
<lang freebasic>' Mutually recursive FUNCTION F(int n) TYPE int
RETURN IIF(n = 0, 1, n - M(F(n -1)))
END FUNCTION
FUNCTION M(int n) TYPE int
RETURN IIF(n = 0, 0, n - F(M(n - 1)))
END FUNCTION
' Get iteration limit, default 20 SPLIT ARGUMENT$ BY " " TO arg$ SIZE args limit = IIF(args > 1, VAL(arg$[1]), 20)
FOR i = 0 TO limit
PRINT F(i) FORMAT "%2d "
NEXT PRINT FOR i = 0 TO limit
PRINT M(i) FORMAT "%2d "
NEXT PRINT</lang>
- Output:
prompt$ ./mutually-recursive 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12
BASIC
<lang qbasic>DECLARE FUNCTION f! (n!) DECLARE FUNCTION m! (n!)
FUNCTION f! (n!)
IF n = 0 THEN
f = 1
ELSE
f = m(f(n - 1))
END IF
END FUNCTION
FUNCTION m! (n!)
IF n = 0 THEN
m = 0
ELSE
m = f(m(n - 1))
END IF
END FUNCTION</lang>
BBC BASIC
<lang bbcbasic> @% = 3 : REM Column width
PRINT "F sequence:"
FOR i% = 0 TO 20
PRINT FNf(i%) ;
NEXT
PRINT
PRINT "M sequence:"
FOR i% = 0 TO 20
PRINT FNm(i%) ;
NEXT
PRINT
END
DEF FNf(n%) IF n% = 0 THEN = 1 ELSE = n% - FNm(FNf(n% - 1))
DEF FNm(n%) IF n% = 0 THEN = 0 ELSE = n% - FNf(FNm(n% - 1))</lang>
- Output:
F sequence: 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 M sequence: 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12
IS-BASIC
<lang IS-BASIC>100 PROGRAM "Hofstad.bas" 110 PRINT "F sequence:" 120 FOR I=0 TO 20 130 PRINT F(I); 140 NEXT 150 PRINT :PRINT "M sequence:" 160 FOR I=0 TO 20 170 PRINT M(I); 180 NEXT 190 DEF F(N) 200 IF N=0 THEN 210 LET F=1 220 ELSE 230 LET F=N-M(F(N-1)) 240 END IF 250 END DEF 260 DEF M(N) 270 IF N=0 THEN 280 LET M=0 290 ELSE 300 LET M=N-F(M(N-1)) 310 END IF 320 END DEF</lang>
BASIC256
<lang BASIC256># Rosetta Code problem: http://rosettacode.org/wiki/Mutual_recursion
- by Jjuanhdez, 06/2022
n = 24 print "n : "; for i = 0 to n : print ljust(i, 3); : next i print chr(10); ("-" * 78) print "F : "; for i = 0 to n : print ljust(F(i), 3); : next i print chr(10); "M : "; for i = 0 to n : print ljust(M(i), 3); : next i end
function F(n)
if n = 0 then return 0 else return n - M(F(n-1))
end function
function M(n)
if n = 0 then return 0 else return n - F(M(n-1))
end function</lang>
Bc
<lang bc>cat mutual_recursion.bc: define f(n) {
if ( n == 0 ) return(1); return(n - m(f(n-1)));
}
define m(n) {
if ( n == 0 ) return(0); return(n - f(m(n-1)));
}</lang>
POSIX bc doesn't have the print statement.
<lang bc>/* GNU bc */ for(i=0; i < 19; i++) {
print f(i); print " ";
} print "\n"; for(i=0; i < 19; i++) {
print m(i); print " ";
} print "\n"; quit</lang>
- Output:
GNU bc mutual_recursion.bc bc 1.06.95 Copyright 1991-1994, 1997, 1998, 2000, 2004, 2006 Free Software Foundation, Inc. This is free software with ABSOLUTELY NO WARRANTY. For details type `warranty'. 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12
BCPL
<lang BCPL>get "libhdr"
// Mutually recursive functions let f(n) = n=0 -> 1, n - m(f(n-1)) and m(n) = n=0 -> 0, n - f(m(n-1))
// Print f(0..15) and m(0..15) let start() be $( writes("F:")
for i=0 to 15 do
$( writes(" ")
writen(f(i))
$)
writes("*NM:")
for i=0 to 15 do
$( writes(" ")
writen(m(i))
$)
writes("*N")
$)</lang>
- Output:
F: 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 M: 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9
BQN
<lang bqn>F ← {0:1; 𝕩-M F𝕩-1} M ← {0:0; 𝕩-F M𝕩-1} ⍉"FM"∾>(F∾M)¨↕15</lang>
- Output:
┌─
╵ 'F' 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9
'M' 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9
┘
Bracmat
<lang bracmat> (F=.!arg:0&1|!arg+-1*M$(F$(!arg+-1)));
(M=.!arg:0&0|!arg+-1*F$(M$(!arg+-1)));
-1:?n&whl'(!n+1:~>20:?n&put$(F$!n " "))&put$\n 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13
-1:?n&whl'(!n+1:~>20:?n&put$(M$!n " "))&put$\n 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12</lang>
Brat
<lang brat>female = null #yes, this is necessary
male = { n |
true? n == 0
{ 0 }
{ n - female male(n - 1) }
}
female = { n |
true? n == 0
{ 1 }
{ n - male female(n - 1 ) }
}
p 0.to(20).map! { n | female n } p 0.to(20).map! { n | male n }</lang>
C
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 explicitly, 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>
- include <stdlib.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(const int n); int M(const int n);
int F(const int n) {
return (n == 0) ? 1 : n - M(F(n - 1));
}
int M(const int n) {
return (n == 0) ? 0 : n - F(M(n - 1));
}
int main(void) {
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 EXIT_SUCCESS;
}</lang>
C#
<lang csharp>namespace RosettaCode {
class Hofstadter {
static public int F(int n) {
int result = 1;
if (n > 0) {
result = n - M(F(n-1));
}
return result;
}
static public int M(int n) {
int result = 0;
if (n > 0) {
result = n - F(M(n - 1));
}
return result;
}
}
}</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>
Ceylon
<lang ceylon>Integer f(Integer n)
=> if (n > 0)
then n - m(f(n-1))
else 1;
Integer m(Integer n)
=> if (n > 0)
then n - f(m(n-1))
else 0;
shared void run() {
printAll((0:20).map(f)); printAll((0:20).map(m));
}</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
Clojure
<lang lisp>(declare F) ; forward reference
(defn M [n]
(if (zero? n) 0 (- n (F (M (dec n))))))
(defn F [n]
(if (zero? n) 1 (- n (M (F (dec n))))))</lang>
CLU
<lang clu>% At the top level, definitions can only see the definitions above. % But if we put F and M in a cluster, they can see each other. mutrec = cluster is F, M
rep = null
F = proc (n: int) returns (int)
if n=0 then return(1)
else return(n - M(F(n-1)))
end
end F
M = proc (n: int) returns (int)
if n=0 then return(0)
else return(n - F(M(n-1)))
end
end M
end mutrec
% If we absolutely _must_ have them defined at the top level, % we can then just take them out of the cluster. F = mutrec$F M = mutrec$M
% Print the first few values for F and M print_first_16 = proc (name: string, fn: proctype (int) returns (int))
po: stream := stream$primary_output()
stream$puts(po, name || ":")
for i: int in int$from_to(0,15) do
stream$puts(po, " " || int$unparse(fn(i)))
end
stream$putl(po, "")
end print_first_16
start_up = proc ()
print_first_16("F", F)
print_first_16("M", M)
end start_up</lang>
- Output:
F: 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 M: 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9
CoffeeScript
<lang coffeescript> F = (n) ->
if n is 0 then 1 else n - M F n - 1
M = (n) ->
if n is 0 then 0 else n - F M n - 1
console.log [0...20].map F console.log [0...20].map M </lang>
- Output:
<lang> > coffee mutual_recurse.coffee [ 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12 ] [ 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12 ] </lang>
Common Lisp
<lang lisp>(defun m (n)
(if (zerop n)
0
(- n (f (m (- n 1))))))
(defun f (n)
(if (zerop n)
1
(- n (m (f (- n 1))))))</lang>
D
<lang d>import std.stdio, std.algorithm, std.range;
int male(in int n) pure nothrow {
return n ? n - male(n - 1).female : 0;
}
int female(in int n) pure nothrow {
return n ? n - female(n - 1).male : 1;
}
void main() {
20.iota.map!female.writeln; 20.iota.map!male.writeln;
}</lang>
- Output:
[1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12] [0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12]
Dart
<lang dart>int M(int n) => n==0?1:n-F(M(n-1)); int F(int n) => n==0?0:n-M(F(n-1));
main() {
String f="",m="";
for(int i=0;i<20;i++) {
m+="${M(i)} ";
f+="${F(i)} ";
}
print("M: $m");
print("F: $f");
}</lang>
Delphi
<lang Delphi> unit Hofstadter;
interface
type
THofstadterFemaleMaleSequences = class public class function F(n: Integer): Integer; class function M(n: Integer): Integer; end;
implementation
class function THofstadterFemaleMaleSequences.F(n: Integer): Integer; begin
Result:= 1; if (n > 0) then Result:= n - M(F(n-1));
end;
class function THofstadterFemaleMaleSequences.M(n: Integer): Integer; begin
Result:= 0; if (n > 0) then Result:= n - F(M(n - 1));
end;
end. </lang>
Déjà Vu
<lang dejavu>F n: if n: - n M F -- n else: 1
M n: if n: - n F M -- n else: 0
for i range 0 10: !.( M i F i )</lang>
- Output:
0 1 0 1 1 2 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6
Draco
<lang draco>/* We need to predeclare M if we want F to be able to see it.
* This is done using 'extern', same as if it had been in a * different compilation unit. */
extern M(byte n) byte;
/* Mutually recursive functions */ proc F(byte n) byte:
if n=0 then 1 else n - M(F(n-1)) fi
corp
proc M(byte n) byte:
if n=0 then 0 else n - F(M(n-1)) fi
corp
/* Show the first 16 values of each */ proc nonrec main() void:
byte i;
write("F:");
for i from 0 upto 15 do write(F(i):2) od;
writeln();
write("M:");
for i from 0 upto 15 do write(M(i):2) od;
writeln()
corp</lang>
- Output:
F: 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 M: 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9
Dyalect
<lang dyalect>func f(n) {
n == 0 ? 1 : n - m(f(n-1))
} and m(n) {
n == 0 ? 0 : n - f(m(n-1))
}
print( (0..20).Map(i => f(i)).ToArray() ) print( (0..20).Map(i => m(i)).ToArray() )</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.
Eiffel
<lang Eiffel> class APPLICATION
create make
feature
make -- Test of the mutually recursive functions Female and Male. do across 0 |..| 19 as c loop io.put_string (Female (c.item).out + " ") end io.new_line across 0 |..| 19 as c loop io.put_string (Male (c.item).out + " ") end end
Female (n: INTEGER): INTEGER -- Female sequence of the Hofstadter Female and Male sequences. require n_not_negative: n >= 0 do Result := 1 if n /= 0 then Result := n - Male (Female (n - 1)) end end
Male (n: INTEGER): INTEGER -- Male sequence of the Hofstadter Female and Male sequences. require n_not_negative: n >= 0 do Result := 0 if n /= 0 then Result := n - Female (Male (n - 1)) end end
end </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
Elena
ELENA 4.x : <lang elena>import extensions; import system'collections;
F = (n => (n == 0) ? 1 : (n - M(F(n-1))) ); M = (n => (n == 0) ? 0 : (n - F(M(n-1))) );
public program() {
var ra := new ArrayList();
var rb := new ArrayList();
for(int i := 0, i <= 19, i += 1)
{
ra.append(F(i));
rb.append(M(i))
};
console.printLine(ra.asEnumerable());
console.printLine(rb.asEnumerable())
}</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
Elixir
<lang elixir>defmodule MutualRecursion do
def f(0), do: 1 def f(n), do: n - m(f(n - 1)) def m(0), do: 0 def m(n), do: n - f(m(n - 1))
end
IO.inspect Enum.map(0..19, fn n -> MutualRecursion.f(n) end) IO.inspect Enum.map(0..19, fn n -> MutualRecursion.m(n) end)</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]
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>
Euphoria
<lang Euphoria>integer idM, idF
function F(integer n)
if n = 0 then
return 1
else
return n - call_func(idM,{F(n-1)})
end if
end function
idF = routine_id("F")
function M(integer n)
if n = 0 then
return 0
else
return n - call_func(idF,{M(n-1)})
end if
end function
idM = routine_id("M")</lang>
F#
<lang fsharp>let rec f n =
match n with | 0 -> 1 | _ -> n - (m (f (n-1)))
and m n =
match n with | 0 -> 0 | _ -> n - (f (m (n-1)))</lang>
Like OCaml, the let rec f .. and m ... construct indicates that the functions call themselves (rec) and each other (and).
Factor
In Factor, if you need a word before it's defined, you have to DEFER: it.
<lang>DEFER: F
- M ( n -- n' ) dup 0 = [ dup 1 - M F - ] unless ;
- F ( n -- n' ) dup 0 = [ drop 1 ] [ dup 1 - F M - ] if ;</lang>
FALSE
<lang false>[$[$1-f;!m;!-1-]?1+]f: [$[$1-m;!f;!- ]? ]m: [0[$20\>][\$@$@!." "1+]#%%]t:
f; t;!"
"m; t;!</lang>
Fantom
<lang fantom> class Main {
static Int f (Int n)
{
if (n <= 0) // ensure n > 0
return 1
else
return n - m(f(n-1))
}
static Int m (Int n)
{
if (n <= 0) // ensure n > 0
return 0
else
return n - f(m(n-1))
}
public static Void main ()
{
50.times |Int n| { echo (f(n)) }
}
} </lang>
FOCAL
<lang FOCAL>01.01 C--PRINT F(0..15) AND M(0..15) 01.10 T "F(0..15)" 01.20 F X=0,15;S N=X;D 4;T %1,N 01.30 T !"M(0..15)" 01.40 F X=0,15;S N=X;D 5;T %1,N 01.50 T ! 01.60 Q
04.01 C--N = F(N) 04.10 I (N(D)),4.11,4.2 04.11 S N(D)=1;R 04.20 S D=D+1;S N(D)=N(D-1)-1;D 4;D 5 04.30 S D=D-1;S N(D)=N(D)-N(D+1)
05.01 C--N = M(N) 05.10 I (N(D)),5.11,5.2 05.11 R 05.20 S D=D+1;S N(D)=N(D-1)-1;D 5;D 4 05.30 S D=D-1;S N(D)=N(D)-N(D+1)</lang>
- Output:
F(0..15)= 1= 1= 2= 2= 3= 3= 4= 5= 5= 6= 6= 7= 8= 8= 9= 9 M(0..15)= 0= 0= 1= 2= 2= 3= 4= 4= 5= 6= 6= 7= 7= 8= 9= 9
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 defer@ 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)
<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>
FreeBASIC
<lang freebasic>' FB 1.05.0 Win64
' Need forward declaration of M as it's used ' by F before its defined Declare Function M(n As Integer) As Integer
Function F(n As Integer) As Integer
If n = 0 Then
Return 1
End If
Return n - M(F(n - 1))
End Function
Function M(n As Integer) As Integer
If n = 0 Then
Return 0
End If
Return n - F(M(n - 1))
End Function
Dim As Integer n = 24 Print "n :"; For i As Integer = 0 to n : Print Using "###"; i; : Next Print Print String(78, "-") Print "F :"; For i As Integer = 0 To n : Print Using "###"; F(i); : Next Print Print "M :"; For i As Integer = 0 To n : Print Using "###"; M(i); : Next Print Print "Press any key to quit" Sleep</lang>
- Output:
n : 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ------------------------------------------------------------------------------ F : 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 13 14 14 15 M : 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12 13 14 14 15
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
It just works. No special pre-declaration is necessary. <lang go>package main import "fmt"
func F(n int) int {
if n == 0 { return 1 }
return n - M(F(n-1))
}
func M(n int) int {
if n == 0 { return 0 }
return n - F(M(n-1))
}
func main() {
for i := 0; i < 20; i++ {
fmt.Printf("%2d ", F(i))
}
fmt.Println()
for i := 0; i < 20; i++ {
fmt.Printf("%2d ", M(i))
}
fmt.Println()
}</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>
Icon and Unicon
<lang Icon>procedure main(arglist) every write(F(!arglist)) # F of all arguments end
procedure F(n) if integer(n) >= 0 then
return (n = 0, 1) | n - M(F(n-1))
end
procedure M(n) if integer(n) >= 0 then
return (0 = n) | n - F(M(n-1))
end</lang>
Idris
<lang idris>mutual {
F : Nat -> Nat F Z = (S Z) F (S n) = (S n) `minus` M(F(n))
M : Nat -> Nat M Z = Z M (S n) = (S n) `minus` F(M(n))
}</lang>
Io
<lang Io>f := method(n, if( n == 0, 1, n - m(f(n-1)))) m := method(n, if( n == 0, 0, n - f(m(n-1))))
Range 0 to(19) map(n,f(n)) println 0 to(19) map(n,m(n)) println</lang>
- Output:
list(1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12) list(0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12)
J
<lang j>F =: 1:`(- M @ $: @ <:) @.* M."0 M =: 0:`(- F @ $: @ <:) @.* M."0</lang>
Example use:
<lang j> F i. 20 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12</lang>
That said, note that numbers are defined recursively, so some other approaches using numbers which give equivalent results should be acceptable.
Java
Replace translation (that doesn't compile) with a Java native implementation. <lang java> import java.util.HashMap; import java.util.Map;
public class MutualRecursion {
public static void main(final String args[]) {
int max = 20;
System.out.printf("First %d values of the Female sequence: %n", max);
for (int i = 0; i < max; i++) {
System.out.printf(" f(%d) = %d%n", i, f(i));
}
System.out.printf("First %d values of the Male sequence: %n", max);
for (int i = 0; i < 20; i++) {
System.out.printf(" m(%d) = %d%n", i, m(i));
}
}
private static Map<Integer,Integer> F_MAP = new HashMap<>();
private static int f(final int n) {
if ( F_MAP.containsKey(n) ) {
return F_MAP.get(n);
}
int fn = n == 0 ? 1 : n - m(f(n - 1));
F_MAP.put(n, fn);
return fn;
}
private static Map<Integer,Integer> M_MAP = new HashMap<>();
private static int m(final int n) {
if ( M_MAP.containsKey(n) ) {
return M_MAP.get(n);
}
int mn = n == 0 ? 0 : n - f(m(n - 1));
M_MAP.put(n, mn);
return mn;
}
} </lang>
- Output:
First 20 values of the Female sequence:
f(0) = 1 f(1) = 1 f(2) = 2 f(3) = 2 f(4) = 3 f(5) = 3 f(6) = 4 f(7) = 5 f(8) = 5 f(9) = 6 f(10) = 6 f(11) = 7 f(12) = 8 f(13) = 8 f(14) = 9 f(15) = 9 f(16) = 10 f(17) = 11 f(18) = 11 f(19) = 12
First 20 values of the Male sequence:
m(0) = 0 m(1) = 0 m(2) = 1 m(3) = 2 m(4) = 2 m(5) = 3 m(6) = 4 m(7) = 4 m(8) = 5 m(9) = 6 m(10) = 6 m(11) = 7 m(12) = 7 m(13) = 8 m(14) = 9 m(15) = 9 m(16) = 10 m(17) = 11 m(18) = 11 m(19) = 12
JavaScript
<lang JavaScript>function f(num) {
return (num === 0) ? 1 : num - m(f(num - 1));
}
function m(num) {
return (num === 0) ? 0 : num - f(m(num - 1));
}
function range(m, n) {
return Array.apply(null, Array(n - m + 1)).map(
function (x, i) { return m + i; }
);
}
var a = range(0, 19);
//return a new array of the results and join with commas to print console.log(a.map(function (n) { return f(n); }).join(', ')); console.log(a.map(function (n) { return m(n); }).join(', '));</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
ES6 implementation <lang JavaScript>var f = num => (num === 0) ? 1 : num - m(f(num - 1)); var m = num => (num === 0) ? 0 : num - f(m(num - 1));
function range(m, n) {
return Array.apply(null, Array(n - m + 1)).map(
function (x, i) { return m + i; }
);
}
var a = range(0, 19);
//return a new array of the results and join with commas to print console.log(a.map(n => f(n)).join(', ')); console.log(a.map(n => m(n)).join(', '));</lang>
More ES6 implementation
<lang JavaScript>var range = (m, n) => Array(... Array(n - m + 1)).map((x, i) => m + i)</lang>
jq
jq supports mutual recursion but requires functions to be defined before they are used. In the present case, this can be accomplished by defining an inner function.
He we define F and M as arity-0 filters: <lang jq> def M:
def F: if . == 0 then 1 else . - ((. - 1) | F | M) end; if . == 0 then 0 else . - ((. - 1) | M | F) end;
def F:
if . == 0 then 1 else . - ((. - 1) | F | M) end;</lang>Example:<lang jq>
[range(0;20) | F], [range(0;20) | M]</lang><lang sh>$ jq -n -c -f Mutual_recursion.jq
[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]</lang>
Jsish
<lang javascript>/* Mutual recursion, is jsish */ function f(num):number { return (num === 0) ? 1 : num - m(f(num - 1)); } function m(num):number { return (num === 0) ? 0 : num - f(m(num - 1)); }
function range(n=10, start=0, step=1):array {
var a = Array(n).fill(0); for (var i in a) a[i] = start+i*step; return a;
}
var a = range(21); puts(a.map(function (n) { return f(n); }).join(', ')); puts(a.map(function (n) { return m(n); }).join(', '));
/*
!EXPECTSTART!
1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12
!EXPECTEND!
- /</lang>
- Output:
prompt$ jsish -u mutual-recursion.jsi [PASS] mutual-recursion.jsi prompt$ jsish mutual-recursion.jsi 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
Julia
<lang julia>F(n) = n < 1 ? one(n) : n - M(F(n - 1)) M(n) = n < 1 ? zero(n) : n - F(M(n - 1))</lang>
- Output:
julia> [F(i) for i = 0:19], [M(i) for i = 0:19] ([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])
Kotlin
<lang scala>// version 1.0.6
fun f(n: Int): Int =
when {
n == 0 -> 1
else -> n - m(f(n - 1))
}
fun m(n: Int): Int =
when {
n == 0 -> 0
else -> n - f(m(n - 1))
}
fun main(args: Array<String>) {
val n = 24
print("n :")
for (i in 0..n) print("%3d".format(i))
println()
println("-".repeat(78))
print("F :")
for (i in 0..24) print("%3d".format(f(i)))
println()
print("M :")
for (i in 0..24) print("%3d".format(m(i)))
println()
}</lang>
- Output:
n : 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ------------------------------------------------------------------------------ F : 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 13 14 14 15 M : 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12 13 14 14 15
Lambdatalk
<lang scheme> {def F {lambda {:n} {if {= :n 0} then 1 else {- :n {M {F {- :n 1}}}} }}} {def M {lambda {:n} {if {= :n 0} then 0 else {- :n {F {M {- :n 1}}}} }}}
{map F {serie 0 19}} -> 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 {map M {serie 0 19}} -> 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 </lang>
The naïve version is very slow, {F 80} requires 3800 ms on a recent laptop, so let's memoize:
<lang scheme> {def cache
{def cache.F {#.new}}
{def cache.M {#.new}}
{lambda {:f :n}
{let { {:f :f} {:n :n}
{:cx {if {equal? :f MF}
then {cache.F}
else {cache.M}}}
} {if {equal? {#.get :cx :n} undefined}
then {#.get {#.set! :cx :n {:f :n}} :n}
else {#.get :cx :n}}}}}
-> cache
{def MF
{lambda {:n}
{if {= :n 0}
then 1
else {- :n {MM {cache MF {- :n 1}}}}}}}
-> MF
{def MM
{lambda {:n}
{if {= :n 0}
then 0
else {- :n {MF {cache MM {- :n 1}}}}}}}
-> MM
{MF 80} -> 50 (requires 55 ms)
{map MF {serie 0 100}} (requires75ms) -> 1 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 19 20 21 21 22 22 23 24 24 25 25 26 27 27 28 29 29 30 30 31 32 32 33 34 34 35 35 36 37 37 38 38 39 40 40 41 42 42 43 43 44 45 45 46 46 47 48 48 49 50 50 51 51 52 53 53 54 55 55 56 56 57 58 58 59 59 60 61 61 62 </lang>
Liberty BASIC
<lang lb> print "F sequence." for i = 0 to 20 print f(i);" "; next print print "M sequence." for i = 0 to 20 print m(i);" "; next
end
function f(n)
if n = 0 then
f = 1
else
f = n - m(f(n - 1))
end if
end function
function m(n)
if n = 0 then
m = 0
else
m = n - f(m(n - 1))
end if
end function
</lang>
- Output:
F sequence. 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 M sequence. 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12
LibreOffice Basic
<lang LibreOffice Basic>'// LibreOffice Basic Implementation of Hofstadter Female-Male sequences
'// Utility functions sub setfont(strfont) ThisComponent.getCurrentController.getViewCursor.charFontName = strfont end sub
sub newline oVC = thisComponent.getCurrentController.getViewCursor oText = oVC.text oText.insertControlCharacter(oVC, com.sun.star.text.ControlCharacter.PARAGRAPH_BREAK, False) end sub
sub out(sString) oVC = ThisComponent.getCurrentController.getViewCursor oText = oVC.text oText.insertString(oVC, sString, false) end sub
sub outln(optional sString) if not ismissing (sString) then out(sString) newline end sub
function intformat(n as integer,nlen as integer) as string dim nstr as string nstr = CStr(n) while len(nstr) < nlen nstr = " " & nstr wend intformat = nstr end function
'// Hofstadter Female-Male function definitions function F(n as long) as long if n = 0 Then F = 1 elseif n > 0 Then F = n - M(F(n - 1)) endif end function
function M(n) if n = 0 Then M = 0 elseif n > 0 Then M = n - F(M(n - 1)) endif end function
'// Hofstadter Female Male sequence demo routine sub Hofstadter_Female_Male_Demo '// Introductory Text setfont("LM Roman 10") outln("Rosetta Code Hofstadter Female and Male Sequence Challenge") outln out("Two functions are said to be mutually recursive if the first calls the second,") outln(" and in turn the second calls the first.") out("Write two mutually recursive functions that compute members of the Hofstadter") outln(" Female and Male sequences defined as:") outln setfont("LM Mono Slanted 10") outln(chr(9)+"F(0) = 1 ; M(0)=0") outln(chr(9)+"F(n) = n - M(F(n-1)), n > 0") outln(chr(9)+"M(n) = n - F(M(n-1)), n > 0") outln '// Sequence Generation const nmax as long = 20 dim n as long setfont("LM Mono 10") out("n = " for n = 0 to nmax out(" " + intformat(n, 2)) next n outln out("F(n) = " for n = 0 to nmax out(" " + intformat(F(n),2)) next n outln out("M(n) = " for n = 0 to nmax out(" " + intformat(M(n), 2)) next n outln
end sub
Output
n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 F(n) = 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 M(n) = 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12 </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>
LSL
To test it yourself; rez a box on the ground, and add the following as a New Script. <lang LSL>integer iDEPTH = 100; integer f(integer n) { if(n==0) { return 1; } else { return n-m(f(n - 1)); } } integer m(integer n) { if(n==0) { return 0; } else { return n-f(m(n - 1)); } } default { state_entry() { integer x = 0; string s = ""; for(x=0 ; x<iDEPTH ; x++) { s += (string)(f(x))+" "; } llOwnerSay(llList2CSV(llParseString2List(s, [" "], []))); s = ""; for(x=0 ; x<iDEPTH ; x++) { s += (string)(m(x))+" "; } llOwnerSay(llList2CSV(llParseString2List(s, [" "], []))); } }</lang>
- Output:
1, 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, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45, 46, 46, 47, 48, 48, 49, 50, 50, 51, 51, 52, 53, 53, 54, 55, 55, 56, 56, 57, 58, 58, 59, 59, 60, 61, 61 0, 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, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45, 46, 46, 47, 48, 48, 49, 50, 50, 51, 51, 52, 53, 53, 54, 54, 55, 56, 56, 57, 58, 58, 59, 59, 60, 61, 61
Lua
<lang lua> function m(n) return n > 0 and n - f(m(n-1)) or 0 end function f(n) return n > 0 and n - m(f(n-1)) or 1 end</lang>
It is important to note, that if m and f are to be locally scoped functions rather than global, that they would need to be forward declared:
<lang lua> local m,n function m(n) return n > 0 and n - f(m(n-1)) or 0 end function f(n) return n > 0 and n - m(f(n-1)) or 1 end</lang>
M2000 Interpreter
A function can call a global function and must be global to call it again by the second function
A group's function can call sibling function from same group. We can use This.F() or simply .f() to use group's f() member.
We can use subroutines, which can call each other, in a module, and we can use the modules stack of values to get results from subs. Subs running as parts of module, and see same variables and same stack of values. Arguments are local to sub, and we can define local variables too.
Last module export to clipboard and that used for output here. <lang M2000 Interpreter> \\ set console 70 characters by 40 lines Form 70, 40 Module CheckSubs {
Flush
Document one$, two$
For i =0 to 20
Print format$("{0::-3}",i);
f(i)
\\ number pop then top value of stack
one$=format$("{0::-3}",number)
m(i)
two$=format$("{0::-3}",number)
Next i
Print
Print one$
Print two$
Sub f(x)
if x<=0 then Push 1 : Exit sub
f(x-1) ' leave result to for m(x)
m()
push x-number
End Sub
Sub m(x)
if x<=0 then Push 0 : Exit sub
m(x-1)
f()
push x-number
End Sub
} CheckSubs
Module Checkit {
Function global f(n) {
if n=0 then =1: exit
if n>0 then =n-m(f(n-1))
}
Function global m(n) {
if n=0 then =0
if n>0 then =n-f(m(n-1))
}
Document one$, two$
For i =0 to 20
Print format$("{0::-3}",i);
one$=format$("{0::-3}",f(i))
two$=format$("{0::-3}",m(i))
Next i
Print
Print one$
Print two$
} Checkit Module Checkit2 {
Group Alfa {
function f(n) {
if n=0 then =1: exit
if n>0 then =n-.m(.f(n-1))
}
function m(n) {
if n=0 then =0
if n>0 then =n-.f(.m(n-1))
}
}
Document one$, two$
For i =0 to 20
Print format$("{0::-3}",i);
one$=format$("{0::-3}",Alfa.f(i))
two$=format$("{0::-3}",Alfa.m(i))
Next i
Print
Print one$
Print two$
Clipboard one$+{
}+two$
} Checkit2
</lang>
- Output:
1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12
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>
MAD
By default, functions in MAD are not reentrant. There is also no variable scope, all variables are always global. Functions can call other functions, but on the old IBM mainframes this was done by storing the return address in a special location (one per function); should a function call itself (either directly or indirectly), the return address would be overwritten.
MAD does include a stack mechanism, but it is entirely manual. The programmer
must allocate memory for it himself and activate it by hand, by default there
is no stack. The command for this is SET LIST TO array.
Once this is done, however, variables can be pushed and popped
(using the SAVE and RESTORE commands).
Furthermore, SAVE RETURN and RESTURE RETURN can
be used to push and pop the current return address, enabling proper recursion,
as long as the programmer is careful.
The downside to this is that it does not play well with argument passing. All variables are still global. This means that passing arguments to a recursive function has to be done either by pushing them on the stack beforehand, or by setting global variables that the functions will push and pop themselves. (This program does the latter.)
At the same time, the language syntax demands that all functions take at least one
argument, so a dummy argument must be passed. To obtain a recursive function that uses the
argument it is given, it is necessary to write a front-end function that uses its argument
to pass it through the actual function in the manner described above. This is also shown.
In this program, F. and M. are the front ends, taking an
argument and using it to set N, then calling either FREC.
or MREC., which are the actual recursive functions, with a dummy zero argument.
<lang MAD> NORMAL MODE IS INTEGER
R SET UP STACK SPACE
DIMENSION STACK(100)
SET LIST TO STACK
R DEFINE RECURSIVE FUNCTIONS
R INPUT ARGUMENT ASSUMED TO BE IN N
INTERNAL FUNCTION(DUMMY)
ENTRY TO FREC.
WHENEVER N.LE.0, FUNCTION RETURN 1
SAVE RETURN
SAVE DATA N
N = N-1
N = FREC.(0)
X = MREC.(0)
RESTORE DATA N
RESTORE RETURN
FUNCTION RETURN N-X
END OF FUNCTION
INTERNAL FUNCTION(DUMMY)
ENTRY TO MREC.
WHENEVER N.LE.0, FUNCTION RETURN 0
SAVE RETURN
SAVE DATA N
N = N-1
N = MREC.(0)
X = FREC.(0)
RESTORE DATA N
RESTORE RETURN
FUNCTION RETURN N-X
END OF FUNCTION
R DEFINE FRONT-END FUNCTIONS THAT CAN BE CALLED WITH ARGMT
INTERNAL FUNCTION(NN)
ENTRY TO F.
N = NN
FUNCTION RETURN FREC.(0)
END OF FUNCTION
INTERNAL FUNCTION(NN)
ENTRY TO M.
N = NN
FUNCTION RETURN MREC.(0)
END OF FUNCTION
R PRINT F(0..19) AND M(0..19)
THROUGH SHOW, FOR I=0, 1, I.GE.20
SHOW PRINT FORMAT FMT,I,F.(I),I,M.(I)
VECTOR VALUES FMT =
0 $2HF(,I2,4H) = ,I2,S8,2HM(,I2,4H) = ,I2*$
END OF PROGRAM</lang>
- Output:
F( 0) = 1 M( 0) = 0 F( 1) = 1 M( 1) = 0 F( 2) = 2 M( 2) = 1 F( 3) = 2 M( 3) = 2 F( 4) = 3 M( 4) = 2 F( 5) = 3 M( 5) = 3 F( 6) = 4 M( 6) = 4 F( 7) = 5 M( 7) = 4 F( 8) = 5 M( 8) = 5 F( 9) = 6 M( 9) = 6 F(10) = 6 M(10) = 6 F(11) = 7 M(11) = 7 F(12) = 8 M(12) = 7 F(13) = 8 M(13) = 8 F(14) = 9 M(14) = 9 F(15) = 9 M(15) = 9 F(16) = 10 M(16) = 10 F(17) = 11 M(17) = 11 F(18) = 11 M(18) = 11 F(19) = 12 M(19) = 12
Maple
<lang Maple>female_seq := proc(n) if (n = 0) then return 1; else return n - male_seq(female_seq(n-1)); end if; end proc;
male_seq := proc(n) if (n = 0) then return 0; else return n - female_seq(male_seq(n-1)); end if; end proc; seq(female_seq(i), i=0..10); seq(male_seq(i), i=0..10);</lang>
- Output:
1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6
Mathematica/Wolfram Language
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>
MATLAB
female.m: <lang MATLAB>function Fn = female(n)
if n == 0
Fn = 1;
return
end
Fn = n - male(female(n-1));
end</lang>
male.m: <lang MATLAB>function Mn = male(n)
if n == 0
Mn = 0;
return
end
Mn = n - female(male(n-1));
end</lang>
- Output:
<lang MATLAB>>> n = (0:10); >> arrayfun(@female,n)
ans =
1 1 2 2 3 3 4 5 5 6 6
>> arrayfun(@male,n)
ans =
0 0 1 2 2 3 4 4 5 6 6</lang>
Maxima
<lang maxima>f[0]: 1$ m[0]: 0$ f[n] := n - m[f[n - 1]]$ m[n] := n - f[m[n - 1]]$
makelist(f[i], i, 0, 10); [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6]
makelist(m[i], i, 0, 10); [0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6]
remarray(m, f)$
f(n) := if n = 0 then 1 else n - m(f(n - 1))$ m(n) := if n = 0 then 0 else n - f(m(n - 1))$
makelist(f(i), i, 0, 10); [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6]
makelist(m(i), i, 0, 10); [0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6]
remfunction(f, m)$</lang>
Mercury
<lang>
- - module mutual_recursion.
- - interface.
- - import_module io.
- - pred main(io::di, io::uo) is det.
- - implementation.
- - import_module int, list.
main(!IO) :-
io.write(list.map(f, 0..19), !IO), io.nl(!IO), io.write(list.map(m, 0..19), !IO), io.nl(!IO).
- - func f(int) = int.
f(N) = ( if N = 0 then 1 else N - m(f(N - 1)) ).
- - func m(int) = int.
m(N) = ( if N = 0 then 0 else N - f(m(N - 1)) ). </lang>
MiniScript
<lang MiniScript>f = function(n)
if n > 0 then return n - m(f(n - 1)) return 1
end function
m = function(n)
if n > 0 then return n - f(m(n - 1)) return 0
end function
print f(12) print m(12)</lang>
- Output:
8 7
MiniZinc
<lang MiniZinc> function var int: F(var int:n) =
if n == 0 then 1 else n - M(F(n - 1)) endif;
function var int: M(var int:n) =
if (n == 0) then 0 else n - F(M(n - 1)) endif;
</lang>
MMIX
<lang mmix> LOC Data_Segment
GREG @ NL BYTE #a,0 GREG @ buf OCTA 0,0
t IS $128 Ja IS $127
LOC #1000
GREG @ // print 2 digits integer with trailing space to StdOut // reg $3 contains int to be printed bp IS $71 0H GREG #0000000000203020 prtInt STO 0B,buf % initialize buffer LDA bp,buf+7 % points after LSD % REPEAT 1H SUB bp,bp,1 % move buffer pointer DIV $3,$3,10 % divmod (x,10) GET t,rR % get remainder INCL t,'0' % make char digit STB t,bp % store digit PBNZ $3,1B % UNTIL no more digits LDA $255,bp TRAP 0,Fputs,StdOut % print integer GO Ja,Ja,0 % 'return'
// Female function F GET $1,rJ % save return addr PBNZ $0,1F % if N != 0 then F N INCL $0,1 % F 0 = 1 PUT rJ,$1 % restore return addr POP 1,0 % return 1 1H SUBU $3,$0,1 % N1 = N - 1 PUSHJ $2,F % do F (N - 1) ADDU $3,$2,0 % place result in arg. reg. PUSHJ $2,M % do M F ( N - 1) PUT rJ,$1 % restore ret addr SUBU $0,$0,$2 POP 1,0 % return N - M F ( N - 1 )
// Male function M GET $1,rJ PBNZ $0,1F PUT rJ,$1 POP 1,0 % return M 0 = 0 1H SUBU $3,$0,1 PUSHJ $2,M ADDU $3,$2,0 PUSHJ $2,F PUT rJ,$1 SUBU $0,$0,$2 POP 1,0 $ return N - F M ( N - 1 )
// do a female run Main SET $1,0 % for (i=0; i<25; i++){ 1H ADDU $4,$1,0 % PUSHJ $3,F % F (i) GO Ja,prtInt % print F (i) INCL $1,1 CMP t,$1,25 PBNZ t,1B % } LDA $255,NL TRAP 0,Fputs,StdOut // do a male run SET $1,0 % for (i=0; i<25; i++){ 1H ADDU $4,$1,0 % PUSHJ $3,M % M (i) GO Ja,prtInt % print M (i) INCL $1,1 CMP t,$1,25 PBNZ t,1B % } LDA $255,NL TRAP 0,Fputs,StdOut TRAP 0,Halt,0</lang>
- Output:
~/MIX/MMIX/Rosetta> mmix mutualrecurs1 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 13 14 14 15 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12 13 14 14 15
Modula-2
<lang modula2>MODULE MutualRecursion; FROM InOut IMPORT WriteCard, WriteString, WriteLn;
TYPE Fn = PROCEDURE(CARDINAL): CARDINAL;
PROCEDURE F(n: CARDINAL): CARDINAL; BEGIN
IF n=0 THEN RETURN 1; ELSE RETURN n-M(F(n-1)); END;
END F;
PROCEDURE M(n: CARDINAL): CARDINAL; BEGIN
IF n=0 THEN RETURN 0; ELSE RETURN n-F(M(n-1)); END;
END M;
(* Print the first few values of one of the functions *) PROCEDURE Show(name: ARRAY OF CHAR; fn: Fn);
CONST Max = 15; VAR i: CARDINAL;
BEGIN
WriteString(name);
WriteString(": ");
FOR i := 0 TO Max DO
WriteCard(fn(i), 0);
WriteString(" ");
END;
WriteLn;
END Show;
(* Show the first values of both F and M *) BEGIN
Show("F", F);
Show("M", M);
END MutualRecursion.</lang>
- Output:
F: 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 M: 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9
Nemerle
<lang Nemerle>using System; using System.Console;
module Hofstadter {
F(n : int) : int
{
|0 => 1
|_ => n - M(F(n - 1))
}
M(n : int) : int
{
|0 => 0
|_ => n - F(M(n - 1))
}
Main() : void
{
foreach (n in [0 .. 20]) Write("{0} ", F(n));
WriteLine();
foreach (n in [0 .. 20]) Write("{0} ", M(n));
}
}</lang>
Nim
<lang nim>proc m(n: int): int
proc f(n: int): int =
if n == 0: 1 else: n - m(f(n-1))
proc m(n: int): int =
if n == 0: 0 else: n - f(m(n-1))
for i in 1 .. 10:
echo f(i) echo m(i)</lang>
Objeck
<lang objeck> class MutualRecursion {
function : Main(args : String[]) ~ Nil {
for(i := 0; i < 20; i+=1;) {
f(i)->PrintLine();
};
"---"->PrintLine();
for (i := 0; i < 20; i+=1;) {
m(i)->PrintLine();
};
}
function : f(n : Int) ~ Int {
return n = 0 ? 1 : n - m(f(n - 1));
}
function : m(n : Int) ~ Int {
return n = 0 ? 0 : n - f(m(n - 1));
}
} </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 <Foundation/Foundation.h>
@interface Hofstadter : NSObject + (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>
Oforth
Oforth can declare methods objects without any implementation. This allows to implement mutual recursion. This does not work with functions (declaration and implementation must be together).
<lang Oforth>Method new: M
Integer method: F
self 0 == ifTrue: [ 1 return ] self self 1 - F M - ;
Integer method: M
self 0 == ifTrue: [ 0 return ] self self 1 - M F - ;
0 20 seqFrom map(#F) println 0 20 seqFrom map(#M) println</lang>
- Output:
[1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13] [0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12]
Ol
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. <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))))))))
(print (F 19)))
- produces 12
</lang>
Order
Since Order is powered by the C preprocessor, definitions follow the same rule as CPP macros: they can appear in any order relative to each other as long as all are defined before the ORDER_PP block that calls them.
<lang c>#include <order/interpreter.h>
- define ORDER_PP_DEF_8f \
ORDER_PP_FN(8fn(8N, \
8if(8is_0(8N), \
1, \
8sub(8N, 8m(8f(8dec(8N)))))))
- define ORDER_PP_DEF_8m \
ORDER_PP_FN(8fn(8N, \
8if(8is_0(8N), \
0, \
8sub(8N, 8f(8m(8dec(8N)))))))
//Test ORDER_PP(8for_each_in_range(8fn(8N, 8print(8f(8N))), 0, 19)) ORDER_PP(8for_each_in_range(8fn(8N, 8print(8m(8N))), 0, 19))</lang>
Oz
<lang oz>declare
fun {F N}
if N == 0 then 1
elseif N > 0 then N - {M {F N-1}}
end
end
fun {M N}
if N == 0 then 0
elseif N > 0 then N - {F {M N-1}}
end
end
in
{Show {Map {List.number 0 9 1} F}}
{Show {Map {List.number 0 9 1} M}}</lang>
PARI/GP
<lang parigp>F(n)=if(n,n-M(F(n-1)),1) M(n)=if(n,n-F(M(n-1)),0)</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>sub F { my $n = shift; $n ? $n - M(F($n-1)) : 1 } sub M { my $n = shift; $n ? $n - F(M($n-1)) : 0 }
- Usage:
foreach my $sequence (\&F, \&M) {
print join(' ', map $sequence->($_), 0 .. 19), "\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
Phix
You should normally explicitly declare forward routines since it often makes things easier to understand (strictly only necessary when using optional or named parameters). There would be no point pre-declaring F, since it is not called before it is defined anyway.
with javascript_semantics forward function M(integer n) function F(integer n) return iff(n?n-M(F(n-1)):1) end function function M(integer n) return iff(n?n-F(M(n-1)):0) end function for i=0 to 20 do printf(1," %d",F(i)) end for printf(1,"\n") for i=0 to 20 do printf(1," %d",M(i)) end for
- Output:
1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12
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>
Picat
Here are two approaches, both using tabling. For small values (say N < 50) tabling is not really needed.
Tabled functions
<lang Picat>table f(0) = 1. f(N) = N - m(f(N-1)), N > 0 => true.
table m(0) = 0. m(N) = N - f(m(N-1)), N > 0 => true.</lang>
Tabled predicates
<lang Picat>table female(0,1). female(N,F) :-
N>0, N1 = N-1, female(N1,R), male(R, R1), F = N-R1.
table male(0,0). male(N,F) :-
N>0, N1 = N-1, male(N1,R), female(R, R1), F = N-R1.</lang>
Test
<lang Picat>go =>
N = 30, println(func), test_func(N), println(pred), test_pred(N), nl. nl.
% Testing the function based approach test_func(N) =>
println([M : I in 0..N, male(I,M)]), println([F : I in 0..N, female(I,F)]), nl.
% Testing the predicate approach test_pred(N) =>
println([M : I in 0..N, male(I,M)]), println([F : I in 0..N, female(I,F)]), nl.</lang>
- Output:
func [0,0,1,2,2,3,4,4,5,6,6,7,7,8,9,9,10,11,11,12,12,13,14,14,15,16,16,17,17,18,19] [1,1,2,2,3,3,4,5,5,6,6,7,8,8,9,9,10,11,11,12,13,13,14,14,15,16,16,17,17,18,19] pred [0,0,1,2,2,3,4,4,5,6,6,7,7,8,9,9,10,11,11,12,12,13,14,14,15,16,16,17,17,18,19] [1,1,2,2,3,3,4,5,5,6,6,7,8,8,9,9,10,11,11,12,13,13,14,14,15,16,16,17,17,18,19]
Larger values
For larger values, tabling is essential and then one can discern that the predicate based approach is a little faster. Here are the times for testing N=1 000 000:
- func: 1.829s
- pred: 1.407s
PicoLisp
<lang PicoLisp>(de f (N)
(if (=0 N)
1
(- N (m (f (dec N)))) ) )
(de m (N)
(if (=0 N)
0
(- N (f (m (dec N)))) ) )</lang>
PL/I
<lang PL/I>test: procedure options (main);
M: procedure (n) returns (fixed) recursive; /* 8/1/2010 */
declare n fixed; if n <= 0 then return (0); else return ( n - F(M(n-1)) );
end M;
F: procedure (n) returns (fixed) recursive;
declare n fixed; if n <= 0 then return (1); else return ( n - M(F(n-1)) );
end F;
declare i fixed;
do i = 1 to 15;
put skip list ( F(i), M(i) );
end;
end test;</lang>
PostScript
<lang> /female{ /n exch def n 0 eq {1} { n n 1 sub female male sub }ifelse }def
/male{ /n exch def n 0 eq {0} { n n 1 sub male female sub }ifelse }def </lang>
<lang postscript> /F { {
{0 eq} {pop 1} is?
{0 gt} {dup 1 sub F M sub} is?
} cond }.
/M { {
{0 eq} {pop 0} is?
{0 gt} {dup 1 sub M F sub} is?
} cond }.
</lang>
PowerShell
<lang powershell>function F($n) {
if ($n -eq 0) { return 1 }
return $n - (M (F ($n - 1)))
}
function M($n) {
if ($n -eq 0) { return 0 }
return $n - (F (M ($n - 1)))
}</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>
<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
Pure
The Pure definitions very closely maps to the mathematical definitions.
<lang pure>F 0 = 1; M 0 = 0; F n = n - M(F(n-1)) if n>0; M n = n - F(M(n-1)) if n>0;</lang>
<lang pure>> let females = map F (0..10); females; [1,1,2,2,3,3,4,5,5,6,6] > let males = map M (0..10); males; [0,0,1,2,2,3,4,4,5,6,6]</lang>
PureBasic
<lang PureBasic>Declare M(n)
Procedure F(n)
If n = 0 ProcedureReturn 1 ElseIf n > 0 ProcedureReturn n - M(F(n - 1)) EndIf
EndProcedure
Procedure M(n)
If n = 0 ProcedureReturn 0 ElseIf n > 0 ProcedureReturn n - F(M(n - 1)) EndIf
EndProcedure
Define i If OpenConsole()
For i = 0 To 19
Print(Str(F(i)))
If i = 19
Continue
EndIf
Print(", ")
Next
PrintN("")
For i = 0 To 19
Print(Str(M(i)))
If i = 19
Continue
EndIf
Print(", ")
Next
Print(#CRLF$ + #CRLF$ + "Press ENTER to exit")
Input()
CloseConsole()
EndIf</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
Python
.
<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).
Quackery
See also Even or Odd#Quackery: With Anonymous Mutual recursion.
<lang Quackery> forward is f ( n --> n )
[ dup 0 = if done
dup 1 - recurse f - ] is m ( n --> n )
[ dup 0 = iff 1+ done
dup 1 - recurse m - ]
resolves f ( n --> n )
say "f = " 20 times [ i^ f echo sp ] cr say "m = " 20 times [ i^ m echo sp ] cr</lang>
- Output:
f = 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 m = 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12
R
<lang R>F <- function(n) ifelse(n == 0, 1, n - M(F(n-1))) M <- function(n) ifelse(n == 0, 0, n - F(M(n-1)))</lang>
<lang R>print.table(lapply(0:19, M)) print.table(lapply(0:19, F))</lang>
Racket
<lang Racket>#lang racket (define (F n)
(if (>= 0 n)
1
(- n (M (F (sub1 n))))))
(define (M n)
(if (>= 0 n)
0
(- n (F (M (sub1 n))))))</lang>
Raku
(formerly Perl 6) A direct translation of the definitions of and : <lang perl6>multi F(0) { 1 }; multi M(0) { 0 } multi F(\𝑛) { 𝑛 - M(F(𝑛 - 1)) } multi M(\𝑛) { 𝑛 - F(M(𝑛 - 1)) }
say map &F, ^20; say map &M, ^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
REBOL
<lang REBOL>REBOL [
Title: "Mutual Recursion" URL: http://rosettacode.org/wiki/Mutual_Recursion
References: [1] ]
f: func [ "Female." n [integer!] "Value." ] [either 0 = n [1][n - m f n - 1]]
m: func [ "Male." n [integer!] "Value." ] [either 0 = n [0][n - f m n - 1]]
fs: [] ms: [] for i 0 19 1 [append fs f i append ms m i] print ["F:" mold fs crlf "M:" mold ms]</lang>
- Output:
F: [1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12] M: [0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12]
REXX
vanilla
This version uses vertical formatting of the output. <lang rexx>/*REXX program shows mutual recursion (via the Hofstadter Male and Female sequences). */ parse arg lim .; if lim= then lim= 40; w= length(lim); pad= left(, 20)
do j=0 for lim+1; jj= right(j, w); ff= right(F(j), w); mm= right(M(j), w)
say pad 'F('jj") =" ff pad 'M('jj") =" mm
end /*j*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ F: procedure; parse arg n; if n==0 then return 1; return n - M( F(n-1) ) M: procedure; parse arg n; if n==0 then return 0; return n - F( M(n-1) )</lang>
- output when using the default input of: 40
Shown at three-quarter size.)
F( 0) = 1 M( 0) = 0
F( 1) = 1 M( 1) = 0
F( 2) = 2 M( 2) = 1
F( 3) = 2 M( 3) = 2
F( 4) = 3 M( 4) = 2
F( 5) = 3 M( 5) = 3
F( 6) = 4 M( 6) = 4
F( 7) = 5 M( 7) = 4
F( 8) = 5 M( 8) = 5
F( 9) = 6 M( 9) = 6
F(10) = 6 M(10) = 6
F(11) = 7 M(11) = 7
F(12) = 8 M(12) = 7
F(13) = 8 M(13) = 8
F(14) = 9 M(14) = 9
F(15) = 9 M(15) = 9
F(16) = 10 M(16) = 10
F(17) = 11 M(17) = 11
F(18) = 11 M(18) = 11
F(19) = 12 M(19) = 12
F(20) = 13 M(20) = 12
F(21) = 13 M(21) = 13
F(22) = 14 M(22) = 14
F(23) = 14 M(23) = 14
F(24) = 15 M(24) = 15
F(25) = 16 M(25) = 16
F(26) = 16 M(26) = 16
F(27) = 17 M(27) = 17
F(28) = 17 M(28) = 17
F(29) = 18 M(29) = 18
F(30) = 19 M(30) = 19
F(31) = 19 M(31) = 19
F(32) = 20 M(32) = 20
F(33) = 21 M(33) = 20
F(34) = 21 M(34) = 21
F(35) = 22 M(35) = 22
F(36) = 22 M(36) = 22
F(37) = 23 M(37) = 23
F(38) = 24 M(38) = 24
F(39) = 24 M(39) = 24
F(40) = 25 M(40) = 25
with memoization
This version uses memoization as well as a horizontal (aligned) output format.
The optimization due to memoization is faster by many orders of magnitude.
<lang rexx>/*REXX program shows mutual recursion (via the Hofstadter Male and Female sequences). */
parse arg lim .; if lim== then lim=40 /*assume the default for LIM? */
w= length(lim); $m.=.; $m.0= 0; $f.=.; $f.0= 1; Js=; Fs=; Ms=
do j=0 for lim+1
Js= Js right(j, w); Fs= Fs right( F(j), w); Ms= Ms right( M(j), w)
end /*j*/
say 'Js=' Js /*display the list of Js to the term.*/ say 'Fs=' Fs /* " " " " Fs " " " */ say 'Ms=' Ms /* " " " " Ms " " " */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ F: procedure expose $m. $f.; parse arg n; if $f.n==. then $f.n= n-M(F(n-1)); return $f.n M: procedure expose $m. $f.; parse arg n; if $m.n==. then $m.n= n-F(M(n-1)); return $m.n</lang>
- output when using the default input of: 99
Js= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 Fs= 1 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 19 20 21 21 22 22 23 24 24 25 25 26 27 27 28 29 29 30 30 31 32 32 33 34 34 35 35 36 37 37 38 38 39 40 40 41 42 42 43 43 44 45 45 46 46 47 48 48 49 50 50 51 51 52 53 53 54 55 55 56 56 57 58 58 59 59 60 61 61 Ms= 0 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 19 20 20 21 22 22 23 24 24 25 25 26 27 27 28 29 29 30 30 31 32 32 33 33 34 35 35 36 37 37 38 38 39 40 40 41 42 42 43 43 44 45 45 46 46 47 48 48 49 50 50 51 51 52 53 53 54 54 55 56 56 57 58 58 59 59 60 61 61
with memoization, specific entry
This version is identical in function to the previous example, but it also can compute and
display a specific request (indicated by a negative number for the argument).
<lang rexx>/*REXX program shows mutual recursion (via the Hofstadter Male and Female sequences). */
/*───────────────── If LIM is negative, a single result is shown for the abs(lim) entry.*/
parse arg lim .; if lim== then lim= 99; aLim= abs(lim) w= length(aLim); $m.=.; $m.0= 0; $f.=.; $f.0= 1; Js=; Fs=; Ms=
do j=0 for aLim+1; call F(J); call M(j)
if lim<0 then iterate
Js= Js right(j, w); Fs= Fs right($f.j, w); Ms= Ms right($m.j, w)
end /*j*/
if lim>0 then say 'Js=' Js; else say 'J('aLim")=" right( aLim, w) if lim>0 then say 'Fs=' Fs; else say 'F('aLim")=" right($f.aLim, w) if lim>0 then say 'Ms=' Ms; else say 'M('aLim")=" right($m.aLIM, w) exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ F: procedure expose $m. $f.; parse arg n; if $f.n==. then $f.n= n-M(F(n-1)); return $f.n M: procedure expose $m. $f.; parse arg n; if $m.n==. then $m.n= n-F(M(n-1)); return $m.n</lang>
- output when using the input of: -70000
J(70000)= 70000 F(70000)= 43262 M(70000)= 43262
- output when using the input of a negative ¼ million: -250000
J(250000)= 250000 F(250000)= 154509 M(250000)= 154509
Ring
<lang ring> see "F sequence : " for i = 0 to 20
see "" + f(i) + " "
next see nl see "M sequence : " for i = 0 to 20
see "" + m(i) + " "
next
func f n
fr = 1
if n != 0 fr = n - m(f(n - 1)) ok
return fr
func m n
mr = 0
if n != 0 mr = n - f(m(n - 1)) ok
return mr
</lang>
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).
Run BASIC
<lang Runbasic>print "F sequence:"; for i = 0 to 20
print f(i);" ";
next i print :print "M sequence:"; for i = 0 to 20
print m(i);" ";
next i end
function f(n)
f = 1 if n <> 0 then f = n - m(f(n - 1))
end function
function m(n)
m = 0 if n <> 0 then m = n - f(m(n - 1))
end function</lang>
- Output:
F sequence:1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 M sequence:0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12
Rust
<lang rust>fn f(n: u32) -> u32 {
match n {
0 => 1,
_ => n - m(f(n - 1))
}
}
fn m(n: u32) -> u32 {
match n {
0 => 0,
_ => n - f(m(n - 1))
}
}
fn main() {
for i in (0..20).map(f) {
print!("{} ", i);
}
println!("");
for i in (0..20).map(m) {
print!("{} ", i);
}
println!("")
}</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
S-lang
<lang S-lang>% Forward definitions: [also deletes any existing definition] define f(); define m();
define f(n) {
if (n == 0) return 1;
else if (n < 0) error("oops");
return n - m(f(n - 1));
}
define m(n) {
if (n == 0) return 0;
else if (n < 0) error("oops");
return n - f(m(n - 1));
}
foreach $1 ([0:19])
() = printf("%d ", f($1));
() = printf("\n"); foreach $1 ([0:19])
() = printf("%d ", m($1));
() = printf("\n");</lang>
- Output:
1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12
Sather
<lang sather>class MAIN is
f(n:INT):INT pre n >= 0 is if n = 0 then return 1; end; return n - m(f(n-1)); end;
m(n:INT):INT pre n >= 0 is if n = 0 then return 0; end; return n - f(m(n-1)); end;
main is
loop i ::= 0.upto!(19);
#OUT + #FMT("%2d ", f(i));
end;
#OUT + "\n";
loop i ::= 0.upto!(19);
#OUT + #FMT("%2d ", m(i));
end;
end;
end;</lang>
There's no need to pre-declare F or M.
Scala
<lang scala>def F(n:Int):Int =
if (n == 0) 1 else n - M(F(n-1))
def M(n:Int):Int =
if (n == 0) 0 else n - F(M(n-1))
println((0 until 20).map(F).mkString(", ")) println((0 until 20).map(M).mkString(", "))</lang>
- Output:
1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12
Scheme
define declarations are automatically mutually recursive:
<lang scheme>(define (F n)
(if (= n 0) 1
(- n (M (F (- n 1))))))
(define (M n)
(if (= n 0) 0
(- n (F (M (- n 1))))))</lang>
If you wanted to use a let-like construct to create local bindings, you would do the following. The define construct above is just a syntactic sugar for the following where the entire rest of the scope is used as the body.
<lang scheme>(letrec ((F (lambda (n)
(if (= n 0) 1
(- n (M (F (- n 1)))))))
(M (lambda (n)
(if (= n 0) 0
(- n (F (M (- n 1))))))))
(F 19)) # evaluates to 12</lang>
The letrec indicates that the definitions can be recursive, and fact that we placed these two in the same letrec block makes them mutually recursive.
Seed7
<lang seed7>$ include "seed7_05.s7i";
const func integer: m (in integer: n) is forward;
const func integer: f (in integer: n) is func
result
var integer: res is 0;
begin
if n = 0 then
res := 1;
else
res := n - m(f(n - 1));
end if;
end func;
const func integer: m (in integer: n) is func
result
var integer: res is 0;
begin
if n = 0 then
res := 0;
else
res := n - f(m(n - 1));
end if;
end func;
const proc: main is func
local
var integer: i is 0;
begin
for i range 0 to 19 do
write(f(i) lpad 3);
end for;
writeln;
for i range 0 to 19 do
write(m(i) lpad 3);
end for;
writeln;
end func;</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
Sidef
<lang ruby>func F(){} func M(){} F = func(n) { n > 0 ? (n - M(F(n-1))) : 1 } M = func(n) { n > 0 ? (n - F(M(n-1))) : 0 } [F, M].each { |seq|
}</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
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>
SNOBOL4
<lang SNOBOL4> define('f(n)') :(f_end) f f = eq(n,0) 1 :s(return)
f = n - m(f(n - 1)) :(return)
f_end
define('m(n)') :(m_end)
m m = eq(n,0) 0 :s(return)
m = n - f(m(n - 1)) :(return)
m_end
- # Test and display
L1 s1 = s1 m(i) ' ' ; s2 = s2 f(i) ' '
i = le(i,25) i + 1 :s(L1)
output = 'M: ' s1; output = 'F: ' s2
end</lang>
- Output:
M: 0 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 F: 1 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
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> SPL<lang spl>f(n)= ? n=0, <= 1 <= n-m(f(n-1)) . m(n)= ? n=0, <= 0 <= n-f(m(n-1)) . > i, 0..20 fs += " "+f(i) ms += " "+m(i) <
F: 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 M: 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12 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))
The <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))
which indicates that the functions call themselves (
> val terms = List.tabulate (10, fn x => x); val terms = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]: int list > map f terms; val it = [1, 1, 2, 2, 3, 3, 4, 5, 5, 6]: int list > map m terms; val it = [0, 0, 1, 2, 2, 3, 4, 4, 5, 6]: int list SwiftIt just works. No special pre-declaration is necessary. <lang swift>func F(n: Int) -> Int { return n == 0 ? 1 : n - M(F(n-1)) } func M(n: Int) -> Int { return n == 0 ? 0 : n - F(M(n-1)) } for i in 0..20 { print("\(F(i)) ")
} println() for i in 0..20 { print("\(M(i)) ")
} println()</lang> Symsyn<lang Symsyn> F param Fn if Fn = 0
1 R
else
(Fn-1) nm1
save Fn
call F nm1
result Fr
save Fr
call M Fr
result Mr
restore Fr
restore Fn
(Fn-Mr) R
endif
return R
M param Mn if Mn = 0
0 R
else
(Mn-1) nm1
save Mn
call M nm1
result Mr
save Mr
call F Mr
result Fr
restore Mr
restore Mn
(Mn-Fr) R
endif
return R
start i if i <= 19 call F i result res " $s res ' '" $s + i goif endif $s [] $s i if i <= 19 call M i result res " $s res ' '" $s + i goif endif $s [] </lang> Tailspin<lang tailspin> templates male when <=0> do 0 !
otherwise def n: $;
$n - 1 -> male -> female -> $n - $ !
end male templates female when <=0> do 1 !
otherwise def n: $;
$n - 1 -> female -> male -> $n - $ !
end female 0..10 -> 'M$;: $->male; F$;: $->female; ' -> !OUT::write </lang>
M0: 0 F0: 1 M1: 0 F1: 1 M2: 1 F2: 2 M3: 2 F3: 2 M4: 2 F4: 3 M5: 3 F5: 3 M6: 4 F6: 4 M7: 4 F7: 5 M8: 5 F8: 5 M9: 6 F9: 6 M10: 6 F10: 6 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> TI-89 BASIC<lang ti89b>Define F(n) = when(n=0, 1, n - M(F(n - 1))) Define M(n) = when(n=0, 0, n - F(M(n - 1)))</lang> TXR<lang txrlisp>(defun f (n) (if (>= 0 n) 1 (- n (m (f (- n 1)))))) (defun m (n) (if (>= 0 n) 0 (- n (f (m (- n 1)))))) (each ((n (range 0 15))) (format t "f(~s) = ~s; m(~s) = ~s\n" n (f n) n (m n)))</lang> $ txr mutual-recursion.txr f(0) = 1; m(0) = 0 f(1) = 1; m(1) = 0 f(2) = 2; m(2) = 1 f(3) = 2; m(3) = 2 f(4) = 3; m(4) = 2 f(5) = 3; m(5) = 3 f(6) = 4; m(6) = 4 f(7) = 5; m(7) = 4 f(8) = 5; m(8) = 5 f(9) = 6; m(9) = 6 f(10) = 6; m(10) = 6 f(11) = 7; m(11) = 7 f(12) = 8; m(12) = 7 f(13) = 8; m(13) = 8 f(14) = 9; m(14) = 9 f(15) = 9; m(15) = 9 uBasic/4tHuBasic/4tH supports mutual recursion. However, the underlying system can't support the stress this puts on the stack - at least not for the full sequence. This version uses memoization to alleviate the stress and speed up execution. <lang>LOCAL(1) ' main uses locals as well FOR a@ = 0 TO 200 ' set the array @(a@) = -1 NEXT PRINT "F sequence:" ' print the F-sequence FOR a@ = 0 TO 20 PRINT FUNC(_f(a@));" "; NEXT PRINT PRINT "M sequence:" ' print the M-sequence FOR a@ = 0 TO 20 PRINT FUNC(_m(a@));" "; NEXT PRINT END
IF a@ = 0 THEN RETURN (1) ' memoize the solution IF @(a@) < 0 THEN @(a@) = a@ - FUNC(_m(FUNC(_f(a@ - 1)))) RETURN (@(a@)) ' return array element
IF a@ = 0 THEN RETURN (0) ' memoize the solution IF @(a@+100) < 0 THEN @(a@+100) = a@ - FUNC(_f(FUNC(_m(a@ - 1)))) RETURN (@(a@+100)) ' return array element</lang>
F sequence: 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 M sequence: 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12 0 OK, 0:199 UNIX 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> UrsalaForward declarations are not an issue in Ursala, which allows any
definition to depend on any symbol declared within the same
scope. However, cyclic dependences are not accepted unless the
programmer explicitly accounts for their semantics. If the recurrence
can be solved using a fixed point combinator, the compiler can be
directed to use one by the <lang Ursala>#import std
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>
( <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>) Vala<lang vala>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)); } void main() { print("n : ");
for (int s = 0; s < 25; s++){
print("%2d ", s);
}
print("\n------------------------------------------------------------------------------\n");
print("F : ");
for (int s = 0; s < 25; s++){
print("%2d ", F(s));
}
print("\nM : ");
for (int s = 0; s < 25; s++){
print("%2d ", M(s));
}
}</lang>
n : 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ------------------------------------------------------------------------------ F : 1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 13 14 14 15 M : 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12 13 14 14 15 VBA<lang vb>Private Function F(ByVal n As Integer) As Integer If n = 0 Then
F = 1
Else
F = n - M(F(n - 1))
End If
End Function Private Function M(ByVal n As Integer) As Integer If n = 0 Then
M = 0
Else
M = n - F(M(n - 1))
End If
End Function Public Sub MR() Dim i As Integer
For i = 0 To 20
Debug.Print F(i);
Next i
Debug.Print
For i = 0 To 20
Debug.Print M(i);
Next i
End Sub</lang>
1 1 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 12 13 0 0 1 2 2 3 4 4 5 6 6 7 7 8 9 9 10 11 11 12 12 Wren<lang ecmascript>var M // forward declaration var F = Fn.new { |n| if (n == 0) return 1 return n - M.call(F.call(n-1)) } M = Fn.new { |n| if (n == 0) return 0 return n - F.call(M.call(n-1)) } System.write("F(0..20): ") (0..20).each { |i| System.write("%(F.call(i)) ") } System.write("\nM(0..20): ") (0..20).each { |i| System.write("%(M.call(i)) ") } System.print()</lang>
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 x86 AssemblySince 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. 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
end</lang> XPL0<lang XPL0>code ChOut=8, CrLf=9, IntOut=11; ffunc M; \forward-referenced function declaration func F(N); int N; return if N=0 then 1 else N - M(F(N-1)); func M(N); int N; return if N=0 then 0 else N - F(M(N-1)); int I; [for I:= 0 to 19 do [IntOut(0, F(I)); ChOut(0, ^ )]; CrLf(0); for I:= 0 to 19 do [IntOut(0, M(I)); ChOut(0, ^ )]; CrLf(0); ]</lang>
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 Yabasic<lang Yabasic>// User defined functions sub F(n) if n = 0 return 1 return n - M(F(n-1)) end sub
if n = 0 return 0 return n - F(M(n-1)) end sub
print F(i) using "###"; next print for i = 0 to 20 print M(i) using "###"; next print</lang> zklThis works if the functions are in a file or on one line (in the REPL) as zkl doesn't like referencing undefined objects. You could also pass/close the other function. <lang zkl>fcn f(n){ if(n==0)return(1); n-m(f(n-1,m),f) } fcn m(n){ if(n==0)return(0); n-f(m(n-1,f),m) } [0..19].apply(f).println(); // or foreach n in ([0..19]){ print(f(n)," ") } [0..19].apply(m).println(); // or foreach n in ([0..19]){ print(m(n)," ") }</lang>
L(1,1,2,2,3,3,4,5,5,6,6,7,8,8,9,9,10,11,11,12) L(0,0,1,2,2,3,4,4,5,6,6,7,7,8,9,9,10,11,11,12) |
|---|