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# Edit detail for Mutual Recursion revision 3 of 3

 1 2 3 Editor: test1 Time: 2018/02/17 16:05:57 GMT+0 Note:

changed:
-Here we give two examples of defining functions by mutual recursion in Axiom
Here we give two examples of defining functions by mutual recursion in FriCAS


Here we give two examples of defining functions by mutual recursion in FriCAS

## Recursion between Separate Domains

Mutual recursion between domains is allowed. However, compiling such domains is a little tricky. Fist one need to compile files in "bootstrap mode". That is one needs to give:

)boot $bootStrapMode := true Then compile domains first time using )compile. Next, compile again in normal way. That is give )boot$bootStrapMode := false

and angain use )compile.

Note, that goal of "bootstrap mode" compilation is to break cycles, so for example if there are two mutially dependent domains, then it is enough to compile in "bootstrap mode" just one.

This process can not be used on the wiki, because wiki software automatically issues compilation commands. Instead one can use manual workaround which we describe below.

First we show how to define two separate domains EVEN and ODD both of which have an attribute represented by the polymorthic function parity.

We must start with a "bootstrap" definition of parity(n)$even. All that is really needed here is a package that exports a function named parity with the right signature. This particular function will never be called and will be re-defined later. It's only purpose is as a placeholder to allow the later definition of ODD. spad )abbrev domain EVEN even even(): E == I where E == with parity: Integer -> Boolean I == add parity(n) == true spad  Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3342984326569348053-25px001.spad using old system compiler. EVEN abbreviates domain even ------------------------------------------------------------------------ initializing NRLIB EVEN for even compiling into NRLIB EVEN compiling exported parity : Integer -> Boolean EVEN;parity;IB;1 is replaced by QUOTET Time: 0.01 SEC. (time taken in buildFunctor: 0) ;;; *** |even| REDEFINED ;;; *** |even| REDEFINED Time: 0 SEC. Cumulative Statistics for Constructor even Time: 0.01 seconds finalizing NRLIB EVEN Processing even for Browser database: --->-->even(constructor): Not documented!!!! --->-->even((parity ((Boolean) (Integer)))): Not documented!!!! --->-->even(): Missing Description ; compiling file "/var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.lsp" (written 17 FEB 2018 04:05:56 PM): ; /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.fasl written ; compilation finished in 0:00:00.012 ------------------------------------------------------------------------ even is now explicitly exposed in frame initial even will be automatically loaded when needed from /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN Now we can define parity(n)$odd. It depends on EVEN.

)abbrev domain ODD odd
odd(): E == I where
E == with
parity: Integer -> Boolean
parity(n:Integer) ==
--    output("ODD",n::OutputForm)$OutputPackage (n>0) => parity(n-1)$even
(n<0) => parity(n+1)$even false spad  Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7392352806104595030-25px002.spad using old system compiler. ODD abbreviates domain odd ------------------------------------------------------------------------ initializing NRLIB ODD for odd compiling into NRLIB ODD compiling exported parity : Integer -> Boolean Time: 0.02 SEC. (time taken in buildFunctor: 0) ;;; *** |odd| REDEFINED ;;; *** |odd| REDEFINED Time: 0 SEC. Cumulative Statistics for Constructor odd Time: 0.02 seconds finalizing NRLIB ODD Processing odd for Browser database: --->-->odd(constructor): Not documented!!!! --->-->odd((parity ((Boolean) (Integer)))): Not documented!!!! --->-->odd(): Missing Description ; compiling file "/var/aw/var/LatexWiki/ODD.NRLIB/ODD.lsp" (written 17 FEB 2018 04:05:56 PM): ; /var/aw/var/LatexWiki/ODD.NRLIB/ODD.fasl written ; compilation finished in 0:00:00.014 ------------------------------------------------------------------------ odd is now explicitly exposed in frame initial odd will be automatically loaded when needed from /var/aw/var/LatexWiki/ODD.NRLIB/ODD But the bootstrap definition of EVEN is incomplete. It really depends (recusively) on ODD. So finally we need the full (re-)definition of parity(n)$even

)abbrev domain EVEN even
even(): E == I where
E == with
parity: Integer -> Boolean
parity(n) ==
--    output("EVEN",n::OutputForm)$OutputPackage n>0 => parity(n-1)$odd
n<0 => parity(n+1)$odd true spad  Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8584362905457126194-25px003.spad using old system compiler. Illegal NRLIB EVEN.NRLIB claims that its constructor name is the domain even but even is already known to be the for package EVEN . EVEN abbreviates domain even ------------------------------------------------------------------------ initializing NRLIB EVEN for even compiling into NRLIB EVEN compiling exported parity : Integer -> Boolean Time: 0 SEC. (time taken in buildFunctor: 0) ;;; *** |even| REDEFINED ;;; *** |even| REDEFINED Time: 0 SEC. Cumulative Statistics for Constructor even Time: 0 seconds finalizing NRLIB EVEN Processing even for Browser database: --->/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3342984326569348053-25px001.spad-->even(constructor): Not documented!!!! --->/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3342984326569348053-25px001.spad-->even((parity ((Boolean) (Integer)))): Not documented!!!! --->/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3342984326569348053-25px001.spad-->even(): Missing Description ; compiling file "/var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.lsp" (written 17 FEB 2018 04:05:56 PM): ; /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.fasl written ; compilation finished in 0:00:00.009 ------------------------------------------------------------------------ even is already explicitly exposed in frame initial even will be automatically loaded when needed from /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN Now we can test the new function: fricas parity(10)$even
 (1)
Type: Boolean
fricas
parity(8)$odd  (2) Type: Boolean fricas parity(-1111)$odd
 (3)
Type: Boolean

## Recursion within a Single Domain

It is possible to write this same recursion as a domain that exports two functions Even and Odd. In this case we do not need to supply any initial bootstrap code because the compiler is able to resolve both functions simultaneously.

)abbrev domain PARITY Parity
Parity(): Exports == Implements where
Exports == with
Even: Integer -> Boolean
Odd: Integer -> Boolean
Odd(n: Integer) ==
n>0 => Even(n-1)
n<0 => Even(n+1)
false
Even(n: Integer) ==
n>0 => Odd(n-1)
n<0 => Odd(n+1)
true
   Compiling FriCAS source code from file
using old system compiler.
PARITY abbreviates domain Parity
------------------------------------------------------------------------
initializing NRLIB PARITY for Parity
compiling into NRLIB PARITY
compiling exported Odd : Integer -> Boolean
Time: 0 SEC.
compiling exported Even : Integer -> Boolean
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |Parity| REDEFINED
;;;     ***       |Parity| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Parity
Time: 0 seconds
finalizing NRLIB PARITY
Processing Parity for Browser database:
--->-->Parity(constructor): Not documented!!!!
--->-->Parity((Even ((Boolean) (Integer)))): Not documented!!!!
--->-->Parity((Odd ((Boolean) (Integer)))): Not documented!!!!
--->-->Parity(): Missing Description
; compiling file "/var/aw/var/LatexWiki/PARITY.NRLIB/PARITY.lsp" (written 17 FEB 2018 04:05:57 PM):
; /var/aw/var/LatexWiki/PARITY.NRLIB/PARITY.fasl written
; compilation finished in 0:00:00.011
------------------------------------------------------------------------
Parity is now explicitly exposed in frame initial
Parity will be automatically loaded when needed from
/var/aw/var/LatexWiki/PARITY.NRLIB/PARITY

Test

fricas
Even(10)$Parity  (4) Type: Boolean fricas Odd(8)$Parity
 (5)
Type: Boolean
fricas
Odd(-1111)\$Parity
 (6)
Type: Boolean