There seems to be a problem with Axiom's ContinuedFractions? domain. The type of the result is shown as Fraction Float but this is nonesense.

Something similar happens if the argument is Fraction Integer

)set functions compile on

nume(a) == cons(1,[((a-i)*i) for i in 1..]);

dene(a,x) == [(x+2*i+1-a) for i in 0..];

cfe(a,x) == continuedFraction(0,nume(a),dene(a,x));

ccfe(a,x) == convergents cfe(a,x);

ccfe(0,2::Float)

Compiling function nume with type NonNegativeInteger -> Stream(
      Integer)

Compiling function dene with type (NonNegativeInteger,Float) -> 
      Stream(Float)

Compiling function cfe with type (NonNegativeInteger,Float) -> 
      ContinuedFraction(Float)

Compiling function ccfe with type (NonNegativeInteger,Float) -> 
      Stream(Fraction(Float))

(1)

ccfe(0,2::Fraction Integer)

Compiling function dene with type (NonNegativeInteger,Fraction(
      Integer)) -> Stream(Fraction(Integer))

Compiling function cfe with type (NonNegativeInteger,Fraction(
      Integer)) -> ContinuedFraction(Fraction(Integer))

Compiling function ccfe with type (NonNegativeInteger,Fraction(
      Integer)) -> Stream(Fraction(Fraction(Integer)))

(2)

Fraction Fraction Integer is also nonsense.

ff1:Fraction Float

   Fraction(Float) is not a valid type.
ff2:Fraction Fraction Integer

   Fraction(Fraction(Integer)) is not a valid type.

• Fraction R returns R when R is a field
• an operation that wants to return Fraction R has to check whether R is a field.

In the case above I see another problem:

n := nume(0.2)

Compiling function nume with type Float -> Stream(Float)

(3)

d := dene(0.2,x)

Compiling function dene with type (Float,Variable(x)) -> Stream(
      Polynomial(Float))

(4)

continuedFraction(0, n, d)

(note that there is no output...)

But is this possible in Axiom? How would I modify [Fraction]? to make this happen?

• Axiom's interpreter should not complain about types like Fraction Float. In this case, denominator will always return 1, of course. Note that you can perfectly well calculate with objects of type Stream Fraction Fraction Integer, no problem occurs. Or
• operations that want to return Fraction R have to check whether R is a field.

The code that excludes constructions like FRAC FRAC and FRAC R when R is a field is in clammed.boot.pamphlet, lines 108-112.

Possibly we want to allow Fraction Field but disallow Fraction Fraction?

Martin

Fraction(S: IntegralDomain): QuotientFieldCategory S with ...
 == if S has Field
      then S add 
         -- additional functions like "denom" etc.
      else ...
         add 
           -- old code goes here

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3103239194962224468-25px004.spad
      using old system compiler.
******** Spad syntax error detected ********
The current line is:

    1> Fraction(S: IntegralDomain): QuotientFieldCategory S with ...

The number of valid tokens is 1.
The prior token was #S(TOKEN
                       :SYMBOL |with|
                       :TYPE KEYWORD
                       :NONBLANK NIL
                       :LINE_NUM 1
                       :CHAR_NUM 53)
The current token is #S(TOKEN
                        :SYMBOL |..|
                        :TYPE KEYWORD
                        :NONBLANK NIL
                        :LINE_NUM 1
                        :CHAR_NUM 58)

Well, that means the representation of FRAC S depends on whether S is already a field or not.

I think Axiom's two-level type system is going to get in the way. Although it is true that types in Axiom are "first order objects" in the sense that we can assign them to variables etc., the kind of things that we can actually do with them is very limited. For example, I see no way in Axiom, SPAD or Aldor to have:

  FRAC FRAC INT = FRAC INT

since FRAC INT is static type and FRAC FRAC INT is another static type. In Axiom there is no way to write a function which returns different types depending on it's parameters:

  fType(x) ==
     x=0 ==> Integer
     Float

  t:fType(1):=1.0

because types like Integer and Float are not members of some domain in the same since in which 1 and -1 are members of Integer. And further there is no equality defined over types.

Or do I not understand something quite fundamental here?

Perhaps what you are suggesting amounts to ensuring that, as types these two are at least functionally equivalent?

Note that you can perfectly well calculate with objects of type Stream Fraction Fraction Integer, no problem occurs.

I don't think you can calculate reliably with these strange types. I get bizarre results and sometimes Axiom crashes when I try things that should work. For example, this works:

cff1:=continuedFraction(0,repeating [2],repeating [3])

(5)

4*cff1

(6)

cff2:=continuedFraction(0,repeating [1/2],repeating [1/3])

(7)

But the following code sends Axiom into an infinite loop:

  1/4*cff2