[Axiom-math] Re: [open-axiom-devel] missing exports in Finite DirectProduct and Product

[Francois Maltey]

Dear Bill, and others.

About Product (i.e. cartesian product) and DirectProduct over finite domains.

You add 'index', 'lookup', and 'hash'.
> The operation 'size' is already available as a default.

> Having 'index' available for products allows one to iterate over the
> elements of a cross product like this:
> 
>   X:=Product(OVAR [a,b],IntegerMod(3))
>   [index(i::PI)$X for i in 1..size()$X]

Right, axiom searchs for the congruent integer with index and lookup.

But is it possible to have an intrinsic loop than an integer loop ?

Dom := OVAR [aa,bb,cc,dd,ee,ff,gg,hh,ii,jj,kk,ll]
[i for i in dd..ee]

> Writing this made me think that it was also desirable the domains in
> Finite also enumerate there members in a more functional manner so
> because FriCAS now allows us to easily make fairly deep changes in the
> algebra code, as an exercise I decided to define the following new
> export of Finite
> 
>   expand: () -> List %
>   expand() == [index(i::PositiveInteger) for i in 1..size()]

A good primitive. The expand name looks as expand (100..110).

For finite domains you enumerate all the cases, not a subset of cases.
I don't find any expand functions in others finite domain.
Others systems uses expand for expanding calculus only.

So I prefer an enumerate function to an expand function, 
and add an enumerate in all the finite structure  
address@hidden,bb,cc,dd,ee,ff,gg,hh,ii,jj,kk,ll] responds
[aa,bb,cc,dd,ee,ff,gg,hh,ii,jj,kk,ll]

So I see 2 functions :

    enumerate without arguments for all the cases.
and expand with a interval for subset.
Is it possible to get this after an expand (5..7, aa..cc) ?

[[5,aa],[6,aa],[7,aa],[5,bb],[6,bb],[7,bb],[5,cc],[6,cc],[7,cc]]

I don't see how to type this expand. 

> So with this patch one can write for example:
> 
> (1) -> map(x+->(selectsecond(x)::Symbol)^(selectfirst(x)::Integer), _
>          expand()$Product(ZMOD(3),OVAR [a,b]))
>               2  2
>    (1)  [a,b,a ,b ,1,1]

It's a progress to allow this map, but it isn't too serviceable.

Why not a 
map (x +-> x.1::Symbol^x.2::Integer 
                     for x in enumerate()$Product(ZMOD(3),OVAR [a,b]))
map (x +-> x.1::Symbol^x.2::Integer 
                     for x in expand(1..2,a..b)$Product(ZMOD(3),OVAR [a,b]))

So axiom has record and cartesian products, 
but the must use of makeprod(,)@... or news(,)@.... isn't friendly.

Mathematics use so often cartesian product it's penalizing to
have to declare a cartesian product with new or makeprod.
Other languages allow lists for this uses because they aren't typed.

A new constructor as [|...,...|] should be a great progress in the
interface. By example caml uses [1;2;3] for lists and [|1;2;3|] for array.

> I would be interested in your opinions about this approach to
> providing Cartesian products for the Axiom library.

A really good idea for finite domains.
 
> I think it is interesting to see how domains in the category Finite
> able to participate like "sets" in the language. 

What are the commom functions in this cases ?

index, lookup, hash, are they present in all finite domains ?

Enumerate in a list all the elements is a good idea.
It may be interesting to expand? a sub-interval.

For cartesian product in general a short constructor will be a 
serious avantage. output [|n, (X-1)^n|] will work.

> Of course Aldor already has implemented some basic abstractions of
> this kind but it is not entirely obvious to me how best to extend
> Axiom's library and still stay within the limitations of the current
> Spad language.

Will axiom interpreter include aldor or not in the future ?

F.