Re: [Axiom-developer] Polynomials, abstract objects, provisos

[Ralf Hemmecke]

> I have no idea what you are talking about, sorry. I was referring to Tim's (?)
> idea that we could say something like:
>
> p: Abstract POLY INT
>
> i.e., not assign p a value,

Sorry, then I did not understand that.

> and still do interesting things with it. And I doubt that this is something
> within reach, as it is, in my opinion, rather orthogonal to the philosophy of
> the current Axiom Algebra.

> Something easier to grasp is probably:
>
> z: Abstract Complex INT
>
> and then be able to say 
>
> w := conjugate z.
>
> In other words: there would be no data representation as we are used to
> currently.

I don't believe your last statement. Somewhere it must be stored that w 
and z are connected via "conjugate", so there must be some storage 
involved. Now the question is who is going to remember these relations?

Since I don't want the interpreter be too smart, I would like to encode 
such things in the library. And that could be done (theoretically at least).

Assume we have a domain A that exports foo: % -> % then you would need 
another domain DelayedA that basicly has the same exports as A and 
additionally two coerce functions A -> DelayedA and back.

In DelayedA function applications would be collected and only executed 
if you really provide an element of A. But there are more elements in 
DelayedA than in A, namely those that have no value in A.

In other words one would have a kind of ExpressionTrees where you carry 
the type information.

But don't quote me on that my ideas are completely underdeveloped. And 
there is certainly something missing that might look like provisos. For 
example, if you declare "a: Integer" and then say "5/a". It might be a 
bit easier to get the type of 5/a right. The value is a bit more 
problematic since at some point when you finally assign a value to "a" 
an exception might happen and your program has to deal with that. But an 
exception is an easy exercise. It is getting more complicated if the 
"unvalued but typed" expression might result in multible values.

Ralf