RE: [Axiom-developer] Re: Defining piece-wise functions and drawing, integrating, ...

[Bill Page]

On June 4, 2007 4:36 AM Ralf Hemmecke wrote:
> ...
> Oh, I never said, that the current Expression(...) should be 
> removed. If I see what people do here at RISC with symbolic
> summation, that domain fits very well in that area. Still,
> Axiom lacks a true ExpressionTree domain. Or have I just not
> found it?

I think that together InputForm and SExpression play that role
except that the coercion to OutputForm is a little to Lisp-ish.
I think the result of 'expr$SExpression' should be the OutputForm.

> ...
> >> Well, what I would dream of is an expression domain (or 
> >> several of them) that lets you define a general expression
> >> tree where you would have control over what is allowed as
> >> nodes. Maybe it would be interesting to be able to give a
> >> grammar G and MyExpression(G) would then describe the 
> >> language generated by G. It would be nice to be able to 
> >> encode the Aldor language in that way.
> 
> > For Axiom this does not seem sufficiently "algebraic" to me.
> 
> Then remove the "Set" domain. Why must everything be "algebraic".

I would agree to replace "Set" with something more algebraic,
e.g. a topos.

http://en.wikipedia.org/wiki/Topos

The short answer to why everything must be algebraic is:
category theory. You can ask why try to base the Axiom library
on category theory, but that is an argument at a very different
level.

In fact I think it is quite wrong that Axiom's library places
SetCategory so near to the top of the algebra hierarchy. It
would be better to start with something more primitive like the
concept of a cartesian close category.
 
> Actually, you could probably turn an ExpressionTree into some
> form of universal algebra (just leave the set of operations
> empty).

That would not make me nearly as happy as category theory. :-(

> 
> Oh, maybe SExpression is near to what I want. But is somehow 
> sounds to LISPish for me. ;-) Anyway, I think it would be a good
> thing to have a very general expression domain (maybe like
> SExpression) and yet others that only allow certain expression
> trees that correspond to a grammar.

Yes. I think Gaby had some ideas along that line. We discussed
some aspects of that here:

http://wiki.axiom-developer.org/SandBoxInductiveType

and on this list.

> ... 
> > I think of this as a more sophisticated replacement for
> > SExpression.
> 
> Maybe. But it is written in Aldor and does not rely on an 
> underlying LISP.
> 

Lisp does not "own" the concept of SExpression

http://en.wikipedia.org/wiki/S-expression

see also

http://en.wikipedia.org/wiki/M-expression

But I do not have any objection to importing the concept
of ExpressionTree into Axiom.

Regards,
Bill Page.