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RE: [Axiom-developer] RE: symbolic manipulation of expressions in Axiom

From: Bill Page
Subject: RE: [Axiom-developer] RE: symbolic manipulation of expressions in Axiom
Date: Sun, 1 Apr 2007 18:43:12 -0400

On April 1, 2007 4:20 PM Gaby wrote:
> ...
> Bill Page wrote:
> | 
> | There are *no* mathematical (algebraic) operations in InputForm!
> and I'm saying that once you've added them, as for Expression T,
> you'll see no difference.  Adding them is precisely what people
> do when they start "symbolic computation" and "simplification".

No. I disagree at a deep level. I think what is done in the
Axiom library is completely different. What Axiom is doing is
fundamentally *algebraic*. Simplification of symbolic expressions
does not any algebra. For example to simplify an expression it
is not necessary to instantiate an object which directly represents
a polynomial in the way polynomials are represented in Axiom.
All that you need to do is to apply a set of re-write rules that
accomplish (or not) your desired goal.

On the other hand creating a polynomial *object* (and some other
algebraic objects) is exactly what Axiom does in the Expression

> [...]
> | > Also, be aware than the Axiom designers, in many places,
> | > thought of Expression as the general domain for symbolic
> | > manipulation and have appropriate hardwired type inference
> | > rules in the interpreter.
> | > 
> | 
> | Of course that is no problem. If you are able to read and
> | understand Stephen Watt's paper on this subject, I am sure
> | that this would be clear to you.
> That has nothing to do with what I'm saying. 

??? But that is exactly what I am saying.
> BTW, it is possible to read Watt's paper and not agreeing with
> him. And it is possible to agree with him and not reading his
> paper.

Certainly. In fact I agreed with him before I read his paper.

> But again, what he said in his paper has nothing to do with
> the distinction between Expression T and InputForm I'm
> concerned with.

It is possible that I do not understand "the distinction between
Expression T and InputForm that you are concerned with", but I
did think that my concern in this regard was the same as yours.
If that is the case then I strongly disagree with your conclusion.
I also feel strongly that understanding this distinction is
critical to understanding what is different about Axiom as a
computer algebra system, so I have a lot of motivation to continue
this discussion until we can find some agreement.

How shall we proceed?

Would it help if I try to described more formally my understanding
of exactly how Axiom captures matematical semantics in terms of
the representation of objects and how that differs from more
syntactical things like term re-writing? Or do continue to insist
that we not talk about representation?

Bill Page.

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