[Axiom-developer] Re: symbolic manipulation of expressions in Axiom (was: SymPy on Axiom Wiki)

[gdr]

> > Why should InputForm be preferable over Expression T?
>
> I'm glad you asked. :-)
>
> I presume that by "Expression T" you mean such domains as
> 'Expression Integer' and 'Expression Float' etc.

Yes.

> For me to answer your question it is important to understand
> what Expression in Axiom really is. One way to do that is to
> look at the internal representation.
>
>  Rep:=Fraction SparseMultivariatePolynomial(T, Kernel %)
>
> So Expression is a mathematical domain of rational functions
> (ratio of two polymonials) where the polynomial variables
> are extended to kernels including various special functions.

Sorry, I don't think I'm satisfied with an answer that has to
look at the internal representation before explaining why I
should not believe the interface.

> It is of course a very general so many symbolic expressions
> can be evaluated as operations on members of this domain.
> But the members of this domain are *not* general symbolic
> expressions! They are quite literally things represented as
> the ratio of two polynomials, which in turn are represented
> by however ratios and polynomials are represented, etc. And
> the OutputForm for the members of this domain correspond to
> a much reduced set of symbolic expressions.
>
> That is the reason why when you enter something like this:
>
> (1) -> 1/sin(x)+y
>
> Axiom responds with:
>
>        y sin(x) + 1
>   (1)  ------------
>           sin(x)
>                                 Type: Expression Integer
>
> Even though perhaps it looks like it, Axiom is not performing
> some kind of arbitrary "simplification" of symbolic expressions.
> Instead what is happening is that Axiom is interpreting the
> symbols in the input expression as operations performed in
> some domain and evaluating those operations *algebraically*
> to ultimately create a member of that domain - in this case
> Expression Integer. Then the Axiom interpreter displays this
> member using the coercion to OutputForm defined by the
> domain.
>
> In contrast the members of the domain InputForm represent
> exactly the symbolic expression that is the *input* to the
> interpreter. Manipulating members of this domain amounts to
> directly manipulating the expressions.
>
> Regards,
> Bill Page