different approaches to correctness / was: Simplifying radicals

Slightly off topic and another shameless plug from an aunt of MuPAD...

JohnCreighton_@xxxxxxxxxxx writes:

> My answer is there should be a default set of rules and we should be able to
> tell the CAS to change these rules if needed. As for things happening
> internally perhaps you could specify rules in a given context. For instance
> we could tell the CAS that you can only cancel roots when doing limit
> computations. If all the contexts are well documented and the programmer
> could create there own context then this might work. A function could even
> override contexts set externally if it was known that this would lead to a
> wrong answer.

All this is solved by the approach Axiom and MuPAD take. Both define various
domains, and the user decides in which you want to do your calculations. For
example, in axiom there are domains for Polynomials, Integers, Algebraic
Numbers, and so on. If the user doesn't decide, the interpreter tries to find
an appropriate domain himself.

The situation in MuPAD is similar, I believe.

Axiom is free software since a few years, and there is quite some development
going on. One of its main strenghts is integration, as far as I know it's
implementation of the Risch algorithm is the most complete among all
CAS. (implemented, in fact, by Manuel Bronstein himself, together with Barry
Trager, Renaud Rioboo and probably some others I forgot (sorry))

Another strength is its language, which is well defined and can be both
interpreted and compiled (to "native" code), so if you care, efficiency can be
achieved. Apart from some "minor" differences, it has two implementations:
one external, called Aldor, another one internal, called Spad. (I know, this is
not quite exact, but the picture should be correct...)

In my opinion the weakest point is its domain for general expressions. But
there is active developement going on to enhance this.

If you are interested, visit

http://page.axiom-developer.org/zope/mathaction/FrontPage

where you can try it online (in the SandBox, please)

Martin
.

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