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Re: [Axiom-developer] RE: Types as values, and Re: Static versusDynamica

From: Ralf Hemmecke
Subject: Re: [Axiom-developer] RE: Types as values, and Re: Static versusDynamicallytyped(was:Philosophy... )
Date: Tue, 27 Sep 2005 12:50:59 +0200
User-agent: Mozilla Thunderbird 1.0 (X11/20041206)

William Sit wrote:
It is useful, for example, to be
able to distinguish whether a field is a quotient field of some integral domain
or just a field to make codes more efficient. The request was not acted upon
because at the time (around 1990 +/- 3 years?), effort was concentrated on
developing the Aldor (A#) compiler, which promised to solve such problems.

However, I somehow do not think that
being able to query the structure of a type is a good idea. You gave a
good reason, but isn't there a better way to achieve efficiency?

Yes, but another reason, in the above example of field, is in allowing
generalizations to work, not just efficiency due to data representation
(reducing computation in the field to that in the underlying integral domain, as
in Grobner bases computation). For example, if I have a constructor that
requires an integral domain R as a parameter, and I constructed in the package
the quotient field of R, as you may suggest as an explicit way to achieve
efficiency, then the package will FAIL if R is actually a field (you can
consider this as a bug if you like), even though a field IS an integral domain,
mathematically. If we COULD ask whether R is a quotient field (not just a field)
and if in case of affirmative answer also find out the underlying integral
domain S of R = FRAC S, we would be able to replace R by S for the package.

Hmm, but one can also achieve what you want by adding a function to the Fraction constructor

Fraction(R: IntegralDomain): with {
  fractionField: Boolean;
  underlyingDomain: IntegralDomain;
} == add {
  fractionField: Boolean == true;
  underlyingDomain: IntegralDomain == R;

If one then gets a field K, one just has to ask

if K has with {fractionField: Boolean} then {
   Groebner(underlyingDomain K, coercedPolys);
} else {
  Groebner(K, polys);


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