Re: [Axiom-developer] aldor stream oddities

[Ralf Hemmecke]

On 08/11/2007 08:55 PM, Franz Lehner wrote:
> > Now, in Axiom we should have a problem with saying that S is an 
> > additive group.
> That's true. All axioms are satisfied, except uniqueness of inverses and 
> the neutral element. What should it be? A FuzzyAbelianGroup?

Well, the problem is that we don't deal with classical mathematics, we 
deal with constructive mathematics. So let's assume that

  T := UnivariateTaylorSeries(Integer,x,0)

is a Ring. Then I can form F := Fraction T.

Then F is a field. And I can form polynomials that take F as a 
coefficient domain. Now I can take several such polynomials in Axiom and 
compute a Gröbner basis for them. You will be surprised that this is not 
what you would expect. Testing whether a coefficient is zero is used 
quite a lot in Buchberger's algorithm. The algorithm is implemented in a 
generic form and works over any coefficient field. Unfortunately, the 
algorithm cannot tell you that it computed the wrong result since you 
promised to give a field as input but in fact, gave something for which

> > (4) -> zero?(t-t)
> >    (4)  false
> >            Type: Boolean

holds. So how do you ensure correctness of results then? Would you like 
to put all the burden onto the user?

We must be very careful here. And I only addressed that problem. The 
library should be very carefully designed. At the moment it contains a 
lot of flaws.

> > The category Ring is reserved for structures with complete equality 
> > testing.
> Is this the plan for Axiom too?

I don't know. I am not the only person who decides this. But I would 
vote for yes.

> What will happen to ExpressionField? I recall there are undecidable 
> problems in the latter as well.

All I say, is that the library must be rethought.

> Here is a proposal (in SPAD) for a recursive stream generator which 
> looks at all previous elements, not just the last one. It could probably 
> be done with the existing generate(S->S,S), but this one looks more 
> natural.

You should take a look at the definition of Stream in LibAldor or also at
http://www.risc.uni-linz.ac.at/people/hemmecke/AldorCombinat/combinatse12.html#x58-9900012

Ralf