Re: Proposal for a new structure of the standard
On 2010-07-18 17:02:02 -0700, Dan Zuras Intervals wrote:
> Vincent,
>
> I will pass on a detailed response to all of your points
> & concentrate on this one. For it seems that the crux
> of our misunderstanding lies here.
>
> You are thinking of the whole of GMP as the basis for a
> single interval datatype along the lines of MPFI.
Yes. Note that's what's some implementations provide in practice
(e.g. mpz_t for GMP).
> I am thinking of these as a countably infinite set (in your
> resource limited sense of the word 'infinite') of possible
> interval datatypes indexed by base, precision, & range.
> Each one of which is finite in both range & precision.
At the end, your datatype represents a finite set.
> This is in analogy to the countably infinite class of
> floating-point datatypes we defined in 754-2008 indexed
> by b, p, & emax.
>
> I think it makes more sense to have the set that lives
> at level 2 be a single instantiation of a GMP object as
> an interval type indexed by base, precision & range as
> well as style (explicit/implicit).
In case of GMP, there is no notion of range (mpz_t is just seen as
a pointer or a structure for the user, and all the memory management
is done by GMP, transparently). Of course, you could say that in
practice, you'll have a (known or unknown) bound on the maximum
value amongst the values you'll get, but in such a case, this bound
will be above the implementation limits (e.g. from the available
memory). But one should not do this, as this can raise problems (for
instance, consider your example below, where the maximum precision
is something like 2^64 - 1).
> That way when we want to project the solution to the Real
> relation x^2 < 2 onto level 2 we have a definite object
> to land on. Without that the mapping is ill defined.
Does it make sense to want something at Level 2, that doesn't
exist at Level 1 (except the cases for which it is difficult to
do otherwise, such that everything related to the exceptions)?
If some user wants this on x^2 < 2, perhaps he would want the
solution to expr(x) < 2, assuming expr(x) corresponds to an
increasing function, but in that case, one would need to round
expr(x) correctly. This is a rather high-level problem, that we
may not want to consider here. Also note that such solutions can
be obtained only with correct rounding, and concerning interval
arithmetic, this would require the tightest mode (and that would
make sense only when the set is discrete).
So, I don't see this as a problem:
* If one wants FTIA only ("valid" mode), the set can be infinite.
* If one wants more ("tightest" mode, which implies reproducibility,
uniqueness, properties needed for your example above...), the
set is required to be discrete (possibly finite).
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / Arénaire project (LIP, ENS-Lyon)