Re: Motion P1788.1/M004.01
On 2016-05-12 12:36:53 -0400, Lee Winter wrote:
> On Thu, May 12, 2016 at 12:28 PM, Vincent Lefevre <vincent@xxxxxxxxxx> wrote:
> > On 2016-05-10 13:17:40 -0400, Lee Winter wrote:
> >> On Tue, May 10, 2016 at 10:46 AM, Ulrich Kulisch <ulrich.kulisch@xxxxxxx> wrote:
> >>
> >> > I think interval arithmetic should not be defined over the IEEE
> >> > 754 binary64 numbers. This more or less pulls all the IEEE 754
> >> > exceptions into interval arithmetic. We shoud not bother all
> >> > users of interval arithmetic with constructs which really do
> >> > not occur and definitly are not needed in interval arithmetic.
> >> >
> >> > Interval arithmetic shoud just be defined as a calculus for
> >> > connected sets of real numbers.
> >>
> >> I disagree. We don't have native real numbers. We only have finite
> >> (small) sets of rational numbers.
> >
> > I don't think that Ulrich meant *all* connected sets of real numbers.
>
> The cardinality of the sets is a separate, and probably lesser, issue
> from the nature of the elements of the set.
The elements of the sets are not just rational numbers. For instance,
[1,2] is the set of all real numbers x such that 1 <= x <= 2.
> >> > They just serve as bounds for the description
> >> > of unbounded real intervals.
> >>
> >> No. There are no unbounded real intervals. Every single computable
> >> interval over rational FP numbers is bounded. We may lose track of
> >> the bound through representation limitations, but there is such a
> >> bound even if it has an unknown, or possibly unknowable, value.
> >
> > There are several possible definitions for "unbounded", e.g. the one
> > from the order theory or the one for metric spaces (which also depends
> > on the chosen metric).
>
> As I understand them, neither applies to intervals for computer
> software or computer hardware.
What matters is that interval arithmetic is based on a mathematical
theory.
> Consider an alternative description of unbounded in the sense that
> there are members of the set that are not enumerable (in the NLC
> sense).
No, this is never related to the enumerability. [1,2] is bounded
while its members are not enumerable.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <https://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)