Re: Motion P1788/M0032.01:MidpointMeaning -- discussion period begins
On 2012-03-15 08:21:23 -0700, Dan Zuras Intervals wrote:
> > AFAIK, "based on a some floating-point system F" has not been defined
> > for implicit interval types (see the discussion about associated /
> > compatible number formats). IMHO, we could just require that each
> > non-empty interval of T must contain an element of F.
>
> Fair point. What I had in mind for implicits was the
> floating-point type of the midpoint field, should that
> differ from the radius. Is that reasonable to you?
"Implicit type" doesn't necessarily mean mid-rad. I think the only
thing you need is that any non-empty interval of T must contain an
element of F.
> > > If, for any X & Y that live at level 2, we define intervals
> > > X1 & X2 to be the smallest representable intervals for which
> > > [inf_F(X),mid_F(X)] \subset X1 & [mid_F(X),sup_F(X)] \subset
> > > X2 we have
> >
> > Such X1 and X2 do not necessarily exist for implicit interval types.
>
> Such intervals always exist for implicits.
>
> What is not necessarily true is that they
> are unique.
>
> I didn't say anything about unique.
"The smallest" is unique by definition. See the difference between
the notion of "greatest element" and the notion of "maximal element":
http://en.wikipedia.org/wiki/Partial_order (there is the same
difference in French, i.e. with similar wording).
Perhaps you meant "minimal".
> > You should remove "smallest" (I don't think this is necessary: if X1
> > and/or X2 are enlarged, the properties below will remain true). So,
> > the only requirements on X1 and X2 should be:
> > [inf_F(X),mid_F(X)] \subset X1
> > [mid_F(X),sup_F(X)] \subset X2
> >
> > In particular, these requirements are valid for:
> > X1 = hull_T([inf_F(X),mid_F(X)])
> > X2 = hull_T([mid_F(X),sup_F(X)])
> > if a hull_T function is defined.
>
> Also a fair point. Is hull any more unique
> among the implicits than the "smallest"
> superset in question?
The proposition was that the hull was unique (thus arbitrary for
implicit types). I'm not sure whether this is useful or a good
idea, though.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
100% accessible validated (X)HTML - Blog: <http://www.vinc17.net/blog/>
Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)