Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications
> Date: Tue, 06 Jul 2010 17:29:03 +0200
> From: Marco Nehmeier <nehmeier@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
> To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>, stds-1788@xxxxxxxx
> Subject: Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus
>
> Am 30.06.2010 23:11, schrieb Dan Zuras Intervals:
> >> From: "Nate Hayes"<nh@xxxxxxxxxxxxxxxxx>
> >> To: "Dan Zuras Intervals"<intervals08@xxxxxxxxxxxxxx>,
> >> "Ralph Baker Kearfott"<rbk@xxxxxxxxxxxx>
> >> Cc: "P-1788"<stds-1788@xxxxxxxxxxxxxxxxx>,
> >> "Dan Zuras Intervals"<intervals08@xxxxxxxxxxxxxx>
> >> Subject: Re: A question Re: Level 1<---> level 2 mappings; arithmetic versus applications
> >> Date: Wed, 30 Jun 2010 15:05:39 -0500
> >>
> >> Dan Zuras wrote:
> >>
> >> . . .
> >>
> >> John has previously made the observation that there is an exact mapping from
> >> Level 2 mid-rad interval to Level 1 interval. Of course, once at Level 1
> >> there is then also an exact mapping from mid-rad to inf-sup (or vice-versa).
> >> So the only conversion that requires care is mapping back from Level 1 to
> >> Level 2. However, it seems there is some Level 1 mid-rad interval
> >> corresponding to some Level 2 mid-rad interval that is provably the tightest
> >> possible Level 2 enclosure, so long as that Level 2 enclosure is represented
> >> by a midpoint and a radius.
> >>
> >> . . .
> >>
> >> Nate
> >>
> >
> > Nate,
> >
> > I am going to pass on most of the content of your note
> > to focus on this one statement because the fact that
> > you state things in this way means I have not been
> > clear.
> >
> > Level 1 is the set of all possible contiguous subsets
> > of the extended Reals.
> >
> > Therefore there ARE NO mid-rad or inf-sups at level 1.
> > Representations have no meaning there.
> >
> > Level 2 is some finite subset of the intervals that exist
> > at level 1.
> >
> > What I am proposing is that the DEFINING characteristic
> > of that subset be that the bounds be exactly (some say,
> > losslessly) extractable as elements of some floating-point
> > type F.
> >
> > Therefore, there are no mid-rad or inf-sups at level 2
> > either. Representations have no more meaning here then
> > they do at level 1.
> >
> > All the formats live at lower levels.
> >
> > And I am proposing an approach that never speaks of them
> > directly while still knowing that they exist& taking
> > care that some agreeable behavior is possible for them.
> >
> > That's all.
> >
> >
> > Dan
>
>
> Dan,
>
> After having been quiet in the long discussion we want to congratulate
> you for clarifying the level structure.
>
> Your new definition of level 2 is completely analogue to the
> specification of an interface (abstract data type) in object orientated
> programming. Where the methods are specified by pre- and
> post-conditions. The internal representation is not yet determined.
>
> For example addition of two intervals A and B
> pre: A, B in IR
> post: (A + B).inf() == rnd_down(A.inf() + B.inf())
> (A + B).sup() == rnd_up(A.sup() + B.sup())
>
> Now implementation (level 3) with inf-sup-intervals is obvious. The
> mid-rad-guys have to proof if they can meet the specification.
>
>
> We will put in a motion that the standard will be written in this manner.
>
> Note that motion 5 already close to this definition.
>
> Best regards
>
> Jürgen & Marco
Gentlemen,
While I thank you for your praise, I fear much of it is
undeserved.
You see, my primary purpose in posting that note was to
criticise Nate for confusing representations with the
(otherwise unrepresented) Real intervals that live at
level 1.
That may have been overly critical of me. I have spoken
to Nate in a more private forum since then & it turns out
that he was merely discussing the Real analogues of
intervals that could be DESCRIBED in inf-sup or mid-rad
terms.
I have apologised to Nate in private & I apologise to him
in public now.
Still, I agree with your point that objects at level 2
should still remain an abstraction divorced from any
representation which may live at level 3. The clarity
of this position is muddied, however, when there is one
abstraction for inf-sup representations & another, quite
different, abstraction for mid-rad forms.
I made a half baked proposal that level 3 representations
be RESTRICTED to representing only those level 2 intervals
with endpoints in a set F of numbers that can be found in
some associated floating-point type. And I showed how
this might be done (primarily by slightly modifying mid-rad
results). But this proposal met with little support & none
from the mid-rad folks. So we are trying something else.
But it will likely force us to have one level 2 abstraction
for intervals that are represented at level 3 by inf-sup
forms. And another level 2 abstraction for intervals
represented at level 3 by mid-rad forms.
Thus, the distinction between levels 2 & 3 will be less
useful than might otherwise be the case.
I am not happy with it but it may be a necessary compromise.
Yours,
Dan