Re: Motion 11 questions & comments...
> From: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
> To: "P1788" <stds-1788@xxxxxxxxxxxxxxxxx>
> Subject: Re: Motion 11 questions & comments...
> Date: Mon, 8 Feb 2010 12:48:20 -0600
Ah, thank you Nate. I see I was confused about the
nature of reverse operations altogether.
Then, Marco, I am also confused as to the purpose
of these operations.
If they are formally equivalent to the existing
forward operations, why not use those instead?
Well, those together with the intersection
operation in an expression?
If they do not narrow the resulting interval like
the Kaucher/modal operations of which Nate speaks,
how are they 'reverse' in any meaningful sense?
In short, what problems can we solve with these
operations that we cannot solve as well already?
I see in your new document that you have chosen to
define things on the power set of the Reals. While
this may be formally correct for what you have in
mind, I have to say that is it not a very useful
domain of discourse for things to be defined in
the world of our oh so very finite computers.
Further, if these are to be some sort of non-modal
substitute for modal operations, how are we to
generalize them to other operations? Are we to
expect reverse sin, for solving the equation sin(X)
= S, to return infinitely many intervals separated
by 2pi that each have the property that their sines
fall into S?
This business of returning a pair of intervals does
not generalize well beyond the simple pole in divide.
Neither does it play well in expressions.
I think my confusion has graduated to wariness & doubt.
Can you allay my fears by explaining further?
Dan
>
> Dan Zuras Intervals wrote:
> > I am confused about the definitions of reverse operations
> > in sections 3 & 4. But this may be due to my confusion
> > about reverse operations in general.
> >
> > . . .
> >
> > But in reverse operations shouldn't the widths get narrower
> > not wider? Indeed, is that not the whole point of such
> > operations?
> >
> > . . .
> >
> > It seems to me I had this discussion with John & Nate
> > about a year ago & Nate's solution involved using both
> > 'there exists' & 'for all' quantifiers.
> >
> > I may have this wrong. Perhaps Nate can help here.
> >
>
> Dan,
>
> My understanding is that reverse mode is different than the
> Kaucher/modal operations, and perhaps this is the point to
> be clarified. For example, reverse mode does not supply the
> additive or multiplicative inverse elements you are speaking
> of. Those are supplied by Kaucher/modal operations (of which
> there has been no formal motions for, although they have been
> discussed at times in this forum).
>
> If you are trying to solve the equation A + B = C, the unique
> algebraic solution is obatined by the Kaucher arithmetic,
> i.e., B = C - Dual(A). In other words, A - Dual(A) = [0,0]
> is an identity. Similarly, the equation A * B = C has unique
> algebraic solution B = C / Dual(A), so long as 0 is not an
> element of A, i.e., A / Dual(A) = [1,1] is an identity.
>
> The system (IR,+) over closed, bounded and non-empty intervals
> IR is a commutative monoid, hence as you notice there are no
> inverse elements. Since (IR,+) also has the cancellative
> property, it is possible to embed the monoid into a group.
> This is where the inverse elements of Kaucher arithmetic
> come from, as well as the notion of improper intervals
> [b,a] such that a <= b.
>
> Strictly speaking, the "for all" or "there exists" quantifiers
> of modal intervals are not necessary. Only the Kaucher
> arithmetic (and improper intervals) are required. However,
> it is all a separate topic from the motion Marco currently
> is putting forward.
>
> Hope this helps.
>
> Nate