RE: Comments on decoration ill, intersection and union
Dmitry Nadezhin wrote:
> 1) The intersect and convexHull return bare intervals in Motion 42.
> Do you mean intersectDec and convexHullDec in your email ?
Yes.
> 2) Do you agree that set-based and Kaucher/modal flavours may have distinct set of
> decorations ?
I agree P1788 *could* decide to give them distinct set of decorations; but I've never seen any mathematical reason to do this. Modal intervals are the unique extension of Moore's Fundamental Theorem of Interval Arithmetic to the set of Kaucher intervals. Moore's classical interval arithmetic may therefore be viewed as the special case of modal interval arithmetic when all inputs are proper intervals [a,b] such that a <= b. For this reason, I don’t see the decoration systems need to be different... in fact they shouldn't be. IMO, having different decoration systems would only cause unnecessary incompatibility between these two flavors of interval arithmetic that should otherwise be very compatible with each-other.
>
> 3) Do you suggest to add EIN to set-based or to Kaucher/modal flavour ?
I suggest the decoration system with EIN is the only decoration system needed for both the set-based and Kaucher/modal flavors.
> Do you want that I make implementation of Kaucher flavour (including Kaucher/modal
> decorations) more complete ? However, I need to ask you a lot of questions in this
> case.
You can contact me offline, if you wish. I'd be happy to help.
Sincerely,
Nate
>
> -Dima
>
> ----- Исходное сообщение -----
> От: nh@xxxxxxxxxxxxxxxxx
> Кому: j.d.pryce@xxxxxxxxxx, wolff@xxxxxxxxxxxxxxxxxxxxxxxxxxx
> Копия: stds-1788@xxxxxxxxxxxxxxxxx
> Отправленные: Пятница, 28 Декабрь 2012 г 18:13:58 GMT +04:00 Абу-Даби, Маскат
> Тема: RE: Comments on decoration ill, intersection and union
>
> John Pryce wrote:
> > So it seems "max(dx,dy)" for intersectionDec should be "min(dx,dy)"; and
> > "min(dx,dy)" for convexHullDec should be "the tightest decoration
> containing dx and dy
> > in the containment order". (I assume by "weakest" you mean the min,
> because you
> > say it is like an arithmetic operation.)
>
> Where do you see "weakest"? It says "the tightest decoration" not "the
> weakest decoration". This is not the same thing as the min.
>
> Anyhow, applying these rules to Motion 42 still leads to problems:
>
> In the Example 2 in section 8.8.8 for X=[-1,1], it gives:
>
> U = f1(intersect([-1,1],[-oo,-2]))
> = f1((Empty,dac)) // (Empty,dac) is contradictory by section
> 8.8.4
> = (Empty,emp)
> V = (f1([-1,1]),dac)
> W = f1(intersect([-1,1],[2,+oo]))
> = f1((Empty,dac)) // (Empty,dac) is contradictory by section
> 8.8.4
> = (Empty,emp)
>
> So there are still the intermediate results (Empty,dac), which by Motion 42
> is a contradiction.
>
> Taking convexHull of U, V and W and propagating the "tightest" decoration by
> the containment order gives:
> (f1([-1,1]),trv)
>
> This at least is not incorrect (since trv decoration is always true), but it
> is not useful or informative.
>
> This demonstrates my point further: to remove the contradictions specified
> in section 8.8.4, and also to compute the decoration that is both correct
> and most informative, one needs to define an ein decoration as I have done
> in the previous e-mails in this discussion. Without these changes, Motion 42
> simply does not define the correct decoration system to make decorated set
> operations useful, informative or free of contradictions.
>
> Nate