Re: Motion 11 questions & comments...
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Dear Dan and all,
Dan Zuras Intervals wrote:
> 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?
As I already pointed out on this list (January 24th), Corollary 1 is
false. The counter-example I gave was:
A=[0,2]
B=[-1,1]
C=[1,1]
with \circ=\times
and compute \times_1^-(B,C,A) = hull({x\in[0,2]\mid\exists b\in[-1,1],
x\times b\in[1,1]})
You should get [1, 2]
Now, compute \times_1^-(B,C) = hull({x\in R\mid\exists b\in[-1,1],
x\times b\in[1,1]})
You should get hull([-\infty,-1]\cup[1,+\infty]), that is [-\infty,
+\infty]
If you intersect the last result with A=[0, 2], you get [0, 2], which is
different from [1, 2].
Reverse division is useful to compute Newton steps. All the other
reverse operations are useful to implement narrowing operators in
constraint programming algorithms. I shamelessly refer you to the first
two sections of the following paper:
Interval Extensions of Multivalued Inverse Functions. F. Goualard, 2007
http://cinerep.free.fr/refbase/publications/goualard-gaol2007.pdf
for a motivation of the introduction of reverse operations, and to the
following paper for an algorithm that relies exclusively on them:
Revising hull and box consistency. Benhamou, F., Goualard, F.,
Granvilliers, L., & Puget, J. - F. 1999.
http://goualard.free.fr/publications/files/BenhamouGoualardGranvilliersPuget-ICLP99.pdf
FG.
- --
Frédéric Goualard LINA - UMR CNRS 6241
Tel.: +33 2 51 12 58 38 Univ. of Nantes - Ecole des Mines de Nantes
Fax.: +33 2 51 12 58 12 2, rue de la Houssinière - BP 92208
http://goualard.frederic.free.fr/ F-44322 NANTES CEDEX 3
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