RE: Comments on decoration ill, intersection and union
Vincent Lefevre wrote:
> On 2012-12-14 08:04:56 -0600, Nathan T. Hayes wrote:
> > Vincent Lefevre wrote:
> > > On 2012-12-13 16:29:52 -0600, Nathan T. Hayes wrote:
> > > > If we evaluate the range of the intersection of X and Y, then
> > > > g([0,1] intersect [2,3])
> > > > = g(([0,1],DAC) intersect ([2,3],DAC))
> > > > = g((Empty,DAC)) // Empty input!!!
> > > > = (Empty,DAC)
> > >
> > > But the initial input is [0,1], which is not empty ([2,3] can be
> > > seen as a constant or another input, depending on the context).
> > > One gets Empty only after the intersect operation, and since this
> > > is after an operation, Empty cannot be an input.
> >
> > Precisely. This is why the (Empty,DAC) result is not meaningless: it may
> > imply the input was nonempty even though the result is empty.
>
> But what does it mean exactly?
It means: for all arithmetic operations in the lengthy computation that
occurred, the restriction of each arithmetic operation to the domain of its
respective input operands was defined and continuous, but the result is
still empty.
Therefore, (Empty,DAC) may occur for a couple reasons:
-- the input is nonempty, but the result is empty because of an
intersection operation (as in the case of this particular example).
-- the input is empty but there is at least one nonempty element of
the empty input box; or a nonempty constant is encountered in the
computation (similar to what you mention above).
When the interval portion of the result is nonempty, one can safely conclude
by the DAC decoration the input was nonempty.
Nate