RE: Promotion of bare decorations & comparisons
John Pryce wrote:
> On 3 Jan 2013, at 22:08, Nathan T. Hayes wrote:
> >> You can construct an equally silly example with that alternative:
> >> [1,2] \subseteq floor([0,6])
> >> ...
> >> = [1,2] \subseteq Empty
> >> = false
> >>
> >> Now we have a false negative.
> >
> > I don't agree it is a false negative.
> >
> > Compressing the decorated interval ([0,6],def) into the bare decoration
def
> > means the user has explicitly indicated anything less than dac is an
error.
> >
> > So returning false in this example is exactly what the user expects,
since
> > [1,2] cannot be a subset of any defined and continuous interval range of
> > floor([0,6]).
>
> I think "Hmm" on this one. What does "any defined and continuous (dac)
interval range
> of floor([0,6])" mean? The only meaning I can see is "since floor() isn't
dac on the
> input [0,6], such a range doesn't exist; and if we insist on treating this
nonexistent
> thing as a set, it must be the empty set".
That is exactly my view... so why the "Hmmm"?
>
> I think the above is for compressed arithmetic with threshold dac. Change
the example
> slightly:
> [0,0] \subseteq floor([0,0.9])
> Here, floor() IS dac on [0,0.9], so the result as a decorated interval
[0,0]_dac, which
> becomes [0,0] as a compressed interval, so we get
> [0,0] \subseteq [0,0]
> = true.
> But now change [0,0.9] to the large interval [0,6] and according to Nate's
scheme
> above
> [0,0] \subseteq floor([0,6])
> ...
> = [0,0] \subseteq Empty
> = false
> I don't think users would expect this: that (A \subseteq f(B)) is true for
some B, but
> becomes false when they make B larger.
???
If, using compressed arithmetic:
-- the user explicitly indicates anything less than dac is an error
(by setting the threshold), and
-- as you agree above "floor() isn't dac on the input [0,6], such a
range doesn't exist"
then please explain why the user would expect
[0,0] \subseteq floor([0,6])
to be true.
That doesn't make any sense to me.
>
> Michel is right: we should emphasise that X = (a compressed interval whose
value is a
> decoration d) is a quite separate object from D = (a decoration whose
value is d).
> Nate, if you want arithmetic on decorations, as Motion 8 said, let them be
X's, not D's.
> In which case they must follow the worst-case scenario rules for
promotion, that
> Arnold stated.
In my view, a bare decoration D is a compressed decorated Empty set of the
form:
(Empty,D)
The Empty set is a set (we both agree about that).
The Empty set is not an interval, i.e., it is not an element of overline-IR
Nate