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)