Re: Revised interval flavors motion (36.02)
On 2012-07-09 15:27:37 +0100, John Pryce wrote:
> The meanings of "same", and of "shared" in the motion text,are specified by
> the paragraph containing (1) and (2) below. For instance a modal interval
> X* and a set-based interval X** are the "same" if X* = m(X)and X** = s(X)
> for some common interval X.
Some spaces are missing: "text,are" and "m(X)and".
> There are "agreed" one-to-one embedding maps [I abbreviate s(x) to sx and m
> (x) to mx when context makes clear]
> s : C -> S }
> and } (1)
> m : C -> M }
> by which C can be regarded as a subset sC of S or as a subset mC of M. The
> crucial point is that for each "agreed" operation "op" we must have (I'm
> assuming a binary op, the general case is similar):
> s(X op Y) = sX op sY }
> and } (operation compatibility) (2)
> m(X op Y) = mX op mY }
> for all intervals X,Y in C such that sX op sY is in sC and/or mX op mY is
> in mC.
>
> Let an instance of an operation (that is, the specific "op" and specific
> arguments X,Y that are common intervals) be called "internal" in a given
> flavor if X op Y is a common interval. So basically (2) means that if an
> operation-instance is internal in one flavor, then it must be internal and
> give a compatible result in the other flavor. Otherwise there should be two
> separate operations.
> [Thanks to Vincent Lefevre 2012/07/05 for this wording, which I have edited
> slightly.]
This is a bit ambiguous. By definition, X op Y is always a common
interval if defined (in C). There are two ways of seeing things:
1. An operation is first defined on C, and you try to extend it to
several flavors. However this is a bit awkward, because the initial
definition may be simpler on a larger set (i.e. the flavor).
An analogy with the real and complex numbers: if you want to define
the power function x^y on the complex numbers, it is better to do
that directly rather than trying to define it on the real numbers
(in particular with negative x).
Trying to define functions by extensions often work, but not always.
I think it is easier to have a global view for each definition:
2. After choosing a flavor (F,f), you define an operation on it
(without taking care of the other possible flavors). Then someone
else can do the same thing with a similar operation on another
flavor, and one can check whether they agree or not (agreement
should not be a primary goal, otherwise you can end up with a
function that can be useless for both kinds of applications).
When you say "if X op Y is a common interval", it seems that this
doesn't depend on the flavor, while this is not true. So, I would
say:
Let an instance of an operation (that is, the specific "op" and
specific arguments arguments X,Y that are common intervals) be
called "internal" in a given flavor if fX op fY is an image of
a common interval.
Now, I would even call it "C-internal".
> The same must hold for any other flavor that might be added in future.
>
> In view of these maps it is immaterial what a common interval "is". If
> you're an M supporter you think it is a particular kind of modal interval.
> If you're an S supporter you think it is a particular kind of set-based
> interval. (2) says you shall get the same results either way. In that sense
> C is "shared" between all flavors.
>
> Cautions
> --------
> In the previous version, I said I suspected hidden traps in specifying
> operations on the common set C, namely about the set of interval inputs
> they operate on. I think the revised definition of (2) helps, but it
> doesn't entirely solve the problem.
>
> I asked, e.g., if "operation compatibility", as in (2), is to hold for
> A = [1,2]/[-1,1] ?
> B = sqrt([-1,1]) ?
> C = sign([0,7]) ?
> In the language introduced above, which of these operation instances are
> internal?
>
> Instance A can't be internal, because its set-based value cannot be a
> common interval (whether we take the exact range, or its hull).
Well, the "internal" (C-internal) notion might be more useful if
one considers the hull as part of the operation (not sure because
in practice, the hull is useful at Level 1 only for discontinuous
functions, and I don't know what Kaucher's arithmetic requires in
this case).
> B has the set-based value [0,1], a common interval. Here, the function is
> continuous wherever defined, but isn't defined everywhere on the input. I
> don't know the modal/Kaucher value, or if it has one.
Same remark about the modal/Kaucher value.
> C also has the set-based value [0,1], a common interval. Here, the
> function is defined everywhere, but isn't continuous everywhere on the
> input. Again, I don't know the modal/Kaucher value, or if it has one.
Ditto.
> Question to modal folk: Is it necessary to make the quite restricted
> definition:
> An operation instance is internal iff the operation is
> everywhere defined and continuous on its inputs
> or can modal theory go beyond that in some circumstances?
>
> Note. Since this is about "exceptional cases", it is also about how we make
> decorations interoperate with flavors.
[...]
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)