Why saf/saf=con (D4/D4=D2)
Nate Hayes wrote: [in: Re: DRAFT position paper]
Arnold Neumaier wrote:
Jürgen Wolff von Gudenberg wrote:
I suggest to use D_0 (Nate) or ILL (Arnold) to only indicate really
illformed intervals
I agree with the above ...
And I still wait for someone to produce a shred of evidence that this
provides any additional benefit...
If one evaluates a function with bare input decorations D4 for every
input argument and gets D4 as the final result, it is known in my
semantics that the function is defined and continuous for all real
inputs. Thus if one performs initially a _single_ bare decoration
calculation and this results in D4, one can safely do subsequent
bare interval computations, and still knows that the results are safe
(i.e., decorated D4) whenever the input is nonempty and bounded.
Such a property may be used to optimize certain computations in
cases where the decorations are handled by software and hence slowly.
With your semantics, one cannot conclude anything from the result D4,
since it is always obtained from D4 inputs, no matter what the
function is.
I further suggest to change formula (33) to:
D_4/D_4 = D_2
because the division provides the decoration D_2 (somewhere undefined)
This is indeed what my decoration semantics does, starting from a
conceptual semantics that is independent of propagation.
...and why I don't trust it:
the division operation also provides decorations D_0, D_1, D_3 and D_4.
But if decorations are attributes of an operation evaluated over an
interval domain, then D_4/D_4 produces no decoration since there is no
interval operand and hence no evaluation of the division operation.
Indeed, this is the difference in our semantics: You make decorations a
property of single operations, while I make them a property of functions
evaluated on boxes.
Typically, the information about an application is in the functions
programmed, not in the individual operations, hence it is important that
the result decorations can be interpreted in terms of the function,
rather than in terms of how it happened to be programmed.
My choice ensures that the final decoration is not better than what one
would get if the interval had not be dropped, and hence says something
valuable about the function evaluation -- information that is lost in
your approach but is important for optimal performance in cases like
those mentioned above.
Nate Hayes wrote:
> Michel Hack wrote:
>> Arnold's scheme lumps certain conditions together that
>> Nates' distinguishes -- but I doubt ALL distinctions are made, so in
>> general operands of suspicious operations have to be inspected anyway,
>> at which point full discrimination is possible.
>>
>> So if Arnold's scheme has other advantages, they should not be discarded
>> because of this quibble between undefined, ill-defined and empty.
>
> Well, we don't see there are other advantages.
Those who care to look for them, see.
I am interested in making the standard maximally useful for the many
applications I know of, not in providing functionality that follows
some theoretical principle but hardly ever is useful.
Arnold Neumaier