Re: Re-submission of motion 5: multiple-format arithmetic.
Nate Hayes schrieb:
John Pryce wrote:
Some issues left open by this motion:
- The existence and functionality of NaI. I think: the interval
functions
that are the subject of this motion should never give a NaI result
from
non-NaI arguments; they should always give NaI if any argument is
NaI;
only constructors (maybe other related functions) should create NaI
(when given nonsense input); some interval mappings may be allowed
to destroy NaI (e.g maybe hull(xx, NaI) should return xx).
But that's my personal view of what NaI "is for". I know other folk
have different views.
Yes.
I support your view of NaI, in general, but also believe users should be
able to obtain NaI as result of some interval operation that is not
well-defined.
- It is left open what exceptional action an implementation may make
available, on evaluating ff(ss) when ff is an extension of
a point function f, and ss is not a subset of the domain of f.
However, my approach implies such exceptional action must be _in
addition_ to returning an interval value (which might be ignored,
by the user or at the language level). Again, I suspect this is
controversial.
If the only way to obtain NaI for an invalid operation, such as division by
interval containing zero, is by traps or flags, this is what I object to.
This implication is still main reason I don't support this motion.
R4. Overall Aims.
-------------
Ramon Moore's Basic Theorem of Interval Arithmetic (BTIA) is
fundamental to interval computation. Roughly, it says that if E is an
explicit real expression defining a real function f(x_1, ..., x_n),
then evaluating E "in interval mode" over any interval inputs
(xx_1, ..., xx_n) is guaranteed to give an enclosure of the range of
f over those inputs.
Moore's theorem uses real analysis, so operations like division by interval
containing zero are undefined (NaI in modern parlance). This is different
than your motion, though.
I don't mind that the c-set extention you advocate in this motion is
included in 1788. However, the classical definitions should also be
included, in my opinion.
This has nothing to do with c-sets.
Moore's theroem remains valid with the proposed definition, since
a real expression defines a real function f(x_1, ..., x_n) only
on domains where no divisor is zero.
Therefore, correctly, for xx=[0,1]
{1/(x^2-x+1) | x in xx}
subseteq 1/(xx^2-xx+1) = 1/([0,1]-[0,1]+1) = 1/[0,2]=[1/2,inf],
while with your definition, division by an interval containing zero
gives NaN), and we'd get NaN, violating Moore's law since
{1/(x^2-x+1) | x in xx} subseteq NaI
does not hold.
Thus to hawe Moore's law for not everywhere defined expressions
_requires_ that NaI cannot arise as the result of an arithmetic
operation if the arguments are intervals.
The abstract objects and operations form interval level 2. To see its
utility, ask yourself the question "How to prove the BTIA, starting
from a procedural definition of the arithmetic operations as given,
for instance, in the Einarsson-Kulisch position paper presently (May
2009) under discussion as Motion 5?"
Moore defines division by zero as undefined result, i.e., NaI. But there is
not a definition in the motion that allows this result to be obtained...
and rightly so.
I believe nothing in this motion and rationale hinders the
implementation of various forms of non-standard intervals -- Kahan,
modal, etc. -- as discussed at the end of Vienna/1.2.
I've mentioned before this is simply not true. If traps or flags are only
way to obtain NaI result from an interval operation such as 1/[-2,3], this
is hinderance to efficient modal interval implementations.
This is another eason why modal intervals should not be part of
the standard. It makes the latter unnecessarily complicated,
only to introduce an error-prone technique that can be safely handled
only by a tiny minority of users.
Not deciding this issue (modal or not) very soon will constitute a
major quarrel in each issue to be decided.
As it stands, I'm not ready to vote "yes" to this motion. As I've mentioned
in the past, I believe a more balanced solution is to provide two sets of
operations:
op1: { x op y | x in X and y in Y }
op2: { x op y | x in X and y in Y and x op y is defined }
The first is Moore's classical definition and leads to NaI if x op y is not
well-defined (this is the definition compatible with modal intervals). The
second is c-set extention which returns {empty} for those values, instead
(this is what your motion provides).
This ultimately amounts to requiring implementors to fully implement two
interval arithmetics - one with a set of op1's suited for the tiny
minority of modal interval users, with a fully functional modlaa
interval arithmetic, and a second one with a set of op2's for the huge
minority of ordinary interval users who want best achievable enclosures
rather than noninformative NaI results.
This would be a very regrettable state of affairs since it puts mostly
unnecessary duties on the shoulders of _all_ implementors.
Arnold Neumaier
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