Shocking claims about modal arithmetic
Nate Hayes wrote
(in: Re-submission of motion 5: multiple-format arithmetic):
Arnold Neumaier wrote:
As our off-line discussion last year had revealed, not even you were
able to interpret the modal theorems in the literature correctly to
give always valid results.
Absolutely untrue.
It is unprofessional breach of confidence to drag elements of
personal and private discussions into a public forum like this.
Strange.
If what I said were indeed untrue, it could not have been a breach
of confidence, since then I hadn't revealed anything you said.
It would simply have been a false accusation.
But what I said is true, and you had contributed it proudly,
without any request for confidentiality.
You knew that I was investigating the possibilities and limitations of
modal arithmetic, and that I tried to understand your corresponding
claims in order to write a sound paper on nonstandard intervals.
And you made shockingly big claims!
Especially to make such a
cheap and selacious point. I hope others that have spoken to you in
confidence will take warning by how you conduct yourself here.
I hope that people telling me something in confidence via email will
mention that in the respective emails, so that I know that I should
not publicly use the information provided.
Since you publicly accused me of making absolutely untrue
statements about you, I feel entitled to defend myself publicly
with some excerpts from our email discussion, which show that your
accusation is not valid.
Discussing the modal paper in RC 2001, I wrote on 2008-11-17 10:29,
''there is no theorem telling that the result must contain the range.
It is by accident, so to say.''
You replied on 2008-11-17 15:35,
<begin quote>
This is one aspect of the modal theory I have not ever run into
troubles with, but yes, I see your point.
[...]
In very complicated equations optimality is not always possible.
In those cases, it usually requires the coercion to *-interpretability
theorems, where the equation is evaluated many times with duals on the
different variables, and then the disjunction is taken. This does
often lead to "shockingly" good results, even on quite complicated
equations.
<end quote>
Nothing confidential here. Just a very bold, unproved claim about the
power of modal intervals for range enclosures. You kept the
confidential part (the proof of your claims) to yourself,
in effect asking me to buy the pig in a poke.
You never provided confidential information - for example, till today
I don't know your claimed high quality solution to my simple interval
challenge, since you regarded that as confidential.
The remaining discussion was in the context of a clarification of a
point relating to my draft paper on nonstandard intervals, which can
hardly be considered to be subject to confidentiality, either.
Between us, however, you know the example I give above for fR(X)
shows your claim about me is false,
No. My claim is correct.
When stating ''not even you were able to interpret the modal theorems
in the literature correctly'', I did not refer to the above example,
but to your use of coercion to *-interpretability to get unsound
bounds on the range.
On 2008-12-11 07:37, you wrote:
<begin quote>
I realize you are running up against a deadline, so I finished my
comments for you as quick as I could.
I still, of course, would like to see you change your mind on this
topic. At the same time, I respect that you may continue to have your
own opinions, even if they don't agree with mine. At this point,
I'm pretty clear on your position. For this reason, it is not
neccessary to respond personally to any feedback I provide in the
attached PDF. Take the comments for what they're worth. If they change
your mind on anything, I'll be happy. If not, I'll appeal to the
committe in my own position paper.
Of course, if there does happen to be something new you wish to
discuss (or something old you wish to clarify), feel free to contact me.
<end quote>
The pdf attached by you contained as comment to page 14 of my draft
paper on nonstandard intervals referring to my statement
''Modal coercion theorems for obtaining more narrow enclosures of
ranges require total monotonicity''
(see p.21 middle of the current version at
http://www.mat.univie.ac.at/~neum/ms/nonstandard.pdf),
the statement:
''It is not true that the modal interpretability theorems require
total monotonicity. This is only true for coercion to optimality.
The coercion to * and ** interpretability do not require monotonicity
to exist. For example, it is possible to dualize instances of a
multi-incident variable for several evaluations of the function and
then take the conjunction or disjunction of all evaluations.
The result is often "shockingly" narrow.''
In the subsequent discussion, I tried to resolve this issue, important
for my paper on nonstandard intervals, to see who is mistaken, you or
I.
On 2008-12-15 16:21, you wrote:
<begin quote>
>>>>>> ''For example, it is possible to dualize instances of a
>>>>>> multi-incident variable for several evaluations of the function
>>>>>> and then take the conjunction or disjunction of all evaluations.
>>>>>> The result is often "shockingly" narrow.''
>>>>>
>>>>> Please provide references describing this technique and a proof of
>>>>> its validity. Your only reference on modal intervals so far was
>>>>> the RC2001 paper of Gardenes et al, who don't discuss this.<<
>>>>>
>>>> This is discussed in the RC2001 paper on pp. 94-95, i.e., it is in
>>>> the context of theorems 4.8 and 4.9. Note it is only the theorems
>>>> in chapter 5 that require monotonicity, but the chapter 4 theorems
>>>> do not.
>>>
>>> By Theorem 4.1, f^*(X) gives only range enclosure information if all
>>> components of X are proper, and in that case, Theorem 4.8 reduces to
>>> ordinary interval evaluation. And f^**(X) is only approximated from
>>> inside by Theorem 4.9, hence gives no range enclosure at all.
>> The coercion to *-interpretability requires multi-incident improper
>> intervals. This means that to begin with, all variables must be
>> improper. Therefore, to make things simple, begin by setting:
>> x1 = [3,2] // improper interval
>> x2 = [4,3] // improper interval
>> Then, by the coercion to *-interpretability:
>> fR1 = dual(x2)-x1*x2
>> fR2 = x2-x1*dual(x2)
>> fR = dual(fR1&fR2) = [-8,-3],
>> which in this case is the optimal enclosure.
>>
> Only by coincidence, since the expression is so simple.
> There is no theorem guaranteeing that one always gets a valid
> enclosure in this way.
>
>>>> I've attached the complete series of the SIAM publications,
>>>> which are the more detailed versions of the summary presented
>>>> in the RC2001 reference. In the "papersiam3.pdf" document,
>>>> pp. 3-4, the same theorems are explained in a little more detail.
>>> But this does not change the fact mentioned above that one cannot
>>> interpret these results als range enclosures. In example 2.3 of
>>> papersiam3.pdf, the range of x_2-x_1*x_2 over x_1 in xx_1 = [2,3],
>>> x_2 in xx_2 = [3,4] is (1-xx_1)*xx_2 = [-8,-3], and the shockingly
>>> narrow results obtained in the example are too narrow to be valid
>>> enclosures.
>>>
>> It computes in this case the optimal enclosure.
>
> In this case, yes.
>
> But when I apply your recipe to compute an enclosure for f(x)=x^2-x
> in [0,1] by your recipe above, I get the 'enclosure' [0,0], which is
> so shockingly narrow that one cannot believe it to be valid.
Interesting.
I checked your results and get the same answer.
The interpretability is correct, i.e., [0,0] is correct for the
semantics. But it obviously is not a range enclosure, either.
> Indeed, using theorem 4.5 and 4.8, it is clear that your recipe
> provides a valid enclosure _only_ if the result is optimal.
> In all other cases it computes the **-extension and hence an inner
> approximation only.
> This explains the shock on seeing these shockingly narrow results.
>
> Thus, in general, your recipe is invald and may give misleading
> results.
It simply comes from the reference, e.g., it is not "my" recipe.
In any case, it does appear if one is seeking a range enclosure,
the monotonicity analyais must be done instead. I believe you are
justified in saying that much.
<end quote>
Thus you agreed that your original bold claim was unsound, and
that modal arithmetic cannot give better results than a monotonicity
analysis would give.
I find it quite understandable that someone misunderstands the content
of the coercion theorems. Most present publications on modal intervals
are written in a somewhat fuzzy manner that makes it difficult to
understand what was really claimed and proved, and the early papers
on the subject even contained serious mathematical errors.
(To those who want to know more about modal intervals:
An easy-to-read critical introduction to the state of the art,
the history, and the main references can be found in my survey paper
Computer graphics, linear interpolation, and nonstandard intervals.
This complements the introduction by Nate Hayes
Introduction to Modal Intervals,
which presents the basics of the subject as seen through rose-colered
glasses. Both papers are available from the official P1788 site
IEEE Interval Standard Working Group - P1788
http://grouper.ieee.org/groups/1788/
together with a third paper by Vladik Kreinovich.)
The point of all this is therefore not to show that you were in error;
we all are at times; this is excusable.
The point is that it leads to the question:
If even experts who hold patents on modal intervals are prone to
such errors, what to expect from less educated users?
It means that it difficult for _anybody_ to understand modal
interval theory in a clear enough way that no spurius unfounded
claims result.
Thus the number of users that can safely handle modal arithmetic in
full generality may well be zero.
Therefore, it is wise if the standard does not commit itself to
modal arithmetic. It is far from being a mature subject matter.
Arnold Neumaier