Re: Computing infinity modulus and zero modulus

Thanks again for answering David, I do appreciate it a lot.

interval arithmetic sounds amazing to me, and I even envision it could
have some deep consequences on undecidability issues (I mean Goedel's
theorems and related) as well as P. vs NP. matters, etc. etc.

I, OTOH, cannot envision any such thing.

Well, I could mention many facts here and there, like Smullyan
starting on Goedel's theorems with the assumption that "by expression,
we mean any finite non-empty string" of the given symbols: so no oo
and no o/o. Do we really need these restrictions? Doesn't this sound
like a system that is not closed to all operations? (Actually, the
whole Cantor's diagonalization idea to me sounds very very suspect,
valid up to a certain point along the "proof", than no more valid, and
so we prove... do we really prove anything? This is actually tied to
deeper questions into the very philosophical issues at the base of
logical reasoning, like what is "true", whether the reductio ad
absurdum is an acceptable way of reasoning, and so on, you guys know
much better than me.)

On the other hand, Walster tells about global optimization made
possible by reversing the approach to problem solving: rather than
trying to find the optimal solution, we discard incorrect ones. And to
me this is bloody it! This is what we engineers are doing day by day
in our problem-solving real needs coming from real people. And, in a
very practical sense, it "works" and actually is the only thing that
works.

And I could go on and on with uncountable more "suggestions" I get
from apparently unrelated fields. You know, I don't care at all about
exposing myself to ridiculous, because I am actually too serious and
never cheating. But you guys, *you* should know what I am talking
about, even if I cannot tell it properly.

Anyway I promise I'll be back in a couple of months, or in a couple of
years maybe, or even in a couple of lives, doesn't matter really, I'll
just sooner or later come back when I'll be ready to talk the "proper
language". The fact is, maybe that day I'll have no more questions to
ask, and - with all respect - that day it will be your turn. ;)

Which theoretical community? Walster's work is well embraced, I think, by
the interval-arithmetic community.

Where is that? I mean, I have found a lot about interval arithmetic,
but nobody seems to care about "extended interval arithmetic" and how
that means a system _closed to any operation_, and indeed *that* is
not something I could find anywhere apart from the few publications
from Walster and few colleagues of him, and even these are very sparse
and, from the late 90's up to today, the only things I could find are
few patents registered by Walster on behalf of Sun on some specific
techniques, along with a couple of books available from Amazon (I have
just ordered the last one from Walster).

To me this doesn't look like "well embraced". Moreover, to speak about
an "interval community", to my naive understanding of "doing
research", sounds like isolated compartments, which simply cannot be
the case when you want to go deep to the foundations of disciplines.

A better question might be:
Why is interval arithmetic not of greater general interest?

Hmm, how "general" you mean? I think any technician like me would
simply love it at first sight if they only knew about it. It's the
"mathematicians" and only them that to me are in question here.

Indeed, while your remark is "correct", and I'd love an answer to the
rephrased question, I hope you won't mind if I dare paraphrase from
you this time: the distinction between "correct although of no use"
and "incorrect" may be a matter of conventions, yet it leads to
dramatically different outcomes.

Julio


On Feb 25, 10:49 pm, David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:
ju...@xxxxxxxxxxxxx wrote:
Hello back,

I have in the meantime made a first "naive" port to managed C#.Net of
the whole of ACM's Algorithm 737 (A737). This algorithm is also the
base for A763, which is its F90 port, and some more interval based
libraries like INTBIS, PROFIL/BIAS, etc. The only technical difficulty
is in properly defining machine related parameters (the default
provided set for 80x86 doesn't seem to work very well on my Vista 32
based PC), but I believe this can be easily fixed at any time...

However, I really hope you can enlighten me on a couple of things I
cannot seem to get:

-------

1) After implementing A737, I realize this is probably not what
Walster meant! (Me idiot, two weeks of work to just get what the code
does...)

Walster, in its "Interval Arithmetic Specification" (spec.ps, please
refer to the link Mr Cantrell has kindly provided above), says: "The
defined set of extended real intervals is closed with respect to
arithmetic operations and interval enclosures of real functions". But
in A737 I still see some unrecoverable error conditions for
"operations that cannot be assigned any meaning", namely for IIPOWR,
ILOG and ISQRT when the argument is zero or negative (depending on the
specific function), and for IACOS and IASIN when the argument is not
in [-1,+1]. For instance, this means I still cannot issue a 0**0 and
get a result.

-- Q: Could you please confirm that I have implemented the wrong
algorithm? :)

That would seem to be the case. (Note, however, that I am not familiar with
ACM's Algorithm 737.)

-------

2) The fact that extended interval arithmetic is closed to any
operation, and the fact that this leads to amazing results in the
realms of non-linear systems analysis and global optimization, makes
me wander why this thing has not become as widespread a new paradigm
as it sounds it should be, and as Walster himself - as far as I get it
- claims it is.

Walster says (as I get it from the manifesto.ps) this is mainly due to
the lack of built-in support for the extended interval type, which
results in inefficient and clumsy implementations. Still, shouldn't
extended interval arithmetic have been already embraced by the
theoretical community at least?

Which theoretical community? Walster's work is well embraced, I think, by
the interval-arithmetic community.

In other words, as a professional programmer and passionate about
computability and related matters, I can just say this extended
interval arithmetic sounds amazing to me, and I even envision it could
have some deep consequences on undecidability issues (I mean Goedel's
theorems and related) as well as P. vs NP. matters, etc. etc.

I, OTOH, cannot envision any such thing.

Then why at least mathematicians do not seem so interested in it?

-- Q: Why is it so?

A better question might be:
Why is interval arithmetic not of greater general interest?

But I'm not the one to answer that.

David



-------

As always, thank a lot in advance for your insights.

Julio

On Feb 12, 12:25=A0am, ju...@xxxxxxxxxxxxx wrote:
That's it... with Fortran 77 specification...

Bingo!!

:)

Julio

P.S. From this point on, the more I say, the more it starts to loose
sense, and I tend to believe this demonstrates that the question is
answered. I desire to thank you, David, for the patience as well as
the invaluable support. I desire to thank you, Mariano, too, because
in a sense you didn't answer, and in that I feel you may have made
this all possible. In a word, this has been a great learning and human
experience for me. I wish you the best.

On Feb 11, 11:51=A0pm, David W. Cantrell <DWCantr...@xxxxxxxxxxx>
wrote:

ju...@xxxxxxxxxxxxx wrote:
I suspect you mean computation beyond what is strictly algebraic

Yes, I guess so...

Walster does exactly that

That fact is, I couldn't manage to find any readily available paper
from Walster.

Hmm. I couldn't find the listing of his papers which used to be
availabl=
e,
so I used the Wayback Machine. Go to
<http://web.archive.org/web/20050410215323/http://www.mscs.mu.edu/~gl
o..=
.>.
As best I can tell, the links there work.

David

I'll try again, as well as re-reading the whole
discussion ground up. At the moment, to be true, I am lost at the
very=

concept of "number", and I even wander what I am after.

BTW, I have also been looking at interval arithmetic on WP, but I
couldn't get the pregnancy as it gets shortly presented as a tool
to manage error propagation. There is a broader treatment on the
German WP, but I can't read German. Anyway, going back to the
German article,=

I now notice it shows intervals such as [-oo, oo], and this makes
me very very confident...

I'll report here in case I make any progress.

Thanks again.

Julio

On Feb 11, 10:03=3DA0pm, David W. Cantrell <DWCantr...@xxxxxxxxxxx>
wr=
ote:
ju...@xxxxxxxxxxxxx wrote:
This is too interesting not to follow up by any means... ;)

you specifically said that you wanted to avoid NaN

Not really! "Undefined" was broader there, trying to define
NaN. T=
his
made Mr Cantrell comment, too.

This said, point taken: _|_ is NaN and floating-point is a
wheel (=
and
I'm depressed).

Yet, the question stands, just more and more perfected:

1. Is there an algebraic system where 0/0 is a number?

Of course, if you consider _|_ (or NaN) to be a "number", then
such =
a
system is a wheel. But as I said, I don't consider _|_ to be a
numbe=
r;
indeed, NaN stands for "Not a Number".

In one of my first posts in this thread, I mentioned a system in
whi=
ch
0/0 =3D3D 0. I didn't mention then, but it certainly is my
opinion, =
that
if =3D
0/0
is to be a real or complex number, 0 is the only reasonable
value. [FWIW, in J, a descendent of APL, 0/0 yields 0.]

2. If so, could you describe its interesting[*] properties in
basi=
c
terms?

Please look again at that earlier post of mine. I suspect that
you c=
an
derive many of the interesting properties/rules yourself.

3. And, could you provide a list of rules for computation?

Ditto.

But when you say "computation", I suspect you mean computation
beyon=
d
what=3D

is strictly algebraic. For example, you want the trig functions
to b=
e
defined always. But suppose we're using R*. There is no
reasonable w=
ay
IMO=3D

to define, say, sin(oo) as an element of R*. OTOH, there is a
reasonable way to define sin(oo) in an interval arithmetic:
sin(oo) =
=3D3D
[-1, 1]. And =3D
I
believe that Walster does exactly that.

David

Note: References always welcome.

Julio

On Feb 11, 8:28=3D3DA0pm, David W. Cantrell
<DWCantr...@xxxxxxxxxx=
t>
wrote=3D
:
ju...@xxxxxxxxxxxxx wrote:
Hi Mariano, nice to hear from you!

Someway, by digging the Wikipedia, the inconsistency you
have shown above led me to "wheel theory", and that looks
like what=
 I
was looking for:http://en.wikipedia.org/wiki/Wheel_theory

I doubt that that is what you're looking for; otherwise, I
would=

already have mentioned it to you. Why is it not what you're
look=
ing
for? Well, in your first post, you specifically said that you
wanted t=3D
o
avoid NaN. But standard floating-point arithmetic is
essentially=
 a
wheel, and the element=3D3D

notated as _|_ in wheel theory is essentially the same as NaN
in=

floating-point arithmetic. For example, 0/0 is _|_ in a wheel
an=
d
NaN in floating point. In neither is it reasonable, IMO, to
call=

0/0 a "number".

Not that I can already envision a way to implement this
extension. I=3D
n
fact, I could restart with my question over, despite in
slight=
ly
mor=3D
e
proper terms, and, for instance, still wander what cos(oo),

It would have to be _|_, I should think.

or oo %
2pi for the sake, could ever be, though now on wheels. But,
at=

least=3D
,
I am confident about the feasibility of the task.

As for the practical perspective, I suppose extending my
algeb=
ra
to wheels implies an extension of the set of meaningful
operations, and=3D
,
iif this extension leads to a "significantly" broader
applicability,=3D

then my struggle is justified. Of course, I can't yet
"prove" =
my
premises.

Moreover, the ideal situation from my perspective is that
some=
one
ha=3D
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