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Re: Motion 31 draft text V04.4, extra notes

Vincent Lefever wrote:

On 2012-04-11 11:43:46 -0500, Nate Hayes wrote:

Vincent Lefevre wrote:
>So, basically, you get something that is equivalent to an unbounded
>interval, but with a more complex definition!

Exactly.


I prefer a simple theory than a complex one.


The reasons for doing it this way are:
   -- unbounded intervals are then completely unnecessary at any level
   -- the absence of unbounded intervals at Level 1 means A = B
when A + X = B + X
   -- valid ranges for things like 1/[0,1]=[1,+OVR] can still be computed
at Level 2 (which is where for a computational standard like P1788 it
matters in the real world).


Similarly, real numbers are unnecessary, because one can use
sequence of rationals. The absence of irrational numbers means
that one can write each rational under the form p/q. And one
can still compute everything at Level 2 with rational numbers.
Now I wonder why one uses real numbers in practice. :)
If you wish to simply admit you have a strong personal preference and opinion for unbounded intervals in the Level 1 model, I can accept and respect that. But in my view you fall flat on your face when you try to prove they are *neccessary*. :)
My model is another option. Maybe you don't like it, but that's not a scientific argument it is invalid or doesn't exist. 





>>>>When restricted to bounded intervals, the Level 1 arithmetic is
>>>>closed and cancellative for addition, subtraction, multiplication
>>>>and division with 0 not in the denominator.
>>>
>>>This is not the definition of a closed arithmetic. If you can get
>>>an interval as a result, say [0,1], you mustn't remove it from the
>>>possible inputs of an operation.
>>
>>At Level 2 its not.
>
>Do you mean that you have an operation at Level 2 with no
>corresponding operation at Level 1? This doesn't make sense.
Sure it does.
It all depends on the underlying axioms and definitions.


A math problem is always expressed at Level 1, not at Level 2.
A computer program can never truly be expressed at Level 1, but it can at Level 2. 




>I've given an example in another mail.
Where?


The range of atan(1/x) over [0,1]. Could you give a complete proof
of the result with your axioms and definitions?
You've already agreed [1,+OVR] as an infinite family of intervals contains 1/[0,1]. 




>>All your points seem really academic to me.
>
>No, really, I need to compute ranges of functions (that can be defined
>by arbitrary expressions), and I need a correct answer.
So do I. I can compute them at Level 2 with 1/[0,1]=[1,+OVR].
I've never written a computer program that operates at Level 1. That's
nonsense.


for you only.


>>The arithmetic can be closed at Level 2, anyhow, with overflown
>>intervals.
>
>My point is that it must be closed at Level 1 too.
Why? I haven't seen you give any compelling reason for this.


Because it makes everything easier.

How? Please be specific!!


Do you have a proof of the FTIA
with your model?
No. But I know you will not help me do this because you have already made up your mind. So I will work on it with someone else. 




At Level 1, I see it is a choice betwee closure and cancellation.
IMO and experience, cancellation is the much more important property.


Silly. You're going nowhere with such claims: similarly I can say
"I've never used cancellation, so IMHO, closure is the much more
important property."

Development of Kaucher arithmetic *requires* cancellation property!!!



2. Cancellation is still *valid* with the P1788 model when you do
not use unbounded intervals (which is your case). So, I don't see
why you're complaining.
There is a story about a Native American Shaman staring out into the bay as the first European ships of Christopher Columbus sat anchored on the horizon. Even though many people in his tribe stared directly at the ships they did not see them, because it was not something they were familliar with. But the Shaman sat on the shore. He stared for days, then weeks. Eventually he convinced himself they were real. 




BTW, cancellation is invalid at Level 2.

I already pointed that out just the other day.

That makes a good reason
not to consider it.
There is no Kaucher arithmetic without cancellation property at Level 1. This is a fact. Perhaps that's not important to you. But if so, lets be clear about what we are really arguing about and stop all this nonsense. 

Nate
I'm still waiting for you or anyone else to show why they are *necessary*. 

See my example (again). And the range of tan over [0,10].
I want to use intervals, not useless complex things such as
familly of intervals.

--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)