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Re: Intersection example (was Re: Motion 26: NO)

Arnold Neumaier wrote:

On 07/28/2011 10:30 PM, Nate Hayes wrote:

Arnold Neumaier wrote:

-- Not giving any specification for how intersection and union handle
decorations means users will invent their own (probably wrong) rules,
i.e.,
what is the standard? Examples were given that show how incorrect
handling
of decorations with intersection or union can lead to catastrophic
failures,
so this is not something that should be left up to users.


I had given an example where Motion 27 gives erroneous results for
decorated intersections:

The expression f(x)= x/((x+1) intersect x^2) is undefined for any x in
[1,3], but Definition 7 claims a safe answer for f([1,3]).


Arnold, this is not any example of erroneous results.

First of all:

( [1,3] + 1 ) intersect [1,3]^2
= [2,4] intersect [1,9]
= [2,4]

and [1,3] / [2,4] = [1/4,3/2]. The function is not undefined anywhere on
[1,3].


Your argument proves nothing. For example, the expression is undefined for
x=1, for x=2, and for x=3, although all these are in [1,3].

The fact that no exception occurred in your calculation shows that the
calculation only says something about this particular interval, not (like
everywhere else in range computations) about the possible values of
elements from that interval.

But to be safe for the application of mathematical theorems, they must
provide information about the functions evaluated, not just about the
history of possible exceptions.



u(1) = (1+1) = 2
v(1) = 1^2 = 1

f(1) = 1/(u(1) intersect v(1))
   = 1/(2 intersect 1)
   = 1/empty
   = empty

The set-theoretic intersection of u(1) and v(1) is empty and is not
undefined (it is safe).

f(1) =1/empty is not undefined, it is safe because the input is empty.

So why is f(1) undefined?






As I've noted several times before, giving another semantics for the
intersection operation leads to catastophic failures in certain
algorithms,
like those depicted in the PDF attached to:
http://grouper.ieee.org/groups/1788/email/msg03570.html


I consider this a misuse of the intersection operation, which happens to
work in this particular case although there is no coherent semantics
behind it.


The coherent semantics are: the set-theoretic intersection operation is safe
if its operands are safe. By this semantic it is a valid application and not
a misuse.

It is only misuse if one chooses to give some other definition/semantic for
the intersection operation. But you and John are not proposing any such
alternative.





The mathematically correct way of treating your example (with
identical results) was given by John Pryce some time ago.


IMO all John did was sidestep the issue by using addition.

Nate