Re: Discussion paper: what are the level 2 datums?
Dan Zuras Intervals wrote:
Date: Sat, 09 Oct 2010 12:41:42 +0200
From: Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>
3. Concerning the values of the other decoration trits, I think that
this is part of a general observation that not all combinations of trit
values make sense. Indeed, assuming the four trits
v=valid, d=-defined, c=continuous, b=bounded
(which are the indispensible ones) and the possible values
+ (True), - (False), and 0 (no claim),
only 10 combinations of trits are computationally relevant and should
be allowed:
v d c b | #cases
- 0 0 0 | 1
+ - 0 0 | 1
+ 0 0 0- | 2
+ + +0 +0- | 6
(In particular, v is never 0 and c is never -.)
I'll let people with more experience argue about
which cases make sense & which don't.
But this observation brings two things to mind.
First, if there are only a small number of valid
cases, can we not reduce the collection of
decorations to an enumeration type of those cases?
Say, 4 bits when all is said & done?
Probably.
But putting 4 decorations into 8 bits simplifies the propagation rules.
Otherwise we need for each binary operation a 16x16 table for
how the 4 bits propagate in that particular operation.
I don't expect decorated interval computations to be applied
in situations where storage is that scares that saving 4 bits
matter.
Maybe that's a level 3 observation but its worth
putting in an informative note.
And second, isn't the signum() function decorated
with ++-+ in any interval containing zero in its
interior?
By signum() I mean the common definition:
signum(x) = (x<0)?-1.0:(x>0)?1.0:0.0;
That is, a function which is valid, defined, &
bounded but not continuous at zero.
It is not continuous at 0, but it is not clear whether this should
suffice to make c=-. Indeed, it is continuous at x=1, say;
so there is an ambiguity in the interpretation for signum([-2,2]).
We require c=+ to mean that the function restricted to the interval
is there everywhere continuous, so shouldn't c=- mean everywhere
discontinuous? This gives the above interpretation.
Or should it mean somewhere discontinuous?
Even when we opt for this alternative, this information cannot be
exploited by an existence theorem. I know of no natural positive
consequence that can be drawn from the fact that a function is
discontinuous at some unknown place in the interval, which is
the additional info that c=- rather than c=0 would provide in
this case.
I think interval computations (and hence decoration properties)
must be safe, easy to understand, and preserve important information.
But as long as this is the case, we do not need to be maximally
pedantic.
Arnold Neumaier