Nate & all
On 25 May 2011, at 21:31, Nate Hayes wrote:
I do admit it remains a mystery to me why, having aknowledged all of the
above, you persist in advocating the v3.01 definitions; At the minimum,
this
will clearly require some disclaimer and/or theoretical excuse to explain
why IEEE 1788 algorithms may experience failure under certain conditions.
So yes, I am greatly confused and troubled by your position.
In the context of your intersection.pdf example, my position is merely
that using the *point* function
h(x) = f(x) + g(x)
is OBVIOUSLY correct, on any reasonable decoration scheme, because h(x) is
defined at some point if and only if both f(x) and g(x) are defined there.
While changing from a point function to an *interval* function
h(X) = f(X) intersect g(X)
needs machinery to MAKE it work, i.e. a particular rule for decorating an
intersection. And all in a context where the value of h(X) has no physical
meaning.
So I'm also "confused and troubled by your position". I continue to think
we really agree but since I'm having a hard time persuading myself of this
I don't know how in h*** I'm going to persuade you of it. On the other
hand I agree with you 100%
in the min & max discussion in your recent posting
(i) Intersection of objects
---------------------------
> We are currently examining the application of decorations to
> Constructive Solid Geometry (CSG):
HOWEVER the following may help us converge.
I have been thinking about
- Dominique's and your proposal to put "X is empty" at the top of the
hierarchy (D4 in your May 20th draft).
- the "problem to which I see NO satisfactory solution" I wrote about on
April 28, extract of email copied below.
I see that I was talking through my hat in the latter. Making "X is empty"
and "X is disjoint from Dom f" into two separate conditions seems to solve
my problem. And means that inclusion isotonicity holds for decorated
intervals, as it should do.
Following discussion with Arnold we propose: do as you and Dominique say,
and put "X is empty" at the top of the hierarchy. We call this decoration
"ein" for "empty input".
Now, to preserve the "partial order by containment" that is essential to
Arnold's scheme, it seems "X is nonempty" clauses have to be switched from
the "higher" decorations to the "lower" ones. That is, in Arnold's
"dec.txt" paper one changes the definitions
p_saf(f,X): X is a nonempty subset of Dom f, and the restriction
of f to X is continuous and bounded;
p_def(f,X): X is a nonempty subset of Dom f;
p_con(f,X): always true;
p_emp(f,X): X is disjoint from Dom f;
p_ill(f,X): Dom f is empty.
to
p_ein(f,X): X is empty
p_saf(f,X): X is a subset of Dom f, and the restriction
of f to X is continuous and bounded;
p_def(f,X): X is a subset of Dom f;
p_con(f,X): always true;
p_emp(f,X): X is nonempty and disjoint from Dom f;
p_ill(f,X): X is nonempty and Dom f is empty.
I got the following relations to Nate's D0 to D4 of (see e.g. his May 20
Property Tracking paper). The boolean triplets are his (D+,D-,C), and a
value "-" means don't care.
ein = D4 = FFT
saf & !ein = D3 = TFT (and conversely D3 = (saf|ein), and
similarly with the others below)
def & !saf = D2 = TFF
con & !def & !emp = D1 = TTF (implies X and Dom f are both nonempty)
emp = D0 = FT-
ill = D0 & (Dom f is empty), so not covered by Nate's
scheme.
If this is correct, it preserve the "partial order by containment" and
converges closely to Nate's scheme.
Arnold had a criticism, and I haven't had time to let him check this over
before sending, so it may yet be blown out of the water.
Must dash ... appointment for dinner with President Obama. Just kidding,
but I have been honoured with an invitation to dinner at high table in St
Katherine's College Oxford this evening, after a lecture by a friend, so
must go.
Regards
John Pryce
On 28 Apr 2011, at 15:13, John Pryce wrote:
(B)
Now a problem to which I see NO satisfactory solution. I think it's
relevant to the recent discussion whether "emp" should be put at the top
of the "quality order" of decorations, above "saf", instead of near the
bottom as at present.
On 19 Apr 2011, at 17:07, Dominique Lohez wrote:
I have tried to formulate a alternative approach... I came back to some
more fundamental points
...
In my understanding of the Nate Hayes' work the the calculation of the F
= f(X)
where F and X are decorated interval ...
According to John Pryce and Arnold Neumaier, the fundamental theorem
of decorated interval arithmetic must be satisfied. Thus the following
property must hold
if bareX \subseteq bareY then f(bareX) <=_DI f(bareY) (*)
Where <=_DI is some order relation between decorated intervals.
Relation (*) is "isotonicity". It is important, but quite different from
the FTDIA, which says
(the exact decorated range of f over bareX) <=_DI f(bareX) (**)
For *bare* intervals, both are true, where <=_DI is replaced by
\subseteq. That is, for bare intervals X,Y
(the exact range of f over X) \subseteq f(X)
and
if X \subseteq Y then f(X) subseteq f(Y).
Unfortunately I think they CAN'T both be true for decorated intervals.
Consider "the decoration of f over X", which Nate (Proposition 4) calls
S(f,X),
and I (eqn (9) in 4.8.3) call
dec(f,X).
I believe they are essentially the same, apart from D0 not being the same
as "ill" as noted above.
There are two "progressions" involved. Let G be the Evaluation Graph of a
function f for a given input box X.
- As one progresses along the nodes v of G, on any path from inputs to
outputs,
the decoration dec(g_v,X) of the sub-function g_v represented by the
current
node v can only decrease, in the quality order
saf>def>con>emp>ill.
That fact is crucial in proving the FTDIA.
- As X itself starts at Empty and "gradually increases", dec(f,X) can
only decrease
in a *different* order
emp>saf>def>con.
(Assuming f is a well-formed function, ill doesn't come into the
picture.)
Namely, suppose dec(f,X)=saf, so f is defined, continuous and bounded on
the
nonempty interval X. Then f has the same properties when restricted to
any smaller interval X', as long as X' is also nonempty. That is,
(X' \subseteq X) and (dec(f,X)=saf) ==> dec(f,X') is either emp or
saf.
Similarly
(X' \subseteq X) and (dec(f,X)=def) ==> dec(f,X') is either emp, saf
or def.
And
(X' \subseteq X) and (dec(f,X)=con) ==> dec(f,X') is either emp, saf,
def or con.
Since these two orders are incompatible, it seems to me it is impossible
to construct a <=_DI, in Dominique's notation, that gives useful
information and for which BOTH isotonicity and the FTDIA are valid.
That's a bummer.
Can anyone square this circle?
Regards
John