Bill and P1788
I have spent the weekend dealing with important family matters
(including my teenage daughter being assaulted as she walked home
from school on Friday), so I come into this debate when it is well
under way. I'll respond to the early parts of the discussion.
In case Bill's misgivings are partly due to the current revision of
Clauses 6 and 7 I say the following.
§6.
The Motion 52 revision of §6 is not due to any theoretical weaknesses
in the standard. 1788 makes requirements at the level of individual
operations. We agreed this early on -- I seem to recall Baker was
particularly strong on this point. So 1788 does not require any
particular interpretation of expressions, but it points out that *if*
a certain interpretation is used *then* a (flavor-dependent)
Fundamental Theorem, FTIA, about enclosures and decorations holds.
Manipulation of expressions -- whether x-x can be replaced by 0, etc.
-- is a language issue IMO, as well as being flavor-dependent. It is
outside our remit by deliberate choice, though I hope we will have
some informative text about it.
§7.
The proposed change to §7 "Flavors" is also not due to theoretical
weaknesses -- well, not of the set-based flavor. It merely requires,
in all flavors, some properties that the set-based flavor already
has. Without them, a clear interface between Levels 1 and 2 cannot be
specified.
Some more general points. Though Bill and Lee are united in this
motion, their basic positions on interval arithmetic (IA) are IMO so
far apart that any alliance between them can only be destructive,
never constructive.
- I think Bill is basically friendly to a model having a mathematical
Level 1, a finite-precision Level 2, and a well defined interface
between the two. I deduce this from my (limited) knowledge of Sun IA:
it is elegantly designed and any incompatibilities between it and
1788 seem to be of surface, not substance.
- Lee, however, is basically unfriendly to such a model, if I
understand his application area at all. Of course every soundly
engineered product admits a mathematical description (sometimes
engineers blaze the trail and mathematicians follow, as with the
Finite Element Method). But in his case the definition of interval
operations seems so closely bound to properties of a particular
floating-point arithmetic, that to divide such a mathematical
description into Level 1 and Level 2 is not practical.
If you think I'm wrong, I challenge you to produce an outline
interval model that is compatible with the requirements of both of you.
On 2013 Nov 22, at 19:15, G. William (Bill) Walster wrote:
Any standard for how to implement computing with mathematically
defined objects will be ambiguous unless the definition of the
objects and the results of operations on, and functions of them are
unambiguously defined. It has been clear, at least to me, from the
outset that the current P1788 effort was doomed because such a
mathematical foundation has yet to be developed. Such a foundation
must be completely independent of any implementation considerations.
Bill, I think you haven't absorbed some parts of the text:
- "Mathematical context" in the Introduction;
- in the set-based flavor, the whole of Clause 10, which makes
quite unambiguous various things you call ambiguous.
The set-based flavor is one thing. Cset flavor is another. Don't say
one of them is wrong, or ambiguous, because it behaves differently
from the other. [1,1]/[0,0] is Empty in the 1788 set-based flavor; it
is hull({-oo,+oo}) = Entire in the simplest cset flavor. No
contradiction.
Instead of concentrating on the mathematical foundation, what has
been attempted is to define the mathematics and its implementation
simultaneously.
Can you give an example of this?
John