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Uncountably infinite datatypes can be computational.

From: V. Bruce Hunt <vbhunt@silverfox.com>
Date: Sat, 30 Oct 1999 18:01:03 +0000
Message-ID: <381B325F.A43A8D15@silverfox.com>
To: www-xml-schema-comments@w3.org
Hi,

In reading along in the XML schema 2 - datatypes specification, I came
across the sentence in Section 2.4.1.3 that says,

"No computational datatype is uncountably infinite. "

I suggest that the editors have a very narrow meaning for "computational
datatype" and are intending to convey the thought that no "finite
numerical computational" datatype is uncountably infinite based on
current finite numerical precision high performance native number system
implementations in todays computers.   There are of course infinite
precision computational packages and symbolic
computational packages; both categories deal effectively with
uncountably infinite datatypes.

The real numbers are the most common form of an uncountably infinite
datatype.  Many symbolic computational packages  such as Macsyma and
Mathematica "compute" with representations of uncountably infinite
datatypes and obtain results that can be delivered to any desired level
of numerical precision and routinely deliver results in terms that are
completely (in the mathematical sense) precise.  Examples include
integration, differentiation, laplace transforms, and algebraic
manipulation.

Clearly these are good examples of uses of a computational datatype that
are uncountably infinite.  I would suggest a correction to more clearly
define your intention.

In addition, the XML-schema group would be well served to expand the
explained domain of discourse with
a fourth bullet  in this section:

value spaces that are uncountably infinite and exact.

It is perfectly reasonable and is likely to become highly desirable to
represent  value spaces that are "uncountably infinte and exact" to
represent exact "symbolic" mathematical computations using XML-schema.
For example, the ability to make room for such extensions now should
permit a future in which MathML contrains a datatype to, for example,
the real numbers (engineering and physics calculations) or the complex
numbers ( electrical engineering ) or probability spaces ( the real
numbers from 0 through 1 inclusive [for queueing theory]) and is quite
satisfied to have a result that is an expression of "pi" ( the ratio of
the circumference of a circle to its diameter ) or "e" ( the base of the
natural logarithms) or even "i"( the square root of  minus one).

To be clear, I am not suggesting that the working group take on defining
value spaces that are uncountably infinite and exact however I believe
it incumbent on the working group to clearly indicate that they can be
defined (and many are computable).  In addition, you may well wish to
mark them for later clarification and a potential area for future
standardization.

sincerely

Bruce Hunt


Received on Saturday, 30 October 1999 13:56:27 UTC

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