Re: Re-submission of motion 5: multiple-format arithmetic.
John
in your last eply to the Kulisch-Einarsson motion 5 you wrote
". Then, your current p4 stuff is more or less saying the same thing as
the definitions in my (under revision) multi-format motion, except you
only mention the 4 basic ops, and I cover general multivariate
point-functions, which _are_ necessary. So it would be nice for us to
converge closer."
and now you again deny this in section R4.
John Pryce schrieb:
P1788 members
R4. Overall Aims.
-------------
Ramon Moore's Basic Theorem of Interval Arithmetic (BTIA) is fundamental
to interval computation. Roughly, it says that if E is an explicit real
expression defining a real function f(x_1, ..., x_n), then evaluating E
"in interval mode" over any interval inputs (xx_1, ..., xx_n) is
guaranteed to give an enclosure of the range of f over those inputs.
This motion has two linked aims. Aim 1 is to define a framework for
finite precision interval arithmetic in multiple formats. In this
framework the BTIA is easily proved -- indeed almost obvious. The finite
precision interval operations are defined to operate on a set of
abstract objects that happens -- except for Not an Interval, NaI -- to
be a finite subset of the infinite set (IR). The definition of
operations is independent of any representation that may be chosen for
intervals.
The framework should support mixing intervals of different formats in
arithmetic expressions, as well as potential compile-time "exact
denotations" of intervals such as pi + [-0.1,0.1]. However, this motion
does not commit P1788 to any stance on such issues.
[Note: To simplify wording I am assuming P1788 defines an NaI object.
Personally I support NaI; editorially I have no preference; and NaI is
peripheral to the argument.]
This is exactly analogous to how 754 defines the finite precision point
operations, on a set of abstract objects that happens -- except for -0,
+0 and NaN -- to be a finite subset of the extended real numbers.
The abstract objects and operations form interval level 2. To see its
utility, ask yourself the question "How to prove the BTIA, starting from
a procedural definition of the arithmetic operations as given, for
instance, in the Einarsson-Kulisch position paper presently (May 2009)
under discussion as Motion 5?" Answer: you cannot, _except_ by passing
through a stage equivalent, or closely related, to the abstract objects
and operations defined here. (This is no criticism of these definitions:
they have a different purpose.)
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