Vote on Motion 5
My vote on Motion 5: NO
Inasmuch as
(1) section 3.11 of the Vienna proposal ("Vienna 3.11" from here
on) covers the same ground as the document proposed in Motion 5 and
is a small fraction of the size, and
(2) it is apparently considered time to vote on the content of
Vienna 3.11, my preferred version of Motion 5 is not to adopt the
document "Arithmetic operations for intervals" by Ulrich Kulisch
but instead to modify (for the need of this, see below) Vienna 3.11
as follows:
=========================================================
3.11 For each operation <circ> equal to +, *, min, max, and pow:
<circ>Hull(xx,yy,zz) = least floating-point interval containing the
set of z in zz such that
x <circ> y = z with x in xx and y in yy
<circ>Inv1(xx,yy,zz) = least floating-point interval containing the
set of z in zz such that
z <circ> y = x with x in xx and y in yy
<circ>Inv2(xx,yy,zz) = least floating-point interval containing the
set of z in zz such that
y <circ> z = x with x in xx and y in yy
The third argument of <circ>Hull, <circ>Inv1, and of <circ>Inv2 is
optional and is to be taken as Entire if absent.
If <circ> is commutative, then <circ>Inv1 and <circ>Inv2 are the same
and will be called <circ>Inv.
[There is no need to spell out in the standard the mathematical
proposition that translates the above specification into tables
with formulas for the end points.]
=========================================================
Notes:
Interval subtraction and division are obtained as inverses of +
and *. Thus this gives the interval versions of +, *, min, max, pow
as well as of their inverses (one of each except two in the case
of pow).
Rationale
---------
(0) To support the Vienna proposal as much as possible.
(1) See last, bracketed, paragraph of Vienna 3.11, of which I have
adopted the first sentence. This is useful in the context of Motion
5, and is to be discarded at the first opportunity.
(2) To ensure an unambiguous definition of interval division. Vienna
3.11 gives two definitions of interval division: one directly and
one as inverse of multiplication. These give different results for
[0,0]/[0,0], about which more below.
(3) Vienna 3.11 is not symmetric with respect to inverse operations.
As a result it needs qualifications, such as "defined" and "finite".
The above modification is symmetric and has no need of these
qualifications.
(4) The method of defining the inverse operations used here is
well-known and of long standing in the interval community, going
back at least as far as "On extended interval arithmetic and inclusion
isotonicity" by D. Ratz, Karlsruhe Technical Report, 1996.
(5) According to the above modification [0,0]/[0,0] is Entire, whereas
it is Empty according to the <circ>Hull definition of Vienna 3.11,
it is Entire according to the <circ>Inv1 and <circ>Inv2 definitions in
Vienna 3.11.
I have the followng argument for taking [0,0]/[0,0] to be Entire:
Consider the equation ax = b in x with a and b as real coefficients.
If a and b are uncertain reals with values in intervals aa and bb,
then the equation determines x to within the interval bb/aa.
According to what could be called the Correspondence Principle of
Interval Arithmetic, the interval bb/aa should revert to the possible
values for x in ax = b when aa = [a,a] and bb = [b,b]. Setting a = 0
and b = 0, ax = b leaves x indeterminate. The corresponding interval
bb/aa is (-oo, +oo). Ergo, [0,0]/[0,0] should equal Entire.
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