Re: Motion 11 questions & comments...
Dan Zuras Intervals wrote:
I am confused about the definitions of reverse operations
in sections 3 & 4. But this may be due to my confusion
about reverse operations in general.
Please be patient for a bit while I explain my confusion.
I have noticed that in the ordinary forward addition of
A + B = C we have that width(A) + width(B) = width(C).
Also, in subtraction A - B = C we have width(A) + width(B)
= width(C). So widths add in the forward add & subtract
operations. Things can get wider but not narrower.
And yet in sections 3 & 4 you define reverse add &
subtract to be formally equivalent to subtract & add
(respectively). The widths must add again.
But in reverse operations shouldn't the widths get narrower
not wider? Indeed, is that not the whole point of such
operations?
That is, if I am trying to solve the equation A + B = C
for B by using the expression B = C - A, I get a B that
is wider than the interval that solves the equation.
The reason for this is that there is no additive inverse
for non-zero width intervals among the ordinary intervals.
A + (-A) is not [0,0] in general.
So in the reverse add needed to solve for B do I not need
a definition of the form reverse-add(C,A) = {all b such
that for all a in A there exists c in C such that a + b
= c}?
I grant that the definition you have appears equivalent
to the third line &, thus, equivalent to ordinary forward
subtract.
But how can that be? If I am using the reverse add to
solve for B in A + B = C (for which width(A) + width(B)
= width(C)), I need a B such that width(B) = width(C) -
width(A). But if I use B = C - A I have that width(B)
= width(C) + width(A).
It seems to me I had this discussion with John & Nate
about a year ago & Nate's solution involved using both
'there exists' & 'for all' quantifiers.
I may have this wrong. Perhaps Nate can help here.
Dan,
My understanding is that reverse mode is different than the Kaucher/modal
operations, and perhaps this is the point to be clarified. For example,
reverse mode does not supply the additive or multiplicative inverse elements
you are speaking of. Those are supplied by Kaucher/modal operations (of
which there has been no formal motions for, although they have been
discussed at times in this forum).
If you are trying to solve the equation A + B = C, the unique algebraic
solution is obatined by the Kaucher arithmetic, i.e., B = C - Dual(A). In
other words, A - Dual(A) = [0,0] is an identity. Similarly, the equation A *
B = C has unique algebraic solution B = C / Dual(A), so long as 0 is not an
element of A, i.e., A / Dual(A) = [1,1] is an identity.
The system (IR,+) over closed, bounded and non-empty intervals IR is a
commutative monoid, hence as you notice there are no inverse elements. Since
(IR,+) also has the cancellative property, it is possible to embed the
monoid into a group. This is where the inverse elements of Kaucher
arithmetic come from, as well as the notion of improper intervals [b,a] such
that a <= b.
Strictly speaking, the "for all" or "there exists" quantifiers of modal
intervals are not necessary. Only the Kaucher arithmetic (and improper
intervals) are required. However, it is all a separate topic from the motion
Marco currently is putting forward.
Hope this helps.
Nate