Re: More on trits & tetrits... (long)
> Reply-To: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
> From: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
> To: "P1788" <stds-1788@xxxxxxxxxxxxxxxxx>
> Cc: "Dan Zuras Intervals" <intervals08@xxxxxxxxxxxxxx>
> Subject: Re: More on trits & tetrits... (long)
> Date: Sun, 11 Apr 2010 00:10:20 -0500
>
> Hi Dan,
>
> Well done!
>
> It is officially no-longer half-baked.
>
> So now let me raise a few observations about the nicely-cooked version:
>
>
> Dan Zuras Intervals wrote:
> > For the decoration named 'domain' (or defined or valid, I don't
> > really care) we want to set 2 bits in the following way:
> >
> > domainFalse = {There exists x in xx such that x is not
> > in the domain of f()}
> >
> > domainTrue = {There exists x in xx such that x is in the
> > domain of f()}
> >
> > ...
> >
> > sqrt([empty]) = {[empty],{}}
> > The result is decorated with {} (the empty set)
> > because there exist no points of any flavor at
> > all in the operand.
> >
> > This last is the cheap way to find empty intervals.
>
> Let's look at another example, e.g.,
> [1,1]/[0,0] = {[empty], {definedFalse}}.
> It seems in this case, testing the tetrit for {} will give the
> misleading impression that the interval is not empty. So I don't
> think we should count on this approach as a method to test for
> empty intervals.
>
> . . .
>
> Nate
>
Interesting.
OK, let me be even more clear: The empty decoration appears
as a result of the empty interval PASSING THROUGH a function
not as a result of the empty interval being created due to
a domain error.
Look closely & you'll see that the decoration bits are defined
in terms of a function's operands not in terms of its result.
So what you have given us is an example of non-empty
intervals passing through the divide function resulting in
both the empty interval & {definedFalse} as an output.
This is as it should be.
Indeed, you deleted the example I gave of exactly this sort
of thing. It was:
sqrt([-2,-1]) = {[empty],{domainFalse}}
The result is decorated with domainFalse only
because there exist only points outside the
domain of sqrt in this case.
So, let's see if we can write the domain decoration definition
for the divide function. I believe it would be:
domainFalse = {There exists x in xx & y in yy such
that x/y is not in the domain of divide}
(i.e. an x & y such that y = 0)
domainTrue = {There exists x in xx & y in yy such
that x/y is in the domain of divide}
(i.e. an x & y such that y != 0)
So, I believe all of the following are true:
[1,2]/[-2,2] = {[entire],{domainFalse,domainTrue}}
[1,2]/[0,2] = {[1/2,oo],{domainFalse,domainTrue}}
[1,2]/[1,2] = {[1/2,2],{domainTrue}}
[1,2]/[0,0] = {[empty],{domainFalse}}
[1,2]/[empty] = {[empty],{}}
[empty]/[-2,2] = {[empty],{}}
[empty]/[empty] = {[empty],{}}
So an empty interval may be created by a function that
refuses to return an interval (numeric) result at some point
but the empty decoration is only created should an empty
interval be used as an operand.
And, as I have a habit of not being painfully clear on these
matters: The empty decoration appears as a result of the
empty interval passing through a function that will not accept
an empty operand at that point in its domain & NOT as a result
of a domain error.
I am unaware of any function that has that property but I
cannot rule out the possibility. After all, some functions
are independent of one operand at some values of the others.
For example, x^y is independent of y at x = 1 since 1^y = 1
for all y. But I don't think it can be said that the power
function accepts a vacuous operand at any point in its domain.
OK, so maybe this is not a cheap way to detect the creation
of an empty interval. At best, it is a way of detecting that
you're trying to use one.
Well, its not what I was shooting for anyway.
I was shooting for making the definition of trits more
concrete.
Maybe I missed.
But I think I'm getting closer.
Dan