Re: Motion P1788/M003.01_Set_of_reals NO
Ian,
although the discussion period is over I think I should give some
comments on your reason to vote against metion 3.
Ian McIntosh schrieb:
I vote NO on Motion 3.
Infinity is needed, not just as a lower or upper bound, but as a value.
Lots of reasons:
I consider this as a special case of "Thou shalt not lie." Treating an
answer that should be +/-Infinity as not having an answer (empty) is a
form of lieing.
As interval arithmetic is defined as arithmetic for real numbers, there
is only one case among the 4 basic operation that is not defined.: The
division by 0. That will be specially treated by setting a flag. The
same happens with functions evaluated on a point which does not belong
to its domain. log([0,0]) may produce [-oo,-maxreal] and set the flag
For I3 = op (I1, I2), it should be true that if x is in I1 and y is in
I2, that x op y is in I3.
for every defined real basic arithmetic operation I3 is empty only if I1
or I2 is empty. dividion by zero is not allowed in a field
Saying I3 is Empty means you aren't answering
the question. Doing that when it's not necessary to is undesirable.
Empty has its own meaning and uses, and we shouldn't mash empty and
infinity together.
we don't do that
It's important for a standard to be inclusive, and not exclude valid
applications unnecessarily. There will be many applications of IA
beyond what is being done now.
I agree there may be and certainly will be extensions, John Pryce's comments
One obvious example of a useful application would be to transform a
non-IA program into an IA one, to do error analysis. A very simple
example might be feeding millions of values into a math library sqrt
implementation (perhaps one that does multiple square roots in
parallel). The specification includes that sqrt (+Infinity) is
+Infinity, and sqrt (any quiet NaN) = the same quiet NaN. If IA doesn't
allow working with those values what do you do? Do those ones by hand?
Use some non-1788 IA system? And if you produce [Empty], how do you
distinguish values that should be infinity from those that should be NaNs?
Infinities and NaN are not real numbers. You may have to convert them,
if you switch from floating-point to intervals.
Any standard will be more successful if it allows more uses. Some of
the IA uses will naturally produce or need to consume infinities. We
shouldn't rule all those out now, before we can even imagine them. A
couple of less obvious places IA could be useful:
- Project planning: Using IA ranges, calculate for example task_time
= 1 to 2.5 pw + 4.2 to 6 pw = 5.2 to 8.5 pw, then that
calendar_time = task_time / workers.
What if workers = 0? An infinite calendar time is more meaningful
and useful than an empty time.
An infinite calendar is noit meaningful for me.
I would suggest to improvve the software to check for exceptions.
Which bar on a graph shows management
where to add people - a tall bar with "oo" at the top, or an empty bar?
- Financial analysis: Using IA ranges for many possible investment
situations, calculate return_on_investment = profit / cost, then sort
them by value.
What if cost = 0? An infinite ROI is a lot better than an empty one.
Which one tells you more - "Take this freebie!" or "Sorry, I couldn't
calculate your results on this one."?
The point isn't that these applications should necessarily be
implemented using 1788 IA (but why shouldn't they?). It's that once
something is available, people will use it in ways we can't anticipate,
and for IA some (many?) of those uses will need infinities. Even
checking "Is the bridge safe?" can involve infinity if the net support
at some point is zero. The engineers need to know that, not that after
all that work they don't have an answer. Having the 1788 standard force
appllications to use some non-1788 system to get useful results isn't
beneficial to anybody.
The argument against including Infinities seems circular to me. If you
decide that IA is based on R, then you have to produce [Empty] for
expressions where the natural result is +/-Infinity.
No I can produce the unbounded interval [maxreal,oo]
You have to simply
because you don't allow [+/-Infinity]. But if you instead base IA on
R*, then you can produce [+/-Infinity]. The set is still closed (yes,
I'm assuming NaI will exist). It's better because it defines an
informative result value for operations for which you would otherwise
say there is no result.
If you have a application that could produce an infinity it's more
useful to get infinity than an ambiguous empty set that could have been
produced for other reasons. If you have a series of operations that
produce infinity and you don't like Infinity, it's easy to just treat it
as Empty. If you produced Empty instead, it's unsafe to treat that as
Infinity when it could have been produced from other operations like an
empty intersection. For at least some applications, you'd want to keep
separate variables saying whether to treat each particular Empty as
really Empty or as Infinity. Some could simply test the result of every
divide etc. and if it poduced Empty when its inputs ween't then convert
it to Infinity. Oh, wait, 1788 doesn't allow Infinity. Hmmm. Maybe
convert the value to [MaxFinite, Infinity], but that's awkward and a
fib. OK, let's use some non-1788 system so we can get the job done right.
IA needs to coexist with regular 754 floating point. That means they
should have the same base, instead of IA being a subset. You don't need
as many guardian dragons or dogs if you don't have as many dangers to
guard against.
Look into the Vienna proposal, there is a long discussion on the caveats
inluding the infinities as numbers
IA will likely be implemented using 754, and the extra checks to ensure
operations don't produce infinity will make the programs bigger and
slower, for negative benefit.
interval operations have to be defind in any case it just happens that
infinity as a number never occurs
I know Wikipedia has flaws, but its IA write-up assumes IA is based on
R* and includes infinities. Somebody uses them, and believes they're
using a consistent system.
I think we as a group of experts should correct wikipedia after our vote
has passed
The bottom line for me is that to reject infinities is to cripple
ourselves, to deliberately shoot ouselves in both feet, and I'm not a
masochist. Let's start with R* and Infinity instead of without them.
I disagree
Juergen
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