Re: Possreps and numeric types

On Mar 25, 6:33 pm, Bob Badour <bbad..._at_pei.sympatico.ca> wrote:
> Marshall wrote:
>
> > "Approximation" is perhaps not the best choice of words.
> > We certainly have resource limits in our finite computers.
> > There are computations that we can't do because
> > we don't have the resources. For example, a computer
> > might be able to add together two one billion digit
> > integers, but not be able to add together two ten billion
> > digit integers because it didn't have enough memory.
> > That doesn't mean the result of adding the two
> > one billion digit integers is approximate; on the
> > contrary, it is precise and exact. Or consider
> > java.util.BigInteger. Any answer you get from it
> > will be precise, and it can handle up to four
> > billion digit numbers. If it can't give you an answer,
> > it'll fail in a way that can't be mistaken for an
> > answer.
>
> Precise for integers but rationals are not integers.

The situation for rationals and integers is the same; one is just a pair of the other. The result of a rational multiply is either 1) the exactly right answer or 2) failure due to exceeding resource limits.

> > In my mind, there is a significant difference between
> > operations that will either produce precise and exact
> > answers or else fail, having consumed all the machine's
> > resources, and operations that will return results that
> > are only guaranteed up to some degree of precision.
>
> Do you prefer square root to fail having consumed all resources?

The question is a good one however it does not address the distinction I am trying to make between precise and approximate operations. To answer your question, I would treat square root as an approximate operation. (The usual floating point implementation.)

A common way of viewing the situation today is that we have some exact numbers and some inexact numbers. I am arguing that there are advantages to a different model, in which all numbers are exact, and some *operations* are exact and some are approximate.

> I still don't see the point in what you wrote. A billion digit integer
> doesn't make a whit of difference to what I wrote earlier: One still has
> a maximum representable integer,

I agree.

> and a maximum representable integer
> defines a minimum range for the representation of each value.

I disagree.

> > I don't think the model of numbers as ranges is a very
> > good one, even for floating point. With rationals it's even
> > less apt.
>
> Nevertheless, that is the best we can do. If we want to pretend
> something is a rational, we must acknowledge that it necessarily stands
> for any rational within some range regardless how large or small the range.

I don't see how that follows. I don't see the "necessarily" part.

> > If you have a rational one and a rational three
> > and you divide, you get the exact rational one-third.
>
> Except that the represention of 1 actually represents a range of
> rational values and the representation of 3 actually represents a range
> of rational values before we even do the divide.

Same objection.

> > It is more useful to say that the operators, rather
> > than the operands, are approximations. Floating
> > point operations return approximate results in
> > the worst case, but may return precise results in
> > the easier cases.
>
> Except that the operands themselves are approximations representing
> ranges. When the rational add operation adds 1 to 1, it doesn't know
> whether 1 is 1 or 1+epsilon/2 or 1-epsilon/5 etc. It uses the same
> representation for all three and for an infinite number of other rationals.

Same objection again.

> > For example, 80 bit floating point
> > add, subtract, and multiply on numbers with less than
> > 32 bits of mantissa are always exact. Floating point
> > sqrt(2) is an approximation; sqrt(4) is not.
>
> Except that 4 itself is an approximation covering the range
> [4-epsilon,4+epsilon] and the result of the operation represents the
> range [2-epsilon,2+epsilon].

Same objection again. Can you say *why* you think this is necessarily true?

> >>>Crazier still, the fact that some rationals are integers
> >>>and some aren't starts to remind me of parsing a
> >>>string as a number. A possrep for an integer could
> >>>be a sequence of ascii digit characters. Then
> >>>constructing (selecting) an integer from a string
> >>>(succeeding only if the string represents an integer)
> >>>is isomorphic to parsing the string into a number,
> >>>a la parseInt or atoi or whatever. Integer isn't a
> >>>subset of String, but the Integer possrep sequence-
> >>>of-digit-chars *is* a subset of the String possrep
> >>>sequence-of-chars.
>
> >>An optional sign and a string of digits is a possible representation for
> >>an integer.
>
> > I stand corrected again. When can I sit down again?
>
> I thought I was agreeing not correcting. Were you objecting to an
> optional sign and a string of digits as a possible representation for an
> integer?

Ah, I just overreacted. I left off the part about the sign; your statement
of the situation was more complete and accurate.

> >>>Anyway, this post is long and confusing enough
> >>>for now. I think I'll go reread the relevant sections
> >>>of TTM now. Again. I really must buy the 3rd ed.
>
> >>What did you find out?
>
> > I found out that shopping used up most of the day. :-(
> > So far I've managed to get the book off the shelf,
> > but that's about it.
>
> No biggie. Now that you have had it off the shelf for a while longer,
> what did you find out?

Bleah, I barely got ten pages done. Lots of distractions. Probably won't attempt it today, what with the hangover and all. Tomorrow while the kids are at school is more promising. I haven't looked at SbyC in a long time; especially wanna reread that part, but not until I've reread the possreps stuff.

Marshall Received on Mon Mar 26 2007 - 22:02:27 CEST