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Re: inexactness vs. exactness



 | From: William D Clinger <will@xxxxxxxxxxx>
 | Date: Mon, 25 Jul 2005 13:51:15 -0400
 | 
 | On 21 July 2005, Aubrey Jaffer wrote:
 | 
 | > You are suggesting that an implementation in which inexact
 | > numbers are not neighborhoods can conform to R5RS.  I will show
 | > that this is impossible.
 | 
 | On 22 July (but delayed until 24 July in this archive), I paraphrased
 | Jaffer's statement above as
 | 
 | > Aubrey Jaffer claims to have proved that the language of
 | > the R5RS not only regards inexact numbers as neighborhoods,
 | > but that no other interpretations of the R5RS are tenable.
 | 
 | On 24 July, Jaffer quoted the first part of my paraphrase and wrote:
 | 
 | > No, it claims that inexact numbers are in one-to-one
 | > correspondence with neighborhoods around their nominal values.
 | 
 | If that was all Jaffer was claiming, then he misspoke when he
 | denied that "an implementation in which inexact numbers are not
 | neighborhoods can conform to R5RS".

After two failed attempts, I will concede that my command of CS is
inadequate to produce proofs about Scheme.  My July 21 statement was
brazen; apologies to everyone for the flamage.

 | Indeed, if that was all Jaffer was claiming, then his alleged
 | proofs are needlessly complex, and he should not object to the
 | simpler interpretation that identifies all finite inexact complex
 | values z with the mathematical closed neighborhood consisting of z
 | itself and nothing else.

A Scheme implementation in which inexact reals are represented by
unbounded integer ratios (and where complex inexacts are pairs of
unbounded integer ratios) is an example where each point is its entire
neighborhood.

Alex Shinn devised a clever way to define real infinities for a
similar system:

  Infinity in this case could be defined as the range of all real
  numbers greater than the largest possible BigFloat using all of
  memory for the exponent.

But if we consider memory limitations for this system, then there are
many mathematical numbers between any two representable inexacts whose
storage requirements are too large to be represented on a particular
physical computer.  Ideally, on input or in the course of calculation,
those mathematical numbers would be rounded to the closest inexacts
with smaller representations.  So even here, I think that each inexact
number corresponds to a mathematical neighborhood containing more than
one point.

			      -=-=-=-=-

The motivation for the inexact neighborhoods was to unify the
interpretation of inexact numbers with infinities; making both
correspond to non-singular simply-connected neighborhoods.  It was a
nice idea; but it is not indispensable.

Mention of neighborhoods in SRFI-70 is localized to the "6.2.2x
Inexactness" section.  If singular neighborhoods can't be ruled out,
then I will change it.

So, does R5RS permit an inexact number which only one mathematical
number rounds to?