From: "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de>
Subject: Re: Ada.Strings.Bounded
Date: Tue, 15 Apr 2008 10:02:31 +0200
Date: 2008-04-15T10:02:31+02:00 [thread overview]
Message-ID: <1wh7cbu67y4wz$.7iu8likx0fct.dlg@40tude.net> (raw)
In-Reply-To: 80c6fdca-1a89-4d98-b61d-9a405e57d8e5@s13g2000prd.googlegroups.com
On Mon, 14 Apr 2008 19:07:25 -0700 (PDT), Eric Hughes wrote:
> On Apr 14, 12:52 pm, "Dmitry A. Kazakov" <mail...@dmitry-kazakov.de>
> wrote:
>> I think it is wrong to consider N and universal_integer equivalent.
>
> Sure. It's {\bb Z} and universal_integer that are equivalent.
>
> Seriously, we just disagree about this. I can't take
> universal_integer seriously as a root class, because it's impossible
> to write down any representation of it.
Yes, because it is not what you wanted it be.
>> Subseting is not a sufficient condition for a
>> successful modeling.
>
> In a discussion that's got a lot of formal logic in it, the word
> "model" already means something pretty specific, usually involving a
> Tarski structure.
That is beside the point. A subset of values is not yet a type. The
requirement to be a subset (of values) alone does not enforce anything
useful.
>> Actually it is the opposite, a perfect subset cannot
>> be a good model.
>
> There was paper from the fifties (sorry, no reference handy), which
> used Turing machines to compute a Dedekind cut. On input a rational
> number, it returned one of the two symbols "<=x" or ">=x". (You
> cannot compute exact trichotomy without solving the halting problem.)
> In any language describing these machines, there's a least one (Kleene
> minimization), so there's a unique representative of such a machine
> for every computable real number, which means there's a subset
> bijection. Addition, multiplication, and their inverses are defined
> in terms of the underlying operand-machine (it's pretty easy coding,
> actually). So there's pretty close to a perfect model of the real
> numbers, whose only real limitation is that run times are horrendously
> slow. But it's also completely exact, with no compromises but
> execution speed and representation size of a machine.
>
> The subset is every real number that's computable. About as good as
> you can do with computers, I'd say.
No. It cannot represent pi and e. Now consider the following:
type A_Subset_Of_R is (pi, e); -- Ready!
The above is a perfect subset of R containing both pi and e. Isn't it good?
>> The best models of R aren't even close to subsets. For
>> example intervals with rational bounds.
>
> In the precise meaning of model, it's just not a model, because
> there's no total ordering on such intervals, so the ordering axioms
> are not satisfied. In an imprecise meaning, it's real numbers plus
> some other concept, which is more than {\bb R}.
For any model there are properties which are not satisfied. Intervals are
good because they preserve *interesting* properties and provide fair
approximations for properties lost. Total ordering is mere a property,
which has a minor importance to numeric computations. In fact, each second
handbook on numeric methods starts with something like "never ever compare
reals."
--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de
next prev parent reply other threads:[~2008-04-15 8:02 UTC|newest]
Thread overview: 108+ messages / expand[flat|nested] mbox.gz Atom feed top
2008-03-31 22:44 Untyped Ada? Phaedrus
2008-04-01 0:16 ` Randy Brukardt
2008-04-01 7:19 ` tmoran
2008-04-01 7:44 ` Dmitry A. Kazakov
2008-04-01 8:28 ` Jean-Pierre Rosen
2008-04-01 21:44 ` Phaedrus
2008-04-01 21:51 ` Ludovic Brenta
2008-04-01 17:09 ` Pascal Obry
2008-04-01 22:00 ` Phaedrus
2008-04-02 7:31 ` Dmitry A. Kazakov
2008-04-04 15:16 ` Graham
2008-04-04 16:10 ` Pascal Obry
2008-04-04 21:09 ` Ada.Bounded_Strings Adam Beneschan
2008-04-04 21:15 ` Ada.Strings.Bounded Adam Beneschan
2008-04-05 4:39 ` Ada.Strings.Bounded Gautier
2008-04-05 9:43 ` Ada.Strings.Bounded Pascal Obry
2008-04-05 10:10 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-05 11:36 ` Ada.Strings.Bounded Gautier
2008-04-05 12:14 ` Ada.Strings.Bounded Pascal Obry
2008-04-06 0:31 ` Ada.Strings.Bounded Randy Brukardt
2008-04-07 14:57 ` Ada.Strings.Bounded Adam Beneschan
2008-04-07 15:23 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-07 16:34 ` Ada.Strings.Bounded stefan-lucks
2008-04-07 17:34 ` Ada.Strings.Bounded (see below)
2008-04-12 18:50 ` Ada.Strings.Bounded Eric Hughes
2008-04-12 19:46 ` Ada.Strings.Bounded Georg Bauhaus
2008-04-13 16:53 ` Ada.Strings.Bounded Eric Hughes
2008-04-13 20:10 ` Ada.Strings.Bounded Robert A Duff
2008-04-13 23:52 ` Ada.Strings.Bounded Eric Hughes
2008-04-14 8:00 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-14 15:25 ` Ada.Strings.Bounded Eric Hughes
2008-04-14 18:36 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-15 1:39 ` Ada.Strings.Bounded Eric Hughes
2008-04-12 21:09 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-13 16:31 ` Ada.Strings.Bounded Eric Hughes
2008-04-13 20:02 ` Ada.Strings.Bounded Robert A Duff
2008-04-13 23:20 ` Ada.Strings.Bounded Eric Hughes
2008-04-14 9:07 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-14 15:50 ` Ada.Strings.Bounded Eric Hughes
2008-04-14 18:52 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-15 2:07 ` Ada.Strings.Bounded Eric Hughes
2008-04-15 8:02 ` Dmitry A. Kazakov [this message]
2008-04-15 14:20 ` Ada.Strings.Bounded Eric Hughes
2008-04-15 15:23 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-16 2:51 ` Ada.Strings.Bounded Eric Hughes
2008-04-16 8:00 ` Ada.Strings.Bounded Dmitry A. Kazakov
[not found] ` <bc3a8b4e-63fe-47a6-b10b-7056f6d7d586@w5g2000prd.googlegroups.com>
2008-04-15 14:58 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-16 2:46 ` Ada.Strings.Bounded Eric Hughes
2008-04-16 8:16 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-16 14:40 ` Ada.Strings.Bounded Eric Hughes
2008-04-16 18:28 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-21 0:44 ` Ada.Strings.Bounded Eric Hughes
2008-04-21 14:08 ` Ada.Strings.Bounded Robert A Duff
2008-04-21 16:35 ` Ada.Strings.Bounded Eric Hughes
2008-04-21 18:04 ` Ada.Strings.Bounded Robert A Duff
2008-04-22 0:19 ` Ada.Strings.Bounded Eric Hughes
2008-04-22 0:49 ` Ada.Strings.Bounded Adam Beneschan
2008-04-22 1:02 ` Ada.Strings.Bounded Adam Beneschan
2008-04-22 15:30 ` Ada.Strings.Bounded Eric Hughes
2008-04-22 16:08 ` Ada.Strings.Bounded Robert A Duff
2008-04-22 15:25 ` Ada.Strings.Bounded Eric Hughes
2008-04-22 15:54 ` Ada.Strings.Bounded Robert A Duff
2008-04-22 15:41 ` Ada.Strings.Bounded Robert A Duff
2008-04-22 17:49 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-22 18:26 ` Ada.Strings.Bounded Samuel Tardieu
2008-04-22 18:59 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-22 18:47 ` Ada.Strings.Bounded Eric Hughes
2008-04-22 19:19 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-22 19:41 ` Ada.Strings.Bounded Robert A Duff
2008-04-22 22:55 ` Ada.Strings.Bounded Eric Hughes
2008-04-23 6:40 ` Ada.Strings.Bounded christoph.grein
2008-04-23 6:54 ` Ada.Strings.Bounded christoph.grein
2008-04-23 10:42 ` Ada.Strings.Bounded Georg Bauhaus
2008-04-23 12:32 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-23 12:52 ` Ada.Strings.Bounded christoph.grein
2008-04-23 13:34 ` Ada.Strings.Bounded Georg Bauhaus
2008-04-23 15:12 ` Ada.Strings.Bounded Adam Beneschan
2008-04-23 15:36 ` Ada.Strings.Bounded (see below)
2008-04-23 17:09 ` Ada.Strings.Bounded Ray Blaak
2008-04-24 0:29 ` Ada.Strings.Bounded Randy Brukardt
2008-04-22 20:15 ` Ada.Strings.Bounded Adam Beneschan
2008-04-23 13:14 ` Ada.Strings.Bounded Peter Hermann
2008-04-23 14:40 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-23 15:03 ` Ada.Strings.Bounded Adam Beneschan
2008-04-22 19:56 ` Ada.Strings.Bounded Adam Beneschan
2008-04-21 18:50 ` Ada.Strings.Bounded Dmitry A. Kazakov
2008-04-22 0:31 ` Ada.Strings.Bounded Eric Hughes
2008-04-14 15:11 ` Ada.Strings.Bounded Adam Beneschan
2008-04-14 16:09 ` Ada.Strings.Bounded Eric Hughes
2008-04-14 18:13 ` Ada.Strings.Bounded Georg Bauhaus
2008-04-15 1:35 ` Ada.Strings.Bounded Eric Hughes
2008-04-15 20:33 ` Ada.Strings.Bounded Georg Bauhaus
2008-04-16 3:11 ` Ada.Strings.Bounded Eric Hughes
2008-04-04 23:35 ` Ada.Bounded_Strings Robert A Duff
2008-04-05 1:46 ` Ada.Bounded_Strings Adam Beneschan
2008-04-05 4:55 ` Ada.Bounded_Strings Randy Brukardt
2008-04-05 7:30 ` Ada.Bounded_Strings Dmitry A. Kazakov
2008-04-06 0:44 ` Ada.Bounded_Strings Randy Brukardt
2008-04-04 16:18 ` Untyped Ada? Adam Beneschan
2008-04-04 16:32 ` DScott
2008-04-04 17:38 ` Dmitry A. Kazakov
2008-04-04 18:52 ` Georg Bauhaus
2008-04-05 8:07 ` Dmitry A. Kazakov
2008-04-04 19:14 ` Graham
2008-04-04 21:06 ` tmoran
2008-04-05 8:44 ` Dmitry A. Kazakov
2008-04-12 16:50 ` Eric Hughes
2008-04-04 18:02 ` adaworks
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