On Mon, 14 Apr 2008 19:07:25 -0700 (PDT), Eric Hughes wrote:
> 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.
On Apr 15, 2:02 am, "Dmitry A. Kazakov" <mail...@[EMAIL PROTECTED]
>
wrote:
> Yes, because it is not what you wanted it be.
I assert that that Ada as currently defined has no bound on the size
of numbers within universal_integer. Here is my argument.
The best definition is in ARM 3.4.1(6/2-7). (Defect re****t: this
doesn't appear in the index entry for universal_integer.) Wherein:
> The set of values of a universal type is the undiscriminated
> union of the set of values possible for any definable type
> in the associated class.
The associated class to universal_integer is the signed integer type
(3.5.4), delineated by ranges given by static simple_expression.
Simple expressions allow for exponentiation, so it's quite easy to
define ranges that take petabytes to implement. There's no length
limitation on an expression in the language, so arbitrarily large
ranges are possible. Thus I can define the following series of types:
>
> >> 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 im****tance to numeric computations. In fact, each
second
> handbook on numeric methods starts with something like "never ever
compare
> reals."
>
> --
> Regards,
> Dmitry A. Kazakovhttp://www.dmitry-kazakov.de
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