Re: Unbounded intervals
Nate Hayes wrote:
> -- No computer (that I'm aware of) can numerically prove hardly anything
> useful about the domain of a function beyond the underlying numeric
> limits of the system; so for this reason alone, truly unbounded
> intervals are never necessary in numeric models or computations
> (you have never answered my original question from long ago to show
> a counter-example of this).
Computers don't prove; programs may. The only thing that can be proved
strictly numerically is a counterexample; other proofs need to assume
properties between the finitely-many representable points (continuity,
monotonicity, etc.) -- and once you take that into account, unbounded
ranges cease to be a problem if the functions under consideration are
known (or assumed) to be monotonic beyond a certain point.
There also is no largest representable number for a *program*, because
even with a range-limited primitive type programs can use scaling to
reach waaay beyond the range of the primitive type. And as I mentioned
before, some number representations are essentially overflow-proof.
The fact that no physical computer implements such a type as a machine
primitive does not restrict what programs (or virtual machines) can do.
I'd love to hear you give a recursion-theory course using a notion of
Turing machines that are in fact families of Turing machines, each with
a bounded tape, parameterized by the size of the tape. I suppose it
could be done, but I'd hate to have to take this class...
Also note that although P1788 may restrict allowable types to those that
have bounded precision (in order to be able to define "tightest"), there
is currently no requirement for a bounded range. Why would you want that?
Michel.
---Sent: 2012-04-25 15:18:13 UTC