FOM: Precision and Turing-computability

[D Richardson]

It may be that exact computation is quite possible for the real numbers of
classical analysis (which are only a small subset of computable reals).
There have been a number of results recently along these lines.  Exact
computation would involve working with the definition of a number, rather
than with part of its decimal representation.  There is not complete
agreement as to what these classical numnbers are.  But they certainly
include the rational numbers, and are closed under field operations, exp
and log.  A lot of analogue computing can be exactly described with
functions and numbers built up in this way, as we know.  This does not
seem to bring into question the notion of computable, but it may well
imply some results about the complexity hierarchy.  I love the idea of a
device which computes the non computable.  I want this to exist, but I
don't think anything inside classical physics will do it.  To see some
work about computation with classical numbers, you could look at my web
site:  www.bath.ac.uk/~masdr
The results depend on number theoretic conjectures, either Schanuel's
conjecture, or the uniformity conjecture (which predicts the precision
needed to distinguish a non zero classical number from zero).
I have tried to  imagine fast computations in neural nets, but not
got anywhere.
Regards,
Daniel Richardson