real numbers and TMs

[Dr Michael G Dyer]

Albert Boulanger, you said:

There is a model of computation using real numbers
that some high-powered mathematicians have developed:

   "Blum L., M. Shub, and S. Smale, "On a Theory of Computation and
   Complexity Over the Real Numbers: NP Completeness, Recursive
   Functions and Universal Machines", Bull A.M.S. 21(1989): 1-49.

   It offers a model of computing using real numbers
   more powerful than a Turing Machine.
=======

But is the physics of our universe only modelable in terms of
real numbers?  e.g. is there actually an infinite amount of ever smaller space
between any two neighboring pieces of close-together space?
The quantum approach seems to say "no".  Also, while there may be
an infinite number of digits in a real number, for us to find out
that the universe requires reals would require us to spend an infinite
amount of time reading off the results of one of our measurements.
(Is this right?)
Consider a cellular automaton models, where the cell is the smallest
quatum of space itself.
In such models, there actually is no "motion".  Motion
is just an illusion -- the result of a similar looking pattern configurations
being reconstructed near the original pattern, in the next
state of the universe, based on the laws of how cell states interact.
I have not come upon any proof that our universe could
not be *some sort of* cellular system (perhaps with some bizarre topography
and bizarre, non-local "neighborhood" function).  In such a case,
(a) it would be Turing computable and (b) real numbers would be merely 
a useful fiction, used by the measurers locked within that universe, but a
fiction
nonetheless and they would never be able to harnass this "extra-TM" power
(other than in the sense that a TM attached to, say, a vehicle can do more, e.g.
it can move thru space , which a TM with just a tape could not, 
(unless we are talking
about THAT Turing machine simulating ITS vehicle on its own tape :-))

-- Michael Dyer