Wei Dai wrote:
> 
> On Tue, Jan 06, 2004 at 05:32:05PM +0100, Georges Quenot wrote:
> > Many other way of simulating the universe could be considered like
> > for instance a 4D mesh (if we simplify by considering only general
> > relativity; there is no reason for the approach not being possible in
> > an even more general way) representing a universe taken as a whole
> > in its spatio-temporal aspect. The mesh would be refined at each
> > iteration. The relation between the time in the computer and the time
> > in the universe would not be a synchrony but a refinement of the
> > resolution of the time (and space) in the simulated universe as the
> > time in the computer increases.
> >
> > Alternatively (though both views are not necessarily exclusive), one
> > could use a variational formulation instead of a partial derivative
> > formulation in order to describe/build the universe leading again to
> > a construction in which the time in the computer is not related at
> > all to the time in the simulated universe.
> 
> Do you have references for these two ideas?

No. They actually came to me while I was figuring some other
ways of simulating a universe than the sequential one that seemed
to give rise to many problems to me. The second one is influenced
by the prossibility to consider the whole universe within a
variational formulation as suggested by Hawking in "A brief
history of time" where he also considered the possibility of a
boundaryless universe (that makes much sense to me) that would
make difficult the use of any (initial or other) boundary condition.
Among other problems are the one of defining a global time within
a universe ruled by general relativity and including time
singularities within black holes for instance. Last but not
least is the problem of the emergence of the flow of time itself
from the gradient of order within the universe.

There might be references which I do not know of and I would say
probably for the case of simple physics (possibly fluid dynamics
or heat transfer for instance) phenomena which could be simulated
in a 3+1 (or 2+1 or 1+1) dimensional meshes as wholes.

I think the refining mesh could be practically experimented in a
1D+1D and possibly up to 2D+1D for heat conduction within a solid
object with various boundary conditions. While it could much less
efficient (but is it even so obvious ?) than a sequential approach,
implementing a finite element mesh including the time dimension and
solving the partial derivative heat diffusion equations by standard
linear algebra on the whole spatio-temporal domain seems perfectly
feasable to me (at least for small amounts of time).

> I'm wondering, suppose the
> universe you're trying to simulate contains a computer that is running a
> factoring algorithm on a large number, in order to cryptanalyze somebody's
> RSA public key. How could you possibly simulate this universe without
> starting from the beginning and working forward in time? Whatever
> simulation method you use, if somebody was watching the simulation run,
> they'd see the input to the factoring algorithm appear before the output,
> right?

I would say there is a strong anthropomorphic bias in this view.
I suggest you to read my other posts in which I comment a bit
about this kind of things.

Indeed, the practical implementation of the simulation of the
whole universe including the considered computer would be very
heavy if a variational formulation and/or a 4D iteratively
refining mesh had to be used. But I do not see why it should fail
to simulate the computer calculation. What is very difficult is
to guarantee that all interactions propagate at the appropriate
level of accuracy through all of the 4D mesh and/or through all
of the action paths which can be very large and interconnected.
No doubt that "close (up to an unimaginable level) to singular"
matrices will be encountered. But is this very different if one
is to simulate the universe from the big bang up to this
computer calculation with the appropriate accuracy needed to
ensure that from the big bang initial conditions through stellar
formation and human evolution this computer would be built and
would run this particular calculation ? I am not so sure.

I do not believe in either case that a simulation with this level
of detail can be conducted on any computer that can be built in
our universe (I mean a computer able to simulate a universe
containing a smaller computer doing the calculation you considered
with a level of accuracy sufficient to ensure that the simulation
of the behavior of the smaller computer would be meaningful).
This is only a theoretical speculation.

Georges Quénot.

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