From: Vladimir Nesov (firstname.lastname@example.org)
Date: Wed Jan 23 2008 - 15:18:33 MST
On Jan 23, 2008 11:52 PM, Matt Mahoney <email@example.com> wrote:
> An example of uncomputable phenomena would be something like classical
> mechanics, in which the outcome of an experiment requires knowledge of the
> position and velocity of particles with infinite precision.
Again: how would you test that?
> > Finite state machine can perfectly well simulate itself, in any
> > natural interpretation that comes to mind (you'd have to additionally
> > define what it means for this formal construct to have a simulation of
> > something).
> A finite state machine with n states cannot model a machine with more than n
Not every machine, but who needs that? And again, what do you mean by modeling?
> > > 2. The universe has finite entropy. It has finite age T, finite size
> > limited
> > > by the speed of light c, finite mass limited by G, and finite resolution
> > > limited by Planck's constant h. Its quantum state can be described in
> > roughly
> > > (c^5)(T^2)/hG ~ 2^404.6 ~ 10^122 bits. (By coincidence, if the universe
> > is
> > > divided into 10^122 parts, then one bit is the size of the smallest stable
> > > particle, even though T, c, h, and G do not depend on the properties of
> > any
> > > particles).
> > So? If anything, it supports knowability of universe, a counterpart of
> > it being simulated from complex unobservable environment.
> Yes, that is my point.
Well, I meant 'counterpart' as in 'opposite'. Problem with simulated
worlds is (supposedly) that complex unpredicatable miracles can
happen. If everything is simple and observable, what is the problem?
'Simulatedness' is not observable and in itself is a meaningless
> > > 3. Occam's Razor is observed in practice. It is predicted by AIXI if the
> > > universe has a computable probability distribution.
> > If you could please finally define what you mean by that. Occam's
> > razor rule corresponds to good choice of notation/representation,
> > which is usually picked to be compressible given distribution of
> > described domain. What plays a role of notation in your argument, and
> > why does its choice signify anything else?
> Hutter proved that the optimal behavior of a reward seeking agent in an
> unknown environment simulated by a pair of interacting Turing machines is to
> guess at each step that the environment is simulated by the shortest program
> consistent with observation so far. Occam's Razor is an example of this
> strategy. The proof is valid only for the case where the environment has a
> computable probability distribution. Of course this is not a proof that the
> environment is simulated, but if Occam's Razor did not work in practice, it
> would be strong evidence that the universe is not simulated.
How would you test if your notion of Occam's razor didn't work?
> > > 4. The simplest algorithm (and by AIXI, the most likely) for modeling the
> > > universe is to enumerate all Turing machines until a universe supporting
> > > intelligent life is found. The most efficient way to execute this
> > algorithm
> > > is to run each machine with complexity n for 2^n steps. We observe that
> > the
> > > complexity of physics (the free parameters in the Standard Model or most
> > > string theories, plus general relativity) is on the order of n = a few
> > > hundred bits, which is the log of its entropy.
> > Complexity is mostly in random content, so I don't see how you move
> > from simulation of universe of given complexity to complexity of
> > physical laws. Physical laws make up a tiniest part of complexity of
> > the world.
> The fastest way to find a universe supporting intelligent life is run the k'th
> universe for k steps. I claim that for our universe, k ~ 10^122.
But how does it relate to complexity of laws of physics which are much simpler?
-- Vladimir Nesov mailto:firstname.lastname@example.org
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