# Fwd: White Rabbits, Measure and Max

From: <GSLevy.domain.name.hidden>
Date: Wed, 14 Jul 1999 01:00:32 EDT

Hal's and Russell's posts are interesting. The minimum information principle
is equivalent to the well known maximum entropy principle or equivalently to
the second law of thermodynamics. I wonder how far we can take this.

I believe that these are some of the facts that we have established:

- The whole MW has zero information and therefore maximum (infinite) entropy.
- The self sampling process raises the information level and lowers the
entropy of the observable worlds.
- Small worlds are more numerous but cannot support consciousness.
- Large worlds can support consciousness but are exponentially less numerous.

I believe that the anthropic principle can tie all these facts very well in
defining the size of universes which can support consciousness.

George

In a message dated 99-07-13 22:21:26 EDT, R.Standish.domain.name.hidden writes:

<<
I was thinking some more on the physical laws issue, and on Wei Dai's
point on what measure to apply to different universes. The solution
given is to weight shorter strings over longer ones.

strings at all in the everything world. Instead, the first n bits of a
string contain information, and the remainder are "don't care"
values. We also assume a uniform measure on all these infinite
strings. In this picture, worlds who are entirely specified by strings
with a smaller value of n will have higher measure than those with
larger n. The measure will fall off exponentially with n, in fact
precisely 2^{-n}, assuming the uniform measure above. This, then is
the universal measure sought by Wei Dai. Clearly, the everything
universe consists of all strings where we don't care what any bit
is. These have zero information, and measure = 1.

So, by the SSA (strong, weak, relative - doesn't matter in this case),
we should expect a Universe with low Kolmogorov complexity, but high
logical depth, which in essence says we should expect simple physical
laws, and a relatively long evolution producing concious entities. We
could call this the "minimum information principle".

In the Tegmark picture, he discusses mathematical structures, which
clearly should be preferred by the minimum information principle over
unstructured worlds. We should also expect to observe the most general
mathematical structure capable of supporting consciousness, as greater
generality => fewer axioms => less information. I have given arguments
in earlier posts why complex valued vector space with unitary
evolution operators (ie quantum mechanics as we know it) should be the
most general such mathematical structure, although at the time I didn't
know why the most general such structure would be preferred.

I disagree (for the moment at least) that the everything universes
should be restricted to programs on a universal turing machine. I
believe that it is information that is at the heart of the matter, and
that one can have information without computation (I'm happy to be
proven wrong if that is the case). However, the Schmidhueber picture
is at very minimum a useful metaphor, even if it turns out not to be
the whole story. >>

attached mail follows:

I was thinking some more on the physical laws issue, and on Wei Dai's
point on what measure to apply to different universes. The solution
given is to weight shorter strings over longer ones.

strings at all in the everything world. Instead, the first n bits of a
string contain information, and the remainder are "don't care"
values. We also assume a uniform measure on all these infinite
strings. In this picture, worlds who are entirely specified by strings
with a smaller value of n will have higher measure than those with
larger n. The measure will fall off exponentially with n, in fact
precisely 2^{-n}, assuming the uniform measure above. This, then is
the universal measure sought by Wei Dai. Clearly, the everything
universe consists of all strings where we don't care what any bit
is. These have zero information, and measure = 1.

So, by the SSA (strong, weak, relative - doesn't matter in this case),
we should expect a Universe with low Kolmogorov complexity, but high
logical depth, which in essence says we should expect simple physical
laws, and a relatively long evolution producing concious entities. We
could call this the "minimum information principle".

In the Tegmark picture, he discusses mathematical structures, which
clearly should be preferred by the minimum information principle over
unstructured worlds. We should also expect to observe the most general
mathematical structure capable of supporting consciousness, as greater
generality => fewer axioms => less information. I have given arguments
in earlier posts why complex valued vector space with unitary
evolution operators (ie quantum mechanics as we know it) should be the
most general such mathematical structure, although at the time I didn't
know why the most general such structure would be preferred.

I disagree (for the moment at least) that the everything universes
should be restricted to programs on a universal turing machine. I
believe that it is information that is at the heart of the matter, and
that one can have information without computation (I'm happy to be
proven wrong if that is the case). However, the Schmidhueber picture
is at very minimum a useful metaphor, even if it turns out not to be
the whole story.

Cheers

----------------------------------------------------------------------------
Dr. Russell Standish Director
High Performance Computing Support Unit,
University of NSW Phone 9385 6967
Sydney 2052 Fax 9385 7123
Australia R.Standish.domain.name.hidden
Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks
----------------------------------------------------------------------------
Received on Tue Jul 13 1999 - 23:06:17 PDT

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