basic questions

From: Wei Dai <weidai.domain.name.hidden>
Date: Mon, 19 Jan 1998 21:03:07 -0800

Before we start talking about philosophical implications, maybe we should
reach a consensus about more basic questions.

What is the set of all possible universes? Max Tegmark says its the set of
all mathematical structures, and Juergen Schmidhuber says its the set of
all Turing machines, but neither gives much justification. I tend to agree
with Schmidhuber, if only because Tegmark's definition does not seem to
lead to an effective theory. For example, what does a uniform distribution
on all mathematical structures mean? However it would be nice to have some
stronger justifications for assuming that only computable universes exist.

If we agree that all Turing machines exist, it's still not clear how they
should be interpreted as physical universes. Schmidhuber suggests that the
output of a Turing machine should be interpreted as the evolution of a
universe. However this is problematic because it does not lead to an easy
way to think about the complexities and probabilities of structures within
a universe. For example, consider a simple Turing machine that enumerates
the natural numbers. The output of this TM includes every possible
configuration of every other universe. What is the probability that I'm
living in this TM? I don't see a straightforward way to answer this
question under Schmidhuber's interpretation. The problem is that even
though this universe is very simple, it would take a lot of extra
information to find anything in it, so the simplicity of the universe as a
whole is deceptive.

As an alternative, I suggest that each Turing machine should be thought of
as taking the coodinates of a region of a universe as input and producing
the content of that region as output. Here region should be broadly
interpreted. It not only refers to volumes of space-time, but may also for
example specify a branch of a quantum superposition.

Let me clarify some terminology. When I say length of a region of a
universe, I mean the length of the Turing machine plus the length of input
specifying the extent of that region. When I say complexity of the content
of region R, I mean the least length of all regions with the same content
as R, which is just the content's Kolmogorov complexity.

To compute the probability that I'm in a particular universe U, I would
first find the universal prior probability PU(m) of my mind state. (I
assume my mind state, which includes my memories and current perceptions,
is the only information I have direct access to) Then I would find the
region R in U that has my mind state as the content (for simplicity let's
assume there is exactly one such region in U). Finally the probability
that I'm in universe U is 2^-l(R)/P, where l(R) is the length of region R.

The Baysian interpretation of the above procedure is that before taking
into account my mind state, the prior probability that I'm any region R is
2^-l(R). After taking into account my mind state, I eliminate all regions
that do not have my mind state as the content, so the posterior
probability is 2^-l(R)/PU(m) if the content of R is m, and 0 if the
content of R is not m.
Received on Mon Jan 19 1998 - 21:03:07 PST