unlikely universes?

From: Juergen Schmidhuber <juergen.domain.name.hidden>
Date: Thu, 26 Mar 1998 10:40:40 +0100

Several previous contributions to this list dealt with interesting
questions such as: What is the probability that I'm living in this
universe? What's a reasonable prior over universes? Are some of my
choices more probable than others? Will the universe shortly cease
to behave lawfully? Here I'll try to rephrase a few related issues
mentioned in my little paper. I apologize for the delayed response
(due to heavy workload).

Sequential dove-tailing of the computable universes suggests interesting
priors. E.g., the first universe-computing algorithm (in a binary
alphabetical ordering) is run for one instruction every second step,
the next for one instruction every second of the remaining steps, and so
on. Several algorithms may compute the same universe. Assume we randomly
point at a step in a large interval of steps. What is the probability
of ending up in a particular universe U? Most of this probability is
due to the earliest algorithm computing U (this algorithm will be among
the shortest). E.g., a universe in which lambs start attacking lions at
some point seems less probable than ours, because apparently it requires
more information than conveyed by the few physical laws causing "normal"
animal evolution. Hence it will appear later in the list.

Assume we know our universe's history so far could have been generated
by a short algorithm. Given priors like the above, it is very likely
that its algorithm is indeed short. We conclude it is unlikely that
it will start behaving unlawfully, because unlawfulness by definition
implies a long shortest algorithm.

All of this is closely related to the universal prior (UP) mentioned
in the paper. The UP is called UP because it dominates all other
semi-computable priors. It is based on guessing a new bit whenever
the proceeding computation of a particular universe requests one. Many
self-delimiting algorithms cease requesting new bits at some point, some
without halting, e.g., by entering into a loop. They represent infinite
but compressible, regular universes. Again, according to Levin's coding
theorems, under the UP most of the probability of some universe is due
to its shortest algorithms.

Hope this is not too redundant.

Juergen Schmidhuber www.idsia.ch
Received on Thu Mar 26 1998 - 04:34:39 PST