Re: on formally describable universes and measures

From: <juergen.domain.name.hidden>
Date: Mon, 18 Dec 2000 15:39:05 +0100

> From everything-list-request.domain.name.hidden Sun Dec 17 13:21:40 2000
> I just got around to reading Schmidhuber's new paper, and noticed there is
> something strange about the Speed Prior S. With all of the candidate
> priors we have seen so far, the probability of a random (incompressible)
> string of length n is about 2^-n. But with the Speed Prior S, the
> probability is about 2^-2n (unless I misunderstood something?). I think it
> might make sense to have a prior that favors strings that are fast to
> compute, but it certainly doesn't make sense that it also makes random
> strings much more unlikely than they have to be. And BTW, I think this is
> the reason that S predicts the universe is run by a pseudo-random number
> generator rather than a true random number generator. The other priors do
> not seem to make this prediction.

Not quite true - S just predicts the pseudorandom number generator is
also fast. Even the most dominant priors mu^G and mu^E (more dominant
than the enumerable priors studied in the classic work on Occam's
razor by Solomonoff and Levin and Gacs) cannot help assigning vanishing
probability to truly random universes - see Theorems 5.3-5.6. But mu^G
and mu^E do assign nonvanishing probability to certain pseudorandom
universes whose pseudorandomness is hard or even impossible to detect
because the algorithm for computing the pseudorandomness consumes
excessive time. Compare Example 2.1 and Section 7. The speed prior S,
however, must necessarily assign low probability to such hard-to-compute
universes. Roughly speaking, with S the collective probability of all
universes incomputable in phase i of the most efficient way of computing
all universes is discounted by 2^-i (Def 6.3). In particular, zero
probability is assigned to universes whose pseudorandomness consumes
more than countably many resources - from a computational point of view
uncountable resources do not make sense, hence infinite random strings
do not make sense either.

Juergen
Received on Mon Dec 18 2000 - 06:42:14 PST