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

Date: Mon, 18 Dec 2000 15:39:05 +0100

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

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