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From: Juergen Schmidhuber <juergen.domain.name.hidden>

Date: Thu, 15 Nov 2001 10:35:58 +0100

Wei Dai wrote:

*>
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*> Thanks for clarifying the provability issue. I think I understand and
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*> agree with you.
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*>
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*> On Tue, Nov 13, 2001 at 12:05:22PM +0100, Juergen Schmidhuber wrote:
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*> > What about exploitation? Once you suspect you found the PRG you can use
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*> > it
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*> > to predict the future. Unfortunately the prediction will take enormous
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*> > time to stabilize, and you never can be sure it's finished.
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*> > So it's not very practical.
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*>
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*> By exploiting the fact that we're in an oracle universe I didn't mean
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*> using TMs to predict the oracle outputs. That is certainly impractical.
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*>
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*> There are a couple of things you could do though. One is to use some
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*> oracle outputs to predict other oracle outputs when the relationship
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*> between them is computable. The other, much more important, is to quickly
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*> solve arbitrarily hard computational problem using the oracles.
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*>
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*> > I prefer the additional resource assumptions reflected
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*> > by the Speed Prior. They make the oracle universes very unlikely, and
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*> > yield computable predictions.
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*>
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*> Why do you prefer the Speed Prior? Under the Speed Prior, oracle universes
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*> are not just very unlikely, they have probability 0, right? Suppose one
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*> day we actually find an oracle for the halting problem, or even just find
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*> out that there is more computing power in our universe than is needed to
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*> explain our intelligence. Would you then (1) give up the Speed Prior and
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*> adopt a more dominant prior, or (2) would you say that you've encountered
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*> an extremely unlikely event (i.e. more likely you're hallucinating)?
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*>
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*> If you answer (1) then why not adopt the more dominant prior now?
*

You are right in the sense that under the Speed Prior any complete

_infinite_ oracle universe has probability 0. On the other hand,

any _finite_ beginning of an oracle universe has nonvanishing

probability. Why? The fastest algorithm for computing all universes

computes _all_ finite beginnings of all universes. Now oracle universes

occasionally require effectively random bits. History beginnings that

require n such bits come out very slowly: the computation of the n-th

oracle bit requires more than O(2^n) steps, even by the fastest

algorithm.

Therefore, under the Speed Prior the probabilities of oracle-based

prefixes quickly converge towards zero with growing prefix size. But in

the finite case there always will remain some tiny epsilon.

Why not adopt a more dominant prior now? I just go for the simplest

explanation consistent with available data, where my simplicity measure

reflects what computer scientists find simple: things that are easy

to compute.

Juergen Schmidhuber

http://www.idsia.ch/~juergen/

http://www.idsia.ch/~juergen/everything/html.html

http://www.idsia.ch/~juergen/toesv2/

Received on Thu Nov 15 2001 - 01:36:54 PST

Date: Thu, 15 Nov 2001 10:35:58 +0100

Wei Dai wrote:

You are right in the sense that under the Speed Prior any complete

_infinite_ oracle universe has probability 0. On the other hand,

any _finite_ beginning of an oracle universe has nonvanishing

probability. Why? The fastest algorithm for computing all universes

computes _all_ finite beginnings of all universes. Now oracle universes

occasionally require effectively random bits. History beginnings that

require n such bits come out very slowly: the computation of the n-th

oracle bit requires more than O(2^n) steps, even by the fastest

algorithm.

Therefore, under the Speed Prior the probabilities of oracle-based

prefixes quickly converge towards zero with growing prefix size. But in

the finite case there always will remain some tiny epsilon.

Why not adopt a more dominant prior now? I just go for the simplest

explanation consistent with available data, where my simplicity measure

reflects what computer scientists find simple: things that are easy

to compute.

Juergen Schmidhuber

http://www.idsia.ch/~juergen/

http://www.idsia.ch/~juergen/everything/html.html

http://www.idsia.ch/~juergen/toesv2/

Received on Thu Nov 15 2001 - 01:36:54 PST

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