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