Re: Turing vs math

From: Juergen Schmidhuber <juergen.domain.name.hidden>
Date: Fri, 22 Oct 1999 09:46:27 +0200

Bruno:

> Honestly it is still not clear. How could ever "S(U)=TRUE" be computable ?
> As a computer scientist I guess you know that even the apparently simple
> question "does that piece of code computes the factorial function" is not
> computable.

Sure, it's not even decidable in general whether a given program will
produce any output at all. Once the output is there, though, you can
easily decide whether it consists of, say, prime numbers. In absence of
evidence to the contrary we assume that the presence of your consciousness
(whatever that may be) is detectable by a computable process, namely, the
one employed by you, the observer, who decides that he exists, via some
sort of computation taking place as part of the evolution of his universe.

George:

> Juergens seems to be talking as if the measure of information is absolute. It
> is not. Information is always conditional information (Shannon), where the
> condition is the observer's a-priori database of information.

Sure, but in the Kolmogorov framework the role of prior knowlwdge vanishes
in the limit. Each prior corresponds to a particular Turing machine
TM with particular inbuilt prior knowledge. But TM2 can simulate TM1
using a compiler whose size is independent of the information content
of the universes they compute. For all U, K1(U)=O(K2(U)), where K1(U)
and K2(U) are U's information contents relative to T1 and T2. Similarly
P1(U)= O(P2(U)), where P1 and P2 are the respective probabilities given
Solomonoff-Levin distributions for T1 and T2. There is a good reason why
Solomonoff-Levin distributions are called universal distributions!

Hal:

> We then invoke the principle that large-program universes are inherently
> less likely than small-program universes, and presto! we have it more
> likely that we live in a universe without flying rabbits, without
> magic, etc. That's the general argument we are striving to achieve.

I agree. That's precisely one of the major points of the 1997 paper.

> I do think that this argument has some problems, but it is appealing and
> if the holes can be filled it seems to offer an answer to the question.
> What do you think?

Where exactly are the holes?

Juergen
Received on Fri Oct 22 1999 - 00:49:22 PDT