Re: everything priors

From: Juergen Schmidhuber <juergen.domain.name.hidden>
Date: Thu, 4 Nov 1999 09:30:23 +0100

Hi Max,

> I agree completely with your point that
> "All Universes are Cheaper Than Just One", and you'll find that
> echoed also in my older paper "Does the Universe in fact contain almost no
> information" (at www.sns.ias.edu/~max/nihilo.html).

I read this one, too! But there you focus on our own particular universe
and its variants based on different wave function collapses. In your
1998 paper, however, you do make the next step: there you further reduce
information content by considering all possible universes, not just
those based on our own's peculiar wave function. I guess that's what
makes the paper interesting to me.

> To me, the interesting difference is that your starting point is
> (universal) computer programs whereas mine is mathematical structures.
> I think the latter are harder to equip with a prior, but I certainly have
> nothing againt some form of complexity-based prior as long as it can
> be justified in some natural way.

Indeed, one of the nicest aspects of the UTM approach are the natural
priors it provides for predictions based on Bayes' rule.

> So here are some questions for you: 1) Has is been proven that such
> an f can compute anything that a TM can? I'm assuming that someone
> has proven that the two are in fact equally powerful/universal.

Well, before Turing came along Goedel already essentially built a Turing
machine from number theory, encoding all computations in integers.
In this sense number theory encompasses UTMs. On the other hand, UTMs
encompass number theory.

>2) If so, should we really limit ourself to this particular kind
>of mathematical structures? My concern is that we may be a bit too
>narrow-minded if we do.

But this sort of narrow-mindedness seems necessary to remain within the
formally describable realm. I'd go beyond computable structures only
if forced by evidence, e.g., if someone shows our universe somehow won't
run without all the real numbers. But for now there isn't any evidence
in this vein.

>For instance, this would automatically give our world a causal
>one-dimensional (discrete) time, even though we know that general
>relativity is perfectly consistent with having more than one
>time-dimension. My concern is that we're limiting ourselves to such
>"1-dimensional" computations simply because our world happens to have
>one time-dimension.

Causality in the computational sense is unavoidable with the UTM
approach. But this purely logical constraint still allows for describing
or simulating as many dimensions (and time dimensions) as you want. It's
all right as long there is a computer program for it, as long as it's
formally describable.

Cheers,
Juergen
Received on Thu Nov 04 1999 - 00:59:07 PST