On Fri, Oct 21, 2011 at 02:14:48PM +0200, Bruno Marchal wrote:
> >
> >So the histories, we're agreed, are uncountable in number, but OMs
> >(bundles of histories compatible with the "here and now") are surely
> >still countable.
> This is not obvious for me. For any to computational states which
> are in a sequel when emulated by some universal UM,there are
> infinitely many UMs, including one dovetailing on the reals, leading
> to intermediate states. So I think that the "computational
> neighborhoods" are a priori uncoutable. 

Apriori, no. The UMs dovetailing on the reals will have only executed
a finite number of steps, and read a finite number of bits for a given
OM. There are only a countable number of distinct UM states making up
the OM.

> That fits with the
> topological semantics of the first person logics (S4Grz, S4Grz1, X,
> X*, X1, X1*). But many math problems are unsolved there.

You will need to expand on this. I don't know what you mean.

> >
> >If we take the no information ensemble,
> You might recall what you mean by this exactly.

It is the set of all infinite binary strings (isomorphic to [0,1)
). It is described in my book. Equation (2.1) of my book (which is a
variant of Ray Solomonoff's "beautiful formula"
http://world.std.com/~rjs/index.html) gives a value of precisely zero
for the information content of this set.

I do still think the universal dovetailer trace, UD*, is equivalent to
this set, but part of this thread is to understand why you might think

> >and transform it by applying a
> >universal turing machine and collect just the countable output string
> >where the machine halts, then apply another observer function that
> >also happens to be a UTM, the final result will still be a
> >Solomonoff-Levin distribution over the OMs.
> This is a bit unclear to me. Solomonof-Levin distribution are very
> nice, they are machine/theory independent, and that is quite in the
> spirit of comp, but it seems to be usable only in ASSA type
> approach. I do not exclude this can help for providing a role to
> little program, but I don't see at all how it could help for the
> computation of the first person indeterminacy, aka the derivation of
> physics from computer science needed when we assume comp in
> cognitive science. In the work using Solomonof-Levin, the mind-body
> problem is still under the rug. They don't seem aware of the
> first/third person description.

Not even if the reference machine is the observer erself? This would
seem to be applying S-L theory to the first person description. I
think I might be the only person to suggest doing this, though, which
I first did in my "Why Occam's razor" paper. I'm not sure, because
Marcus Hutter suggested something similar in a recent paper (quite
independently of me, it appears).

> >This result follows from
> >the compiler theorem - composition of a UTM with another one is still
> >a UTM.
> >
> >So even if there is a rich structure to the OMs caused by them being
> >generated in a UD, that structure will be lost in the process of
> >observation. The net effect is that UD* is just as much a "veil" on
> >the ultimate ontology as is the no information ensemble.
> UD*, or sigma_1 arithmetic,  can be seen as an effective
> (mechanically defined) definition of a zero information. It is the
> everything for the computational approach, but it is tiny compared
> to the first person view of it by internal observers accounted in
> the limit by the UD.

But isn't first person view of the UD given by a slice of UD*?


Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au

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