On 23 Oct 2011, at 04:41, Russell Standish wrote:

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.

The 3-OM. But the first person indeterminacy depends on all the (infinite) computations going through all possible intermediary 3-OMs states.

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.

I have explained this to Stephen a long time ago, when explaining why the work of Pratt, although very interesting fails to address the comp mind body problem. Basically Pratt's duality is recover by the "duality" between Bp (G) and Bp & Dt (Z1*) or Bp & Dt & p (X1*). You might serach what I said by looking at Pratt in the archive, with some luck.

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,

How? UD* structure relies on computer science, and give a non random countable sets, or strings. The set of binary strings is the set of reals, and it appears in UD*, but only from a first person views, with the real playing the role of oracles.

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?

What do you mean by the reference machine? What is an observer? How would S-L distribution be applied to the first person expectancy?

This would
seem to be applying S-L theory to the first person description.

How will you avoid huge programs accessing your current states.
It might work if we were able to justify why little programs multiply much more observer's state than huge programs, but I doubt S-L could explain this. Any idea?

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*?

UD* is a countable structure, but the math of the first person involves a continuum, so I doubt it can be a slice of UD*. This makes the measure problem very difficult, and that is why I tackle it by the self-reference modal logic, which gives the complete math of the propositional logic of observation (together with belief, knowledge, feelings, etc.). If such logics behaves well, as they should if comp is true, the whole physics can be extracted with a complete bypassing of the measure problem. In a sense the comp-physics is the solution of the measure problem, in that approach. We have already that the bottom of the physical reality behaves symmetrically and linearly. It harder to derive the Hamiltonian reality, and may be here could the S-L provides some help (but this would make the Hamiltonians more geographical than physical). The other reason to use the self-reference logics is that it distinguish automatically the quanta (sharable, communicable at least in a first person plural way) from the qualia (not sharable, purely individual), all this by the Gödel-Löb-Solovay proof/truth splitting of the modal logics.



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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