On 10/22/2011 10:41 PM, 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.
[SPK]
   Hi Russell,

Does this countable number of distinct UM states account for all possible versions of the OM? How do we deal with the set of transformations of the OM that represent rotations, dilations, translations, reflections and boosts? These form a set of smooth functions that imply a continuum, no? Or are you considering each and every POV to be a OM where the moment (duration of time) is an irreducible unit?

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.
[SPK]
It would be wonderful to have an easily accessible glossary/wiki of these terms. I have a hard time keeping track of them myself.



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.
[SPK]
Is it possible that the zero value of the information content of the set is an indication of the inability to distinguish proper subsets of the set? I am thinking of information in the "differences that make a difference" sense...

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
otherwise.
[SPK]
I am not sure that I have a good verbal/visual handle on what a trace is. :-( Is it like the trace of a matrix?



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).
[SPK]
Is "the observer erself" that you are considering here the generator of the OM or the description/representation of the machine that Löb's theorem induces? It seems to me that the fixed point is the "self", but the "self" identifies/represents itself with the boundary of the set of objects over which transformations induce the fixed point. We humans do this when we identify the surface of our skin and all that it contains with our self and yet somehow still have the "sense of self ' as separate from that skin bag of ... .

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*?


[SPK]
    "Slice" analogous to a portrait of a phase space of a dynamical system?

Onward!

Stephen

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