On Thu, Sep 13, 2007 at 03:04:34PM +0200, Bruno Marchal wrote:
> Le 13-sept.-07, à 00:48, Russell Standish a écrit :
> > These sorts of discussions "No-justification", "Zero-information
> > principle", "All of mathematics" and Hal Ruhl's dualling All and
> > Nothing (or should that be "duelling") are really just motivators for
> > getting at the ensemble, which turns out remarkably to be the same in
> > each case - the set of 2^\aleph_0 infinite strings or histories.
> Once you fix a programming language or a universal machine, then I can 

You don't even need a universal machine. All you need is a mapping
from infinite strings to integers. And that can be given by the
observer, where the integers are an enumeration of the oberver's
possible interpretations.

> imagine how to *represent* an history by an infinite string. But then 
> you are using comp and you know the consequences. Unless like some 
> people (including Schmidhuber) you don't believe in the difference 
> between first and third person points of view.
> (Youness Ayaita wrote:
> > When I first wanted to capture mathematically the Everything, I tried
> > several mathematicalist approaches. But later, I prefered the
> > Everything ensemble that is also known here as the Schmidhuber
> > ensemble.
> Could you Youness, or Russell, give a definition of "Schmidhuber 
> ensemble", please.

The set of all infinite length strings in some chosen alphabet. 

> Also I still don't know if the "physical universe" is considered as an 
> ouptut of a program, or if it is associated to the running of a 
> program.)

No, it is considered to be the stable, sharable dream, as you
sometimes put it. It is the interpretation of the observer, but it
isn't arbitrary.

> Russell Standish wrote :
> > Where differences lie is in the measure attached to these strings. I
> > take each string to be of equal weight to any other, so that there are
> > twice the measure of strings satisfying 01* as 011*. This leads
> > naturally to a universal prior.
> I don't understand. If all infinite strings have the same measure, what 
> is the meaning of "universal prior"?

The universal prior is a measure on certain sets of strings.

> > Neither Bruno's nor Max's theories give a measure,
> > but remarkably the
> > Occam's razor theorem and White Rabbit result is fairly insensitive to
> > the measure chosen (so long as it's not too pathological!).
> I don't understand this either.

The measure induced by the process of observation is enough to turn a
uniform measure, which is wabbity into one that is not (universal
prior). If the ensemble measure chosen was less wabbity (eg
Schmidhuber's speed prior for instance), then the observer measure
will also be non-wabbity. It is hard to imagine a more wabbity
distribution than the uniform one, but perhaps a delta function on an
extremely wabbity string might do the trick. 

> > On your comment on permitting infinite strings - the ensemble I
> > describe in my book has only infinite strings, which belong to
> > syntactic space.
> ?

I explain syntactic and semantic spaces in my book - its better to
read that than to try to reproduce it here. These concepts are known
by other names microscopic/macroscopic, L_1/L_2 and so on, but
syntactic/semantic seemed to capture the concept best in the most generality.

> > It would be possible to construct an ensemble of purely finite strings
> > (all strings of length googol bits, say). This wouldn't satisfy the
> > zero information principle, or your no-justification, as you still
> > have the finite string size to justify (why googol and not googol+1,
> > for instance). I suspect the observable results would be
> > indistinguishable from the infinite string ensembles for large enough
> > string string size, however.
> Hmmm... I think that once we do care about the difference between 3-pov 
> and 1-pov, such difference (between ensemble of finite and infinite 
> strings) does become palpable (empirically), unless you take special 
> infinite set of arbitrarily long (but finite) strings, but then all 
> will depends on the chosen representations.

As they say in "Grease" - "Tell me more, tell me more...." I suspect
that it would only be detected empirically if your instruments were
accurate enough, which is why I chose a googol, rather than say a
hundred million (which Borges chose for his Library of Babel).

> Bruno
> http://iridia.ulb.ac.be/~marchal/

A/Prof Russell Standish                  Phone 0425 253119 (mobile)
UNSW SYDNEY 2052                         [EMAIL PROTECTED]
Australia                                http://www.hpcoders.com.au

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