On Wed, Sep 12, 2007 at 07:32:32AM -0700, Youness Ayaita wrote:
> 
> The two concerns, how to give a precise notion of the Everything, and
> how to deduce predictions from a chosen notion, lie at the very heart
> of our common efforts. Though, I did not go into them for the simple
> reason that I wanted to avoid discussions that are not directly linked
> to the topic.
> 
> 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. I assume that the no-justification naturally leads to this
> ensemble. This comes from the development of the (degenerate) property
> of existence which is then assigned to all imaginable things. I don't
> think that a metaphysical discussion of the term "imaginable thing" is
> necessary now, I'm satisfied with the idea that an imaginable thing
> can be completely described by means of language. For further
> research, it is then natural to identify imaginable things with their
> descriptions and to choose a simple alphabet for expressing the
> descriptions (e.g. strings of 0 and 1). In the past I assumed these
> strings to be of finite length. I read that Russell Standish also
> permits infinite strings.
> 

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.

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.

Schmidhuber has a different measure, assuming that the strings are
generated in real time from a machine with bounded resources. This is
his "speed prior", and leads to a quite different measure on the
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!).

On your comment on permitting infinite strings - the ensemble I
describe in my book has only infinite strings, which belong to
syntactic space. A finite string corresponds to a set of infinite
strings all having the same finite prefix, and as such belongs to
semantic space. 


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.

-- 

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