>
>
> ----- Original Message -----
> From: Russell Standish <R.Standish.domain.name.hidden>
> > 1) As you say yourself, a non-wff is just a meaningless string of
> > symbols, i.e. one of the bitstrings making up the Schmidhuber
> > plenitude. As pointed out, the Schmidhuber plenitude does seem to have
> > a natural measure defined on it through K-complexity and Turing
> > interpretability. The non-wff strings, as simple the K-incompressible,
> > or random ones, however, they are still formal entities, and so
> > contribute to the discussion of issues related to measure. I also
> > belief that they are interpretable as well, just as Rorschach plots
> > are interpretable, even when they have no information content at all.
> > The reason that they're interpretable, is that they are close to a wff
> > that has information.
>
> This appears to me to be some kind of conflation of (at least) the two
> schemes discussed earlier (my axiom-based one, and our earlier bit-based
> one) - it is certainly distinct from the basic axiom-based one that you have
> been criticising. Since this (axiom-based) scheme is not bit-based as it
> stands, the 'Schmidhuber plenitude' assumption does not apply to it, and so
> I can legitimately dismiss non-wffs as irrelevant to the dragon analysis as
> far as my scheme is concerned.
>
> However, let me follow your route (as best I can) for a little. I presume
> you must be choosing a particular interpretation of bits corresponding
> simply to symbols. Some such strings lead to wff's, theories, and, for a
> minority of these, (the specification of) universes. Others are meaningless
> strings (non-wffs). I think you would need to clarify:
> 1) How any *additional* interpretation of bits (over and above that leading
> to symbols) could lead to actual universes.
In the Schmidhuber plenitude, non-wff bitstrings are perfectly
reasonable universes. It is not clear whether they would have
self-aware substructures, and what the SASes would make of their universe.
> 2) What actual role (if any) TM's play.
I believe that we use TM's as a crutch to help our thinking, but that
they could be dispensed with an appropriate formalism (which hasn't
been properly worked out yet).
> 3) Whether bitstrings could be immediately interpreted as anything other
> than symbol strings.
You could interpret them as anything you like - UTM programs, axioms
of a mathematical theory, the works of Shakespeare. I'm not sure why
you are asking this though.
> 4) How K-complexity (if relevant) could be compatible with nonsense bits.
> (I must say I also have a severe problem with a bitstring corresponding to a
> non-wff being interpretable as anything other than the non-wff itself.)
>
The K-complexity of a nonsense string is equal to the length of the
string itself. This is the definition of randomness in the
Kolmogoraff-Chaitin theory.
> > 2) There is a curious duality here relating the Tegmark plenitude with
> > the Schmidhuber one, that I've not yet figured out. On one hand, all
> > consistent mathematical systems are based on a finite set of axioms,
> > and a finite set of transition rules for deriving theorems. These can
> > be encoded as a bitstring and a UTM, so all members of the Tegmark
> > ensemble must be contained within the Schmidhuber one.
>
> Note that many other systems (for example pre-programmed microchips, brains,
> other mathematical mappings) can implement transitional rules - there is
> nothing special about UTM sequential processors.
>
Only that a UTM can implement any set of transitional rules. This is
not necessarily true of the other entities you mention.
> > > There is no 'defining away' of the white rabbit problem. It sounds to me
> as
> > > if you are confusing white rabbit universes with logically inconsistent
> > > universes (the latter should not be able to exist in anyone's
> > > book).
> >
> > I guess this is the contentious point. Dreams appear to be examples of
> > logically inconsistent universes. I would explain them in terms of the
> > brain's capacity to interpret meaningless data streams generated in
> > the brain during sleep. I guess you would need to explain either that
> > dreams are in fact logically consistent, or that somehow this example
> > doesn't count.
>
> I can only think that we have fundamentally differing conceptions of what
> constitutes a logically inconsistent universe. For me, a logically
> inconsistent universe would be one where (P AND (NOT-P)) is true somewhere
> within it (and from which *any* statement is derivable); this would
> presumably have to fall out from some logically inconsistent theory. Unless
> we are straying beyond the realms of science here, dreams are quite
> comfortably compatible with physics (apart conceivably from the 'qualia'
> problem - which would apply at least as much to the waking state and
> shouldn't be relevant to logical consistency anyway). If physical laws can
> accommodate dreams there is not even physical inconsistency, let alone
> logical inconsistency. (If by chance you are referring to dream *content*,
> these can hardly be considered as universes, but dream images anyway are
> logically consistent, though dreamed flying rabbits obviously don't have to
> conform to our physical laws.)
>
OK - I was using the dream idea as analogy, not asserting that dreams
are real universes. My point was that human beings, and by somewhat
iffy generalisation all SASes will attempt to interpret random (or
nonsense) data (in the K-C sense), and will partially succeed, even at
the expense of logical consistency. I'm not convinced that the
universe has to be logically consistent, or even logical at all. (It
is, of course, a precondition of the Tegmark plenitude - but I'm not
satisfied that this is correct, or even the only way of constructing a
plenitude). Sorry, I have Douglas Adam's image here of God
disappearing in a puff of his own logic! :)
> > > > a) The dragon universe is the outcome of a mathematical
> > > > description. In the sense we use dragon here of being non-lawlike, we
> > > > may suppose that they are the outcome of a very complex mathematical
> > > > description. As we well know, the measure of such universes is much
> > > > smaller than the very lawlike universe we inhabit.
> > >
> > > We don't know this unless we can find a proof. In fact the analysis that
> we
> > > developed earlier can provide a strong indication of the small measure
> of
> > > dragon universes. (See my web pages.)
> >
> > This result follows directly from the universal measure or universal
> > prior employed in the Schmidhuber plenitude.
>
> If you are referring to our earlier derived explanation for the paucity of
> dragons, that was what I was referring to as well. If not, please elucidate.
There is an obvious mapping of the Tegmark plenitude into the
Schmidhuber one (axioms as bitstrings, transition rules as an
interpreter running in a UTM). To each member of the Tegmark plenitude
(ie a self-consistent mathematical system) corresponds a K-complexity
measure such that the greater the number of axioms, the greater the
K-complexity, and the smaller the measure. eg the set of Reals has
greater complexity than the set of complex numbers. Thus the
Schmidhuber plenitude induces a measure on the Tegmark plenitude. Now,
it may be possible to induce a measure on the Tegmark plenitude
without the scaffolding of the Schmidhuber one, but I sure as hell
don't know how to do it.
Now dragon universes will have a more complex mathematical
description, and hence have smaller measure than non-dragon
universes. []
This argument is completely different to the one we worked out in
July, but has much more in common with your "new argument".
>
> Thanks
>
> Alastair
>
>
>
>
>
>
>
>
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Dr. Russell Standish Director
High Performance Computing Support Unit,
University of NSW Phone 9385 6967
Sydney 2052 Fax 9385 6965
Australia R.Standish.domain.name.hidden
Room 2075, Red Centre
http://parallel.hpc.unsw.edu.au/rks
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Received on Tue Nov 09 1999 - 16:49:34 PST