>
> This thread has, IMHO, gotten more and more confused.
>
> Russell Standish wrote:
> >
> > The first thing I'd like to add, is that I don't believe we're
> > philosophically opposed to each other. There must be some kind of
> > misunderstanding on one of our parts - either of what each other is
> > saying, or of the problem we're trying to solve. It'd be great for
> > someone else with an external perspective to butt in and tell us where
> > we might be going wrong.
>
> I'll try.
>
> > 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.
>
> Let's start with this. The "Schmidhuber plenitude" is the universal
> program, computing all possible *computable* universes. This is an
> entirely different scheme than Tegmark's, which is based on formal
> logic. So far, only vague analogies have been offered to link the
> formal systems of Tegmark, which describe "mathematical structures",
> to the bitstrings corresponding to a state of a particular universe
> in Schmidhuber's scheme. Unless you can make this analogy more
> explicit, I think we're all better served by emphasizing the
> distinction.
They're not vague analogies. There are two ways to link the two
plenitudes, as I had pointed out before, but will try to make more
explicit now. Each member of the Tegmark plenitude is characterised by
a finite set of axioms (given as a symbol strings), and a finite set
of transition rules (the rules of logic). These may be encoded as a
Schmidhuber bitstring as an interpreter describing how to generate all
theorems given the axiom set, and the axiom set itself. It doesn't
matter what encoding is chosen, the compiler theorem ensures that
everything is self consistent. As in my previous post this morning,
this induces a Universal measure on the Tegmark plenitude.
Now the other link is that a UTM operating on bitstrings is in itself
a self-consistent mathematical system.
On the surface, these two links would imply that each plenitude is a
proper subset of each other. On the surface...
My own conviction is that the former way of linking the two systems is
the relevant one, and that the two systems aren't in fact
isomorphic. (ie that Tegmark \subset Schmidhuber). However, strong
that conviction is though, I am of course amenable to reasonable
argument. I also believe that the Tegmark plenitude is the important
"core" of the Schmidhuber from an anthropic perspective - ie
Schmidhuber universes containing SASes will be in most cases
arbitrarily close to the Tegmark one. Sorry for thowing in another
analogy, but its a bit like the Feynman path integral - the measure of
quantum trajectories will peak around the classical trajectory - in
this case the anthropic universes will peak around the Tegmark ensemble.
>
>
> > 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.
>
> Very incorrect. In Tegmark's scheme, you explicitly exclude non well-
> formed formulas. Alistair has emphasized this point quite often, yet
> you keep getting it confused, Russell. As an example, would you say
> that the following sentence is true or false?
>
> jfkduudnneeonnv.c.xdjnaklq..w]=-9u
>
> It's *neither*! It doesn't *mean* anything!
>
> In Tegmark's scheme, you consider only well-formed formulas, and then
> by assigning certain ones as axioms, and certain rules for deriving
> theorems, you separate them into bins for "true" and "false". In that
> way, you then describe a mathematical model.
>
I'm not getting confused on this matter, and I agree perfectly with
what you say here, except for the two words "very incorrect". My
utterings must somehow be confusing to other people in order fo you to
come to this conclusion.
Of course non-wff's do not exist in the Tegmark plenitude. However
they do exist in the Schmidhuber one, in the sense of the embedding I
discussed above. And they are perfectly well defined entities, and
need to be considered.
Another analogy (sorry for this) - it is almost like you are saying
that only rational numbers exist, and we don't need to worry about
things in between like sqrt(2).
>
> > 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.
>
> I can't make heads or tails of this.
>
>
> > 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. Contrariwise,
> > the UTM structure, interpreting bitstrings is but one such
> > self-consitent mathematical structure, so the Schmidhuber ensemble is
> > contained with the Tegmark one. So are the two plenitudes equivalent?
> > If not, then why not? Is it something to do with noncomputable
> > mathematical objects, such as noncomputable real numbers?
>
> I agree that there are interesting similarities, but as I mentioned
> above, I think there's more danger in carrying the analogy too far.
> They are fundamentally different, and I'd wager that they are not
> isomorphic, and that therefore, one is incorrect.
>
>
> > >
> > > ----- Original Message -----
> > > From: Russell Standish <R.Standish.domain.name.hidden>
> > > > I agree, that if you restrict your attention to wffs only, then there
> > > > is no problem. It is also no surprise. However, I don't believe one
> > > > can solve the White Rabbit problem by defining it away.
> > >
> > > It is most certainly NOT just being 'defined away'.
> > >
> > > The whole point is that we are confining ourselves to *logically consistent*
> > > universes (and perhaps other entities), that is, universes with some kind of
> > > logical structure. The only assumption in my analysis is that these must be
> > > definable with some kind of formal system (I don't even assume that it would
> > > be contained within fig 1 of Tegmark's table). A non-wff is just a
> > > meaningless string of symbols - we can't even *consider* any logically
> > > consistent universe without at least a wff-string. Once we have the
> > > wff-string, then we can start to build towards defining logically consistent
> > > universes (via axiom sets, *consistent* theories, metric spaces (probably),
> > > and so on). It is *then* that the analysis of axioms comes into play (see my
> > > original post), and flying rabbits are banished to a very small minority of
> > > universes.
> > >
> > > 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.
> >
> > Perhaps
> > > if I add some comments to an earlier post of yours, it may help to clarify
> > > the wider issue, but remember that what follows below is mainly based on our
> > > earlier scheme and not the same as one described at the start of this
> > > thread.
> > >
> > > > I don't think anyone said they don't exist. Its the measure of the
> > > > worlds that contain that is interesting. There are two possibilities:
> > > >
> > > > 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. It may be more
> > problematic if your ensemble of mathematical systems is not a proper
> > subset of the Schmidhuber plenitude.
> >
> > > > b) The dragon universe corresponds to one of the "nonsense
> > > > bitstrings", which we know to vastly outnumber the lawlike ones. In
> > > > this case, we must apply the argument I worked out with Alistair
> > > > Malcolm to show that indeed these dragon universes are of small
> > > > measure compared with lawlike universes, and that the remaining
> > > > "nonsense" universes are like noise - unobservable.
> > >
> > > If the bits are interpretable as anything, they would normally be
> > > interpretable mathematically (or at least as part of some formal system,
> > > otherwise we would be ranging beyond the formality of information theory
> > > itself), in which case what they represent will be coherent, but not in
> > > general visible to us (usually because what is represented will be in
> > > another universe) - again the general argument we derived holds as required
> > > for their small measure. For totally nonsense bitstings, or for any parts of
> > > bitstrings which are non-interpretable - these are ontologically
> > > problematical (as are non-wffs); I think that we have to remember that we
> > > are just trying to find generic schemes that can subsume all logically
> > > possible universes - the particular set of building blocks used to ground
> > > each scheme ought not to affect the relative measures of various types of
> > > universes.
> >
> > No problems with the above.
> >
> > Neither non-wffs nor non-interpretable bits can represent
> > > anything, and to include them in any analysis is giving them unwarranted
> > > ontological credibility. Fortunately neither are required to ascribe low
> > > measure to dragon universes.
> >
> > However, they exist in the Schmidhuber "all bitstrings" version of the
> > plenitude.
> >
> > >
> > > Alastair
> > >
> > >
> > >
> > >
> >
> > ----------------------------------------------------------------------------
> > 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
> > ----------------------------------------------------------------------------
>
> --
> Chris Maloney
> http://www.chrismaloney.com
>
> "Donuts are so sweet and tasty."
> -- Homer Simpson
>
>
----------------------------------------------------------------------------
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
----------------------------------------------------------------------------
Received on Tue Nov 09 1999 - 17:17:33 PST