On 21 Oct 2011, at 15:08, Stephen P. King wrote:
On 10/21/2011 8:14 AM, Bruno Marchal wrote:
On 19 Oct 2011, at 05:30, Russell Standish wrote:
On Mon, Oct 17, 2011 at 07:03:38PM +0200, Bruno Marchal wrote:
This, ISTM, is a completely different, and more wonderful beast,
than
the UD described in your Brussells thesis, or Schmidhuber's '97
paper. This latter beast must truly give rise to a continuum of
histories, due to the random oracles you were talking about.
All UDs do that. It is always the same beast.
On reflection, yes you're correct. The new algorithm you proposed is
more efficient than the previous one described in your thesis, as
machines are only executed once for each prefix, rather over and
over
again for each input having the same prefix. But in an environment
of
unbounded resources, such as we're considering here, that has no
import.
Note that my programs are not prefixed. They are all generated and
executed. To prefix them is usefulm when they are generated by a
random coin, which I do not need to do.
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. That fits with the
topological semantics of the first person logics (S4Grz, S4Grz1, X,
X*, X1, X1*). But many math problems are unsolved there.
Hi Bruno and Russel,
I would like to better understand what "topological semantics"
means. Are you considering relations defined only in set theoretical
sense, ala the closed or open or clopen nature of the sets relative
to each other?
I guess you know the topological semantics of intuitionist logic.
Instead of interpreting the propositions by sets in a boolean algebra,
you interpret it by open set in a topological space. You have
soundness and completeness theorem for that. I think this came from
semantics for the semantics for the modal logic S4, which mirrors well
intuitionist logic.
What about the form of the axiom of choice for the set theory?
?
How do you induce compactness?
Why do you want the space being compact. The point of the topological
semantics is that it works for all topological spaces, like any
boolean algebra can be used for classical logic.
How is a "space" defined in strictly arithmetic terms?
Why do you want to define it in arithmetic. With comp, arithmetic can
be used for the ontology, but the internal epistemology needs much
more. Remember that the tiny effective sigma_1 arithmetic already
emulate all Löbian machines like PA, ZF, etc. Numbers can use sets to
understand themselves.
If we take the no information ensemble,
You might recall what you mean by this exactly.
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.
S-L seems to assume 1p = 3p or no 1p at all!
I think they just ignore 1p.
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.
How do we define this notion of size?
There are many ways. Tiny in the sense of the ordering of sets of
provable proposition of arithmetic, for example. Robinson theory is
tiny compared to PA which is tiny compared to ZF, in that sense.
Tiny as opposed to ???
To big! Or strong. ZF proves much more arithmetical propositions than
PA.
Unless I'm missing something here.
Lets leave the discussion of the universal prior to another
post. In a
nutshell, though, no matter what prior distribution you put on
the "no
information" ensemble, an observer of that ensemble will always
see
the Solomonoff-Levin distribution, or universal prior.
I don't think it makes sense to use a universal prior. That would
make sense if we suppose there are computable universes, and if we
try to measure the probability we are in such structure. This is
typical of Schmidhuber's approach, which is still quite similar to
physicalism, where we conceive observers as belonging to computable
universes. Put in another way, this is typical of using some sort
of
identity thesis between a mind and a program.
I understand your point, but the concept of universal prior is of
far
more general applicability than Schmidhuber's model. There need
not be
any identity thesis invoked, as for example in applications such as
observers of Rorshach diagrams.
And as for identity thesis, you do have a type of identity thesis in
the statement that "brains make interaction with other observers
relatively more likely" (or something like that).
yes, by the duplication (multiplication) of populations of
observers, like in comp, but also like in Everett.
But we also need something that acts to code the "no preferred
reference frame" of GR. Everett does not solve this problem, it only
compounds it. :-(
I am not sure I follow you. Everett makes clear that the choice of
base is a false problem.
There has to be some form of identity thesis between brain and mind
that prevents the Occam catastrophe, and also prevent the full
retreat
into solipsism. I think it very much an open problem what that is.
This will depend on the degree of similarity between between
quantum mechanics and the comp physics, which is given entirely by
the (quantified) material hypostases (mainly the Z1* and X1*
logics). Open but well mathematically circumscribed problem.
From what I have studied so far, the (static) relationship
between QM and COMP physics is the relationship between Boolean
algebras and Orthocomplete lattices.
Let us hope. Even just that has not been proved. Open problem.
I think that a partial solution might require a weakening of the
definition of a countable model of arithmetic and application of the
Tennenbaum theorem (that would allow for a variational principle,
something like q-deformed theories where q is a measure of the
relative recursiveness of the model of the theory).
?
The idea is that every 1p would observe itself, in the Lob sense,
to be recursive.
?
The proof would require showing that a Lobian machine on a non-
standard model of arithmetic would *not* be able to "see" its non-
standardness and thus it would bet that only it is recursive, thus
it's Bp&p would be 1p and not 3p truth.
? (Bp&p) is the definition of 1-p in, well not really in arithmetic,
but in terms of arithmetic. We cannot define "p" (p is true) in any
effective theory.
Bruno
http://iridia.ulb.ac.be/~marchal/
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