On 10/25/2011 4:40 PM, Russell Standish wrote:
On Mon, Oct 24, 2011 at 04:08:38PM +0200, Bruno Marchal wrote:
On 23 Oct 2011, at 04:41, Russell Standish wrote:
On Fri, Oct 21, 2011 at 02:14:48PM +0200, Bruno Marchal wrote:
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.
Apriori, no. The UMs dovetailing on the reals will have only executed
a finite number of steps, and read a finite number of bits for a given
OM. There are only a countable number of distinct UM states making up
the OM.
The 3-OM. But the first person indeterminacy depends on all the
(infinite) computations going through all possible intermediary
3-OMs states.
So does the OM I'm referring to. Does that still make is a 3 OM?
That fits with the
topological semantics of the first person logics (S4Grz, S4Grz1, X,
X*, X1, X1*). But many math problems are unsolved there.
You will need to expand on this. I don't know what you mean.
I have explained this to Stephen a long time ago, when explaining
why the work of Pratt, although very interesting fails to address
the comp mind body problem. Basically Pratt's duality is recover by
the "duality" between Bp (G) and Bp& Dt (Z1*) or Bp& Dt& p (X1*).
You might serach what I said by looking at Pratt in the archive,
with some luck.
This is above my level of understanding at present. Hopefully, there
will be some quiet time soon to study this, as it sounds interesting!
Hi Russell and Bruno,,
I recommend that you read Steve Vickers' "Topology Via Logic"
first. Pratt's ideas are a bit more abstract.
If we take the no information ensemble,
You might recall what you mean by this exactly.
It is the set of all infinite binary strings (isomorphic to [0,1)
). It is described in my book. Equation (2.1) of my book (which is a
variant of Ray Solomonoff's "beautiful formula"
http://world.std.com/~rjs/index.html) gives a value of precisely zero
for the information content of this set.
I do still think the universal dovetailer trace, UD*, is equivalent to
this set,
How? UD* structure relies on computer science, and give a non random
countable sets, or strings. The set of binary strings is the set of
reals, and it appears in UD*, but only from a first person views,
with the real playing the role of oracles.
Exactly!
but part of this thread is to understand why you might think
otherwise.
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.
Not even if the reference machine is the observer erself?
What do you mean by the reference machine? What is an observer? How
would S-L distribution be applied to the first person expectancy?
The S-L distribution relies upon a universal machine for its
definition, called the reference machine.
Observer is exactly what you and I mean by it. The person with
subjective experience, attaching meaning to experiential data.
The observer map o is a map from data to meaning, the former being
strings of some alphabet (eg binary), the latter being a countable set
- can be modelled by the whole numbers N.
The S-L distribution arises naturally if you ask the question: "What
is the probability of a given meaning being attached to the data by an
observer if the data strings were distributed uniformly"
I think it probably still arises if the data strings were distributed
in other ways a priori - eg being the output of a universal machine
acting as an oracle, for instance. But I haven't sat down to work out
what the limits are to this. Presumably some priori distributions will
affect the final result.
Why does the distribution have to exist a priori? What if it
obtains from interactions of many machines? Looking at just one UTM wil
never show this.
This would
seem to be applying S-L theory to the first person description.
How will you avoid huge programs accessing your current states.
It might work if we were able to justify why little programs
multiply much more observer's state than huge programs, but I doubt
S-L could explain this. Any idea?
You don't avoid huge programs accessing your current states. They are
exponentially suppressed, AFAIC see.
But isn't first person view of the UD given by a slice of UD*?
UD* is a countable structure, but the math of the first person
involves a continuum, so I doubt it can be a slice of UD*.
Then we are completely lost by terminology. I thought the UD* was the
trace of the dovetailer, as seen from inside the dovetailer.
This
makes the measure problem very difficult, and that is why I tackle
it by the self-reference modal logic, which gives the complete math
of the propositional logic of observation (together with belief,
knowledge, feelings, etc.). If such logics behaves well, as they
should if comp is true, the whole physics can be extracted with a
complete bypassing of the measure problem.
That still seems a big "if". I appreciate that some of these modal
logics give something like the quantum logics of von Neumann, but
which ones correspond to our world? Neither the Theatetus knowledge
definition, nor the match with our world is so startlingly obvious as
to say "This must be it!". Also the lack of Kripke frames in X and Z
bothers me a bit with this approach.
In a sense the
comp-physics is the solution of the measure problem, in that
approach. We have already that the bottom of the physical reality
behaves symmetrically and linearly. It harder to derive the
Hamiltonian reality, and may be here could the S-L provides some
help (but this would make the Hamiltonians more geographical than
physical).
Most of the Hamiltonian structure comes from considerations of
symmetry (see Vic Stengar's book Comprehensible Cosmos). But why this
symmetry, and not that is harder to answer. (A bit like why this modal
logic, not that :) If all symmetries applied to observed reality, it
would be too simple.
I suspect that the logic is forced to be bivalent by the mutual
non-contradiction of interactions over many machines. Like a language's
words are forced by the habituation of the speakers.
The other reason to use the self-reference logics is that it
distinguish automatically the quanta (sharable, communicable at
least in a first person plural way) from the qualia (not sharable,
purely individual), all this by the Gödel-Löb-Solovay proof/truth
splitting of the modal logics.
Yes - that is interesting, but is true of any modal logic (apart from
S4Grz, it would appear).
But how do you obtain the mutual orthogonality of observables on a
quantum logic? We must address the relationship between orthocomplete
lattices and Boolean algebras at some point!
Onward!
Stephen
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