Global measure and "one structure, one vote"
Jesse Mazer
"everything?" It seems to me that a lot of the TOE's I've seen make an
assumption like "one structure, one vote." For example, if one assumes that
"everything" is the set of all computations, then one strategy might be to
look at the behavior of an average large turing machine and see what the
computation might look like "from the inside", treating it as a simulation
of a universe of some kind. But the notion of "average" seems to assume
that each possible turing machine is given equal weight...how do we know
they shouldn't be weighed by kolmogorov complexity or something else?
Similarly, Max Tegmark's TOE involves looking at all possible mathematical
structures, dividing them into equivalence classes, and then seeing what
kind of universe the majority of self-aware observers will find themselves
in. But again, this assumes that if one possible mathematical structure
contains 10 observers and another contains 100, then an observer is ten
times more likely to find himself in the second structure than the
first...but why should this necessarily be the case? Are we assuming that
the land of Platonic forms contains exactly one "copy" of each distinct
structure? Again, isn't it possible that some other measure would make
sense?
The main appeal of TOE's is that they reduce the amount of arbitrariness in
our basic assumptions about reality. If all possible universes are real to
observers inside them (or all possible observer-moments are real, to cut out
the middleman), then we can escape the problem of "why these laws of physics
and not some others?" But I think we do need some kind of global measure on
the set of "everything", since everything obviously includes worlds (or
observer-moments) that seem to be identical to this one up to a certain
point but in which the laws suddenly break down, and we want to be able to
say that this is less probable somehow (I've never been sure what people
were talking about when they referred to 'white rabbits' but I think it's
another version of this problem...isn't white actually a pretty common color
for rabbits though? Is it an Alice in Wonderland reference?) The problem
is that in introducing a global measure we run the risk of bringing back
exactly the same arbitrariness that we had before--"why this global measure
and not some other?" It seems to me that this is really the central problem
in divising a good TOE.
One solution is to say there is no global measure...this is what James Higgo
believes, if I understand him correctly, and possibly Hans Moravec as well.
James Higgo's picture of reality is a pretty honest look at what "no global
measure" implies--basically we can't talk about the probabilities of any
future events at all, and our knowledge is limited to the particular things
we're experiencing in this observer-moment and the statement "all possible
thoughts exist." Another solution is the "one distinct structure, one vote"
idea that Max Tegmark seems to use, and possibly some others as well. A
third solution might be to try to show that given some other more basic
assumptions, there is only one possible measure consistent with the
assumptions--this is the one I'm in favor of, and I have a rough idea about
how a kind of formalized version of anthropic reasoning might provide the
necessary constraints. The last solution I can think of would be to treat
the many-worlds theory as a measure on the set of all computations (assuming
that all computations actually end up being instantiated in some branch or
another) and then work backwards to see what the properties of this measure
are...perhaps it will be elegant enough that we can think of some kind of
philosophical "justification" for it.
A lot of people have a lot of different ideas about TOE's on this list, so
maybe the global measure issue could help clarify where we all stand in
relation to each other...do people have specific proposals about this? I
guess the other relevant question is, what is the set of "everything" that
you're putting the measure on...all computations? All mathematical
structures? All observer-moments?
Let me know what you think...
Jesse
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Michiel de Jong
> What are people's ideas about the problem of a global measure on
> "everything?"
If I summarize correctly, you give the following 4 possibilities:
1) there is no global measure
2) "one distinct structure, one vote"
3) only one measure is consistent with some other more basic assumptions
4) work backwards from the computations occurring _inside_ a universe
I suggest to distinguish a 5th:
5) although there is no global measure (as in option 1), Solomonoff's
universal prior allows us to make predictions _as_if_ there were one,
because it approximates any candidate measure within O(1).
Do you feel this is distinct from option 1?
For me, the importance is in the distinction between choosing the
universal prior as _the_ measure, and taking it as an approximation of
_any_ measure.
Kinds regards,
Michiel de Jong.
http://www.cwi.nl/~mbj
Jesse Mazer
I hadn't heard of the universal prior before, but I found a short
description on Russell Standish's site:
http://parallel.hpc.unsw.edu.au/rks/docs/occam/node2.html
It seems to me that this is still a measure of sorts, based on dividing the
set of turing machines into equivalence classes and then adopting a rule of
"one equivalence class, one vote" (correct me if I'm wrong on this). What
do you mean when you say it approximates any measure? Do you mean that any
algorithmically generated measure will be approximated by the universal
prior in some limit?
Russell Standish's page also mentions another option I hadn't thought of: an
observer-relative measure. Different types of information-processing SAS's
might have different preferred measures depending on what type of UTM they
are...although this would not solve the problem of why I find myself as this
type of SAS rather than some other, or why I find myself in this type of
universe as opposed to a completely different kind, it could at least allow
for future predictions and an elimination of the white rabbit problem. But
we'd still need a single rule to tell us which SAS's should use which
measure.
Russell Standish
.. Discussion deleted for compactness ...
>
>
> A lot of people have a lot of different ideas about TOE's on this list, so
> maybe the global measure issue could help clarify where we all stand in
> relation to each other...do people have specific proposals about this? I
> guess the other relevant question is, what is the set of "everything" that
> you're putting the measure on...all computations? All mathematical
> structures? All observer-moments?
>
> Let me know what you think...
>
> Jesse
> _________________________________________________________________
> Get your FREE download of MSN Explorer at http://explorer.msn.com
>
My thoughts are fairly well contained in my "Why Occam's Razor"
paper. Basically, I show that a global measure is not required, since
the question should be "Given I am the observer I am, what is the most
likely universe I will see?". Since many different descriptions can be
equivalenced, the most likely universe will be the one with the
largest equivalence class.
The question of what is the most likely form of consciousness may
never be answered, and the issue of whether there really is a global
measure may end up being no more important than "how many angels can
dance on a pin", but we can start by making assumptions about what
are necessary requirements for consciousness, since the curdest
approximation to a probability distribution is its support function
(S(p,x)={0: p(x)=0, 1: p(x)>0}).
Cheers
----------------------------------------------------------------------------
Dr. Russell Standish Director
High Performance Computing Support Unit, Phone 9385 6967
UNSW SYDNEY 2052 Fax 9385 6965
Australia R.Sta...@unsw.edu.au
Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks
----------------------------------------------------------------------------
Russell Standish
>
>
> I hadn't heard of the universal prior before, but I found a short
> description on Russell Standish's site:
> http://parallel.hpc.unsw.edu.au/rks/docs/occam/node2.html
> It seems to me that this is still a measure of sorts, based on dividing the
> set of turing machines into equivalence classes and then adopting a rule of
> "one equivalence class, one vote" (correct me if I'm wrong on this). What
> do you mean when you say it approximates any measure? Do you mean that any
> algorithmically generated measure will be approximated by the universal
> prior in some limit?
>
> Russell Standish's page also mentions another option I hadn't thought of: an
> observer-relative measure. Different types of information-processing SAS's
> might have different preferred measures depending on what type of UTM they
> are...although this would not solve the problem of why I find myself as this
> type of SAS rather than some other, or why I find myself in this type of
> universe as opposed to a completely different kind, it could at least allow
> for future predictions and an elimination of the white rabbit problem. But
> we'd still need a single rule to tell us which SAS's should use which
> measure.
>
> Jesse
> _________________________________________________________________
> Get your FREE download of MSN Explorer at http://explorer.msn.com
>
Thanks for the rap! I sent my previous post before coming across this
one. Yes your summary is correct. However section 4 goes on to show
what sort of universe we expect to see ourselves in (ie the Multiverse
we find ourselves in - its the best explanation I've come across yet
for Quantum Mechanics), based on some pretty simple, and one would
hope uncontroversial assumptions about what it means to be conscious.
To go further on the measure problem would require attaching a
particular property of our observed universe to the anthropic
principle - eg why we find ourselves in 4D Minkowski
spacetime. Tegmark has some speculations on this matter, but it
doesn't go far enough.
Hal Ruhl
information at all either absolute or relative.
I am trying to display the various model threads in a FAQ for the list.
Hal
Jesse Mazer
>Thanks for the rap! I sent my previous post before coming across this
>one. Yes your summary is correct. However section 4 goes on to show
>what sort of universe we expect to see ourselves in (ie the Multiverse
>we find ourselves in - its the best explanation I've come across yet
>for Quantum Mechanics), based on some pretty simple, and one would
>hope uncontroversial assumptions about what it means to be conscious.
>
>To go further on the measure problem would require attaching a
>particular property of our observed universe to the anthropic
>principle - eg why we find ourselves in 4D Minkowski
>spacetime. Tegmark has some speculations on this matter, but it
>doesn't go far enough.
Doesn't your scheme assume something like "one turing machine, one vote"
though? On the universal prior page you say:
"the natural measure induced on the ensemble of bitstrings is the uniform
one, i.e. no bitstring is favoured over any other."
On the other hand, Michiel de Jong said:
"although there is no global measure (as in option 1), Solomonoff's
universal prior allows us to make predictions _as_if_ there were one,
because it approximates any candidate measure within O(1)."
When he says it "approximates any candidate measure", does this mean there
is some general class of measures for which the universal prior is a "good
enough" approximation in some sense? Obviously not *all* measures would
work, since I could pick a measure that was 100% concentrated on a
particular bitstring and 0% on all the others, and that'd yield predictions
quite different from those based on the universal prior. Juergen
Schmidhuber's paper goes into more detail on the class of measures that the
universal prior is a "good enough" approximation for, right? Maybe I need
to go read that...
Anyway, it may be that for most "plausible" measures the universal prior is
a good approximation, in which case using it is perfectly justified. But it
still seems that for a complete TOE we must address the measure problem in a
more direct way, in order to rule out weird measures like the one I
mentioned...using the universal prior might turn out to be a bit like
"renormalization" in quantum field theory, i.e. a tool that's useful for
making calculations but probably isn't going to be the basis of our final
TOE.
Michiel de Jong
> Obviously not *all* measures would
> work, since I could pick a measure that was 100% concentrated on a
> particular bitstring and 0% on all the others, and that'd yield predictions
> quite different from those based on the universal prior.
Yes, you've pointed out a big problem there. (it's connected to Russell
Standish's remark "The trouble with this argument, ..." on page 4 of
"Why Occam's Razor").
> Juergen Schmidhuber's paper goes into more detail on the class of
> measures that the universal prior is a "good enough" approximation
> for, right?
Yes. This class of measures corresponds to the set of all universal
programming languages (i.e., all programming languages that are
equivalent to the Turing Machine).
(by the way, your particular example that is 100% concentrated on a
particular bitstring, is not necessarily in this class; only if the
chosen bitstring is computable).
> Maybe I need to go read that...
Yes, it's very good stuff. Read "A Computer Scientist's View on Life,
the Universe, and Everything" first. After that, if you have a bit of
background in computer science, then "Algorithmic Theories of
Everything" is also very good reading. In this last paper, Schmidhuber
differentiates between two ways to derive a measure from a universal
programming language:
1) by program length (the standard way in algorithmic information
theory)
2) by program execution time (a very interesting alternative).
> using the universal prior might turn out to be a bit like
> "renormalization" in quantum field theory, i.e. a tool that's useful for
> making calculations but probably isn't going to be the basis of our final
> TOE.
You're right. Your question was about the "basis of our final TOE",
and I gave you an answer about a "tool". So, after some more thinking
and reading, I want to change my answer to option 1: there is no
global measure. As Russell Standish writes in "Why Occam's Razor":
"The conscious observer is responsible, under the Anthropic Principle,
for converting the potential into actual, for creating the observed
information from the zero information of the ensemble".
If I interpret him correctly, this means that we should not try to
make any specific predictions on the basis of a global measure,
because all computable universes are possible anyway.
I think this is also what Hal Ruhl means:
> My particular approach is to produce an Everything that contains no
> information at all either absolute or relative.
I think that trying to establish a single superior global measure is
already a violation of the zero information principle, because it
excludes other possible measures. The probability of being in a
certain universe is only determined by the SAS making
observations. His universe is determined by the outcomes of his
observations, not by some a priori global measure.
Then, the logical next question would be:
What is the prior probability over the different outcomes of such
observations?
It looks like the problem of choosing a measure has been shifted from
the choice of universe to the repeated choice of
observation-outcomes. But I think the important difference here is that
an observation has only 2 possible outcomes, "yes" or "no", whereas
the choice of universe has infinitely many outcomes. It seems to be
much more defendable to use the "one-structure-one-vote" principle for
observation-outcomes, than for a global measure on universes.
I think this must have been Wheeler's motivation for taking on the
it-from-bit view. See http://suif.stanford.edu/~jeffop/WWW/wheeler.txt
for a good summary of it.
Cheers,
Marchal
> [...]
>But I think we do need some kind of global measure on
>the set of "everything",
Global? Only a relative one is necessary. But a relative
one needs a "global" definition. (If we want a TheoryOE).
>since everything obviously includes worlds (or
>observer-moments) that seem to be identical to this one up to a certain
>point but in which the laws suddenly break down, and we want to be able to
>say that this is less probable somehow
Indeed.
> (I've never been sure what people
>were talking about when they referred to 'white rabbits'
Basically what you said above. It is also the skeptical inductivism
Davis Lewis talk about in his "the plurality of worlds".
> but I think it's
>another version of this problem...isn't white actually a pretty common color
>for rabbits though? Is it an Alice in Wonderland reference?)
Sure. It is *the* Wonderland's white rabbit. I'm used to that
expression and I have been happily surprised by its success in the list,
because there is nothing wrong with a "white rabbit" indeed.
Schmidhuber has talked about "lambs eating wolves" somewhere in relation
with that problems.
>A lot of people have a lot of different ideas about TOE's on this list, so
>maybe the global measure issue could help clarify where we all stand in
>relation to each other...do people have specific proposals about this? I
>guess the other relevant question is, what is the set of "everything" that
>you're putting the measure on...all computations? All mathematical
>structures? All observer-moments?
>
>Let me know what you think...
.. Jesse Mazer wrote in the same thread:
>Russell Standish's page also mentions another option I hadn't
>thought of: an observer-relative measure.
The question "if the measure should be absolute or relative"
has been discussed a lot in the thread around Bostrom
self-sampling assumption + some remarks by Wei Dai.
Russell Standish, George Levy and myself
and (hopefully) some others
seems to agree on the
relative use of that measure, although I argue that both aspect
could be important, i.e. there could be an interplay between
some absolute universal prior and relative conditional information.
If I remember correctly Jacques Mallah critisizes the nuances about
relativisation and conditionalisation: you can have an absolute
measure and consider that its relativisation are produced by
classical conditionalisation. (I basically agree if we keep
in mind the difference of point of view).
In my thesis there is two parts: one where I show that if we are
machine then "physics" is reduced to the search of an "absolute
definition" of an observer-relative (first person plural) measure.
In the second part I have developed a road toward a purely mathematical
isolation of that measure based on the incompleteness phenomena.
I got the logic of the certainty-case P=1, and show it to be closely
related with quantum logic. And I take *that* as a confirmation
of both the comp-reversal and the way I have chosen for interviewing
the universal machine.
Schmidhuber's solution is based on a belongness relation between
observer and universes which is impossible to keep once we take
comp seriously. But even if we make sense to such a relation, it
would only eliminates third person white rabbits and not the
first person white rabbits: remember that the great programmer
emulates all (semi)computable "universe" but also all possible
dreams.
In fact Schmidhuber assume a solution of the mind body problem
which is just incompatible with comp. Technically that makes
his work incomplete (at least).
Bruno
Hal Ruhl
However, I view a focus on computable universes as a selection. IMO it
injects a type of absolute information into the Everything.
IMO the Everything needs to be selection and relationship free as well as
contain no information such as a resolution of the question of its own
stability or a global measure etc.
As to SAS type vs universe type to me its self referential. To pick a SAS
type is to pick a universe type. Here I think I agree with Russell Standish.
However, for the various universes to have relative properties such as
unequal equivalence classes is to inject the Everything with net
relationship information. This I feel is unacceptable.
IMO to accomplish a lack of relationship information all universes and all
relationships between universes need to be repeated an infinite number of
times. I think this results in the loss of any and all uniqueness to all
elements of the Everything which is what I am looking for.
The real objective as I see it is to construct a zero information
Everything that one can demonstrate is compatible with our universe's
existence and the existence of all the others.
As to what type of universe/SAS pair we are I prefer to think we are in a
universe that has some degree of required true noise content. In this I
believe I again agree with Russell Standish. Taking a lesson from
steganography [re hiding coded messages in images] up to about 30% of our
universe could be junk true noise and we might never notice even if we did
recognize the existence of any additional and required true noise.
Can this question of our SAS type be resolved? I think so. I think the
answer lies in the study of the information budget and logistics of our
universe.
Hal
Russell Standish
>
> Russell Standish wrote:
>
> >Thanks for the rap! I sent my previous post before coming across this
> >one. Yes your summary is correct. However section 4 goes on to show
> >what sort of universe we expect to see ourselves in (ie the Multiverse
> >we find ourselves in - its the best explanation I've come across yet
> >for Quantum Mechanics), based on some pretty simple, and one would
> >hope uncontroversial assumptions about what it means to be conscious.
> >
> >To go further on the measure problem would require attaching a
> >particular property of our observed universe to the anthropic
> >principle - eg why we find ourselves in 4D Minkowski
> >spacetime. Tegmark has some speculations on this matter, but it
> >doesn't go far enough.
>
> Doesn't your scheme assume something like "one turing machine, one vote"
> though? On the universal prior page you say:
>
> "the natural measure induced on the ensemble of bitstrings is the uniform
> one, i.e. no bitstring is favoured over any other."
No - its "one description, one vote". Nonuniform measures arise out of
equivalencing descriptions through interpretation. The universal prior
is what you get when you use a universal turing machine as your
interpreting device.
My key point was that the obvious interpreting device is the observer
erself. I fudge the details a bit in "Occam" by claiming that
observers should be capable of universal computation (which appears to
be true of homo sapiens), in which case the universal prior is what
one should observe. But really, any equivalencing mechanism will do,
even ones that generate your pathological measue distributions below.
In a later paper "On Complexity and Emergence", I show that an
observer's concept of complexity is inextricably related to this
equivalencing process. One distinct difference between how homo
sapiens does things and how Turing machines do things is that random
strings (or at least patternless strings) have almost zero complexity
to humans (they are all meaningless strings, and equivalent), whereas
to a Turing machine, they are all distinct and have maximum complexity.
>
> On the other hand, Michiel de Jong said:
>
> "although there is no global measure (as in option 1), Solomonoff's
> universal prior allows us to make predictions _as_if_ there were one,
> because it approximates any candidate measure within O(1)."
>
> When he says it "approximates any candidate measure", does this mean there
> is some general class of measures for which the universal prior is a "good
> enough" approximation in some sense? Obviously not *all* measures would
> work, since I could pick a measure that was 100% concentrated on a
> particular bitstring and 0% on all the others, and that'd yield predictions
> quite different from those based on the universal prior. Juergen
> Schmidhuber's paper goes into more detail on the class of measures that the
> universal prior is a "good enough" approximation for, right? Maybe I need
> to go read that...
>
> Anyway, it may be that for most "plausible" measures the universal prior is
> a good approximation, in which case using it is perfectly justified. But it
> still seems that for a complete TOE we must address the measure problem in a
> more direct way, in order to rule out weird measures like the one I
> mentioned...using the universal prior might turn out to be a bit like
> "renormalization" in quantum field theory, i.e. a tool that's useful for
> making calculations but probably isn't going to be the basis of our final
> TOE.
>
> _________________________________________________________________
> Get your FREE download of MSN Explorer at http://explorer.msn.com
>
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