Re: why not high complexity?
[EMAIL PROTECTED] wrote:
OO O O
Othen lim i-inf e(i) = inf.
O
O This will not give you a uniform
O distribution on infinitely many things.
O O
O O yes. i agree.
O O
OO hm.
O O O
OO let's try a thought experiment.
O O O O
O O there is a universe which contains a large
O O machine. a factory. which produces self-aware
OO O robots. they are all identical, aside from their
O O O serial number. it takes a year to make such a
O OO robot. this universe exists for 100 years, in
OOOwhich time 100 robots are produced.
OO O
O O O O when the robots come alive - they are able to
O OOO reason about their serial number. a robot may
OOO O
OO O O Osay there is a 90% chance my serial number is
OO O O O O greater than ten.
OO O O O
O O O O O now consider a similar universe - but this one
O O O O exists for 200 years. in it the robots can
OOO O O O reason similarly, and can say there is a 95%
OOO O O O chance my serial number is greater than ten.
OO O O O
OO OO O O now consider a similar universe - one which
O O O O OO lasts forever.
O OO O OO
O OO OOO by your argument - the robots cannot reason
OO OO OO about their serial numbers - because there are
OO OO OOO infinitely many robots, and there is no uniform
OO OO OOO distribution over them.
OO OOO OO
OO OO OOO poppycock i say. although YES, I AGREE, THERE IS
OO OO OOO NO UNIFORM DISTRIBUTION OVER AN INFINITE SET - i
believe it is still possible for the robots to
O OOO reason about their serial numbers. but they need
OO OOO OOO to use tricks, like the one i presented earlier.
O OO
OOO O similarly, i think it is possible to reason
OO OOO about the complexity of a uniformly chosen
OOO OO OOO universe - and in doing so, i find that it's
OO OOO complexity is extremely high.
OO OOO OO
OO OO
OOO OO
OO OO -k
Re: why not high complexity?
From [EMAIL PROTECTED] Thu May 31 18:14:55 2001
From: Karl Stiefvater [EMAIL PROTECTED]
O
O Maybe you'd like to write down formally
OO what you mean.
O O
O O sure. i suspect we're talking past each other.
O O
OO O let M be the set of universes. let A_i be a
O O O sequence of finite subsets of M, such that A_i
O O is a strict subset of A_(i+1). define e(i) be
O O O the expected complexity of a uniformly chosen
OOO member of A_i.
O O O
O O Othen lim i-inf e(i) = inf.
This will not give you a uniform distribution on infinitely many
things. For simplicity, consider integers instead of universes. Assign
something like probability P(n)=6/(n^2 pi) to integer n. This yields
nonvanishing probability for infinitely many integers. But it's not
uniform. Uniformness and nonzero limits are incompatible.
O OO O Practical unpredictability due to
OO OO deterministic chaos and Heisenberg
O OO O O etc is very different from true
O O O O unpredictability. For instance, despite of
O O O OO chaos and uncertainty principle my computer
OOO O O probably will not disappear within the
O OO O O next hour. But in most possible futures it
OO O O O won't even see the next instant - most are
OO O O O maximally random and unpredictable.
OO O OO O
OOO O O O yes - i think i understand what you're saying
OO O O O here. a universe with high complexity is a very
OOO O OO messy place indeed - computers disappear, etc.
O OOO O OO however, i think you'll agree, that our universe
O OO (unless it *is* using a pseudo-random number
OO O generator) is quite messy.
Not at all. It seems extremely regular. Whatever appears messy
may be due to lack of knowledge, not to lack of regularity.
OOO i'm wondering if perhaps a different force is
OO OO keeping the complexity low. an anthropic force
OO OO - if complexity is too high, then life doesn't
OOO OOO OO evolve - and we don't see it.
According to the weak anthropic principle, the conditional probability
of finding ourselves in a universe compatible with our existence equals
1. But most futures compatible with our existence are complex. So why
is ours so regular? Algorithmic TOEs explain this, and add predictive
power to the weak anthropic principle.
http://www.idsia.ch/~juergen/everything/html.html
http://www.idsia.ch/~juergen/toesv2/
Re: why not high complexity?
oops. my last message didn't make it to the full list. O O Maybe you'd like to write down formally OO what you mean. O O O O sure. i suspect we're talking past each other. O O OO O let M be the set of universes. let A_i be a O O O sequence of finite subsets of M, such that A_i O O is a strict subset of A_(i+1). define e(i) be O O O the expected complexity of a uniformly chosen OOO member of A_i. O O O O O Othen lim i-inf e(i) = inf. O OO O OOO O O OO O Practical unpredictability due to OO OO deterministic chaos and Heisenberg O OO O O etc is very different from true O O O O unpredictability. For instance, despite of O O O OO chaos and uncertainty principle my computer OOO O O probably will not disappear within the O OO O O next hour. But in most possible futures it OO O O O won't even see the next instant - most are OO O O O maximally random and unpredictable. OO O OO O OOO O O O yes - i think i understand what you're saying OO O O O here. a universe with high complexity is a very OOO O OO messy place indeed - computers disappear, etc. O OOO O OO however, i think you'll agree, that our universe O OO (unless it *is* using a pseudo-random number OO O generator) is quite messy. O OO O OOO OOO i'm wondering if perhaps a different force is OO OO keeping the complexity low. an anthropic force OO OO - if complexity is too high, then life doesn't OOO OOO OO evolve - and we don't see it. OOO O OO O OOO O OO O i apologize for not having fully read/understood OOO O OO O your paper. i'm on vacation for the next week, OOO O i'll do so then. OOO OO OOO OOO OO OOO OO OOO OO -k
Re: why not high complexity?
O O ??? - There is no way of assigning equal OO O O O nonvanishing probability to infinitely O O O O many mathematical structures, each being O O O represented by a finite set of axioms. OO O O O O okay - strictly speaking, you are correct. but a OOOcommon trick is to compute equal-probabilities O on finite subsets of the infinite set. and then O O O you can take the limit as those subsets grow O O O to the size of the infinite set. OOO O OOOthe growing here is important - very often the O OO O order in which you add members to the set change O how the series converges. but for the case of O OOO expected complexity, it does not. but in the limit uniform probabilities vanish. Maybe you'd like to write down formally what you mean.
Re: why not high complexity?
[EMAIL PROTECTED] wrote: Even if He completely ignores runtime, He still cannot assign high probability to irregular universes with long minimal descriptions. Lee Smolin wrote about some Darwinian super-selection rule, among trees of universes. Do you think there is a possible connection? He also wrote: we will have to extend the Darwinian idea that the structure of a system [universe] must be formed from within by natural processes of self-organization - to the properties of space and time themselves. The key step in this somewhat Darwinian argument is that the parameters determining the shape of the new cosmos must resemble those of its parent, rather than being picked at random. Granted this, after a while most universes in the mega-cosmos will cluster around certain apparently arbitrary values - the kind that produce observers like us. wrote Damien Broderick. - Scerir
Re: why not high complexity?
From: Karl Stiefvater [EMAIL PROTECTED] Date: Mon, 28 May 2001 00:11:33 -0500 O OO OO Max Tegmark suggests that .. all mathematical O O structures are a priori given equal statistical OOOO O weight and Jurgen Schmidhuber counters that O O OOO there is no way of assigning nonvanishing OOO Oprobability to all (infinitely many) OO O mathematical structures Not quite - the text in Algorithmic Theories of Everything says ... of assigning nonvanishing _equal_ probability ... and he then goes on O (i think) to assign a weighting based upon OO OOO time-complexity. Most of the weightings discussed in Algorithmic Theories of Everything completely ignore time, except for one: the speed prior S derived from the assumption that the universe-generating process is not only computable but also optimally efficient. Concerning time-independent weightings: Different computing devices (traditional Turing Machines, Enumerable Output Machines, General Turing Machines) reflect different degrees of computability (traditional monotone computability, enumerability, computability in the limit). This causes various weightings. All favor short descriptions, given the device. O O OOOi have to say i find Tegmark's argument more OO O persuasive - i can't see why the great O O programmer should be worried about runtime. Even if He completely ignores runtime, He still cannot assign high probability to irregular universes with long minimal descriptions. O O OO furthermore, i feel intuitively that the O O universes ought to have equal weight. Some intuitively feel the sun revolves around the earth. OO OOsuch a sort of probability can be defined, of O O O course, by taking the limit as finite subsets O approach the full infinite set. as long as we OO OOO get the same answer regardless of the order in O O O which we grow the subset, the limit can be said OOO O O O to be defined. ??? - There is no way of assigning equal nonvanishing probability to infinitely many mathematical structures, each being represented by a finite set of axioms. Maybe the intention is to assign measure 2^-n to all histories of size n. That would imply our environment will dissolve into randomness right now, because almost all continuations of its rather regular history so far are random. But instead the universe keeps following the nonrandom traditional laws of physics, thus offering evidence against this measure. O O O the problem is - such a view predicts that we O O OO live in a universe of high Kolmogorov complexity OO OO O - not low complexity. O O O OO O but i don't see why this is such a surprise O O- living in such a universe, we ought to see OO OO O events occur which we cannot effectively O OOO O O predict. but that is exactly what we do see. Practical unpredictability due to deterministic chaos and Heisenberg etc is very different from true unpredictability. For instance, despite of chaos and uncertainty principle my computer probably will not disappear within the next hour. But in most possible futures it won't even see the next instant - most are maximally random and unpredictable. Any irregular future, however, must have small measure, given the rather harmless assumption of formal describability or computability in the limit. http://www.idsia.ch/~juergen/everything/html.html http://www.idsia.ch/~juergen/toesv2/
