Re: White Rabbits, Measure and Max

From: Russell Standish <>
Date: Thu, 15 Jul 1999 09:38:00 +1000 (EST)

> ----- Original Message -----
> From: Russell Standish <>
> To: <>
> Sent: 14 July 1999 03:23
> Subject: White Rabbits, Measure and Max
> > I was thinking some more on the physical laws issue, and on Wei Dai's
> > point on what measure to apply to different universes. The solution
> > given is to weight shorter strings over longer ones.
> >
> > One way of thinking about this is to state that there are _no_ finite
> > strings at all in the everything world. Instead, the first n bits of a
> > string contain information, and the remainder are "don't care"
> > values. We also assume a uniform measure on all these infinite
> > strings. In this picture, worlds who are entirely specified by strings
> > with a smaller value of n will have higher measure than those with
> > larger n. The measure will fall off exponentially with n, in fact
> > precisely 2^{-n}, assuming the uniform measure above. This, then is
> > the universal measure sought by Wei Dai. Clearly, the everything
> > universe consists of all strings where we don't care what any bit
> > is. These have zero information, and measure = 1.
> I think this may have possibilities, but it drew me into a quagmire when I
> looked at it a little while ago. If n is the information-relevant string
> length of our (presumed) universe, m is an exponential mean of the
> equivalent for all possible contrived universes subjectively similar to ours
> (including flying rabbit universes), then it seems to me that if the number
> of different possible such contrived universes is less than 2^(m-n), then we
> should not expect to be in a contrived universe, with or without flying
> rabbits - the rabbit paradox would be solved. This might seem to be

Consider universes implemented by a UTM by way of concreteness. Then
our regular universe corresponds to finite length programs that do not
access the data beyond the finite bit position n. Now, the contrived
universe, presumably consists of a similar program, but with a bug in
it causing the program to process the "don't care" bits lying beyond
the string. Now this contrived universe will have the same measure as
the one we observe. (What is the ratio of correct programs to buggy
ones that work well enough to support consciousness?)

However, what will we see when these "don't care" bits are processed -
flying rabbits, fairies, non-mendacious politicians, decaying protons,
whatever. Most of the time, we won't even recognise it as being
something wrong, it'll just be noise. When we do - eg the flying
rabbit incident, there will be a lot of information involved - special
bit patterns - that we recognise as being unusual. I would hazard a
guess that the K-complexity of a flying white rabbit (or for that
matter, anything grossly unusual like that) is hugely greater than the
fundamental mathematical theories of physics (be it M-theory or
whatever), plus any exceptional fine-tuned quantities such as the fine
structure constant. Why? Because the description of the white rabbit
cannot be compressed, as it didn't arise through any physical
process. To enable something akin to a white rabbit to appear would
probably require of order Avogadro's number of bits, giving it a
2^{-{10^23}} chance of being observed.

> conceivable if we only had to concern ourselves with the single type of
> contrived universe that mimics ours, plus visible deviations (such as flying
> rabbits) from it - we can only perceive a limited number of such within our
> lifetimes; however it would seem we also have to consider other larger
> mathematical structures that would present to us contrived universes with
> (additional) invisible deviations as well (rabbits under the floorboards, in
> the Andromeda galaxy and so on). These have even longer, but finite-length,
> information-relevant bit strings, so although making a small contribution
> individually, their vast quantity may make them dominate - but I am not
> sure.
> Weighting shorter strings over longer ones may turn out to provide a useful
> universal measure, but I haven't been able to see a way through to it
> providing a solution to the flying rabbit problem.
> Alastair

Dr. Russell Standish Director
High Performance Computing Support Unit,
University of NSW Phone 9385 6967
Sydney 2052 Fax 9385 7123
Room 2075, Red Centre
Received on Wed Jul 14 1999 - 16:37:32 PDT

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