The White Rabbit Conjecture

From: Jacques Bailhache <>
Date: Thu, 18 Mar 1999 16:56:58 -0000

Bruno Marchal said :
>Remember I give argument for saying that eventually we will be able to
>derive (part of) physics from comp.
and :
>5O% of my thesis ( makes that obligation
>precise. I mean 50% of my thesis explain why, if we believe in comp, we
>really must and can derive a non trivial part of QM from classical
>computation theory. The other 50% gives a precise attempt toward such a

It seems to me that we cannot derive only OUR physics from classical
computation theory because we can also derive other physics of other
universes; we can derive only the part of our physics which says that our
universe is regular, in the sense I explained in :
"The informations that we perceive are not random, they contain
Mathematically, that means that these informations could be described by an
expression smaller than the one describing them trivially. For exemple, the
characters can be described by the expression "ABC repeated 10 times" which
contains only 21 characters. Perceived regularities permit us to suppose
that raw
perception is the unfolding of a smaller germ (as
"ABCABCABCABCABCABCABCABCABC" is the unfolding of "ABC repeated
10 times") and that there exists an outside world which gives our
depending on the actions that we give it. This outside world is not directly
accessible but only through this information exchange.
>From this information exchange, we build a mental representation of this

Juergen Schmidhuber gave an explanation in :
> What is the probability
>of ending up in a particular universe U? Most of this probability is
>due to the earliest algorithm computing U (this algorithm will be among
>the shortest). E.g., a universe in which lambs start attacking lions at
>some point seems less probable than ours, because apparently it requires
>more information than conveyed by the few physical laws causing "normal"
>animal evolution. Hence it will appear later in the list.
>Many self-delimiting algorithms cease requesting new bits at some point,
>without halting, e.g., by entering into a loop. They represent infinite
>but compressible, regular universes. Again, according to Levin's coding
>theorems, under the UP most of the probability of some universe is due
>to its shortest algorithms.

If our consciousness supervenes on outputs of programs of the Universal
Dovetailer, then perceptions corresponding to outputs which often appears
are more probable.

We could explain the regularity of our universe by formulating and proving a
conjecture which we could call "Conjecture of probabilities of program
outputs", or "White Rabbit Conjecture" (because it explains why we don't see
a white rabbit walking on the ceiling : such a world would be much more
complicated, i.e. described by a bigger theory, as our world) :

My first idea was :

- Consider a given program P.
- Chose randomly a program P' (for example generate a random string, analyze
it, if it is not a syntaxically correct program generate another one)
- Consider the probability f(P) that the output of P is included in the
output of P' (i.e. the proportion of programs P' in the Universal Dovetailer
whose output contains the output of P).
- The conjecture says that f(P) is greater for small programs P than for big

Small programs appear often as parts of other programs in the Universal
Dovetailer, so their output are often produced by other programs.
But there could be big programs whose output appears often in the Universal
Dovetailer, for example a program with a big piece of code which is never
So instead of considering the size of P, we could consider size'(P) =
smallest n such that n = size(P'') and P'' produces the same output as P.
The conjecture says that f(P) is greater for small size'(P) than for big
size'(P), i.e. the repartition of programs looks like :

    f(P) ^
       1 |
   often | **
         | ***
         | *****
         | *******
  rarely | *******
         | ********
       0 +---------------------->
          small big size'(P)

Exercise :
- Formulate precisely the conjecture;
- Prove it.

I don't believe in COMP because I don't believe that consciousness
supervenes on a finite program, because consciousness is the perception of
an outside by an inside, but this inside must also be perceived by an inside
of this inside, and so on ad infinitum, as I explained in :

 | +------------------------------+ |
 | | perception | perception | perception
 | | spirit3 <---------- matter3 |<---------- matter2 |<---------- matter1
 | | ----------> | ----------> | ---------->
 | | action | action | action
 | +------------------------------+ |
 | spirit2 |
These successive "I" (spirit1, spirit2, spirit3...) are not the same but
nested kernels of our spirit, getting smaller and smaller but never empty,
and my idea is that consciousness and free will may emerge from this
infinite nesting."
(see also

But even without believing in COMP, the White Rabbit Conjecture is still
interesting because it could be generalized to infinite mathematical models
defined as limit of series of finite models.

Jacques Bailhache
Y2K Centre of Expertise (BRO)
DTN: 856 ext. 7662
Tel: +32-2 729.7662, Fax: +32-2 729.7985
Visit my home page :
Received on Thu Mar 18 1999 - 09:07:41 PST

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