I submit this link to Shmidhuber's second paper, which discusses various
probability distributions on the set of computable Universes.
Sorry if this has been already covered. I'm not a mathematician, and I'm not
entirely "into" hardcore computer science.
This other site contains the links to Shmidhuber's other works.
----- Original Message -----
From: "David Barrett-Lennard" <[EMAIL PROTECTED]>
To: "'Hal Finney'" <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]>
Sent: Monday, November 03, 2003 11:52 PM
Subject: RE: Is the universe computable?
> An interesting idea.
> Where can I read a more comprehensive justification of this
> If a number of programs are isomorphic the inhabitants naturally won't
> know the difference. As to whether we call this one program or lots of
> programs seems to be a question of taste and IMO shows that "probability
> calculations" are only relative to how one wants to define equivalence
> classes of programs.
> I would expect that the probability distribution will depend on the way
> in which we choose to express, and enumerate our programs. Eg with one
> instruction set, infinite loops or early exits may occur often - so that
> there is a tendency for simplistic programs. On the other hand, an
> alternative instruction set and enumeration strategy may lead to a
> distribution favoring much longer and more complex programs. Perhaps it
> tends to complicate programs with long sequences of conditional
> assignment instructions to manipulate the program state, without risking
> early exit. Importantly such "tampering" doesn't yield a program that is
> isomorphic to a simple one. We seem to have a vast number of
> complicated programs that aren't reducible to simpler versions. This
> seems to be at odds with the premise (of bits that are never executed)
> behind the Universal Distribution.
> - David
> -----Original Message-----
> From: Hal Finney [mailto:[EMAIL PROTECTED]
> Sent: Tuesday, 4 November 2003 2:24 PM
> To: [EMAIL PROTECTED]
> Subject: RE: Is the universe computable?
> IMO the best idea we have discussed for why the universe is and remains
> lawful is that the set of descriptions (equivalently, programs) for
> the universes are governed by the Universal Distribution. This is the
> description where a string whose shortest description has length n bits
> is given measure 1 / 2^n.
> An heuristic argument for this distribution is that if programs are
> self delimiting, then there are 2^x more programs of length n+x than
> of length n, created by appending the 2^x x-bit strings to each n-bit
> program. Since the appended x bits are never executed, all 2^(n+x)
> of these programs are the same as the basic 2^n programs.
> A program which says "obey these simple laws" is shorter than a program
> which says "obey these simple laws for a zillion steps, then start
> obeying these other laws", or a program that says "obey these simple
> everywhere except where this incredibly complicated configuration
> and then do this complicated other thing."
> Hal Finney