I submit this link to Shmidhuber's second paper, which discusses various probability distributions on the set of computable Universes. ftp://ftp.idsia.ch/pub/juergen/toesv2.pdf
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. http://www.idsia.ch/~juergen/computeruniverse.html Peace ----- 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 > distribution? > > 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 > laws > everywhere except where this incredibly complicated configuration > occurs, > and then do this complicated other thing." > > Hal Finney >
