You are being a dualist when you talk of a turing machine and a program. The
difference is in our minds only. It is silly to talk of 'hard wireing'
aspects of the universe as opposed to having them in the software. This is
not a 'hole'.
> -----Original Message-----
> From: Christopher Maloney [SMTP:dude.domain.name.hidden]
> Sent: Saturday, October 23, 1999 4:17 AM
> To: everything-list
> Subject: Re: Turing vs math
>
> hal.domain.name.hidden wrote:
> >
> > Juergen Schmidhuber, juergen.domain.name.hidden, writes, quoting Hal:
> > > > I do think that this argument has some problems, but it is appealing
> and
> > > > if the holes can be filled it seems to offer an answer to the
> question.
> > > > What do you think?
> > >
> > > Where exactly are the holes?
> >
> > One is what I mentioned earlier, that a trivial program which enumerates
> > and executes (in dovetailing, interleaved form) all possible programs
> > will create every mind in every possible situation. This is a very
> > short program and hence is the most likely universe for us to live in.
>
> I don't see this as a hole at all. Maybe I'm missing something, but I
> thought the whole point of postulating a universal dovetailer was that
> it creates "everything" from zero information (or as near as dammit).
>
> This, combined with Bruno's computational indeterminism (thanks for
> quoting your previous post Bruno -- I hadn't read that) provides the
> basis for predictions of our future observations.
>
>
> > You can try to say that this program doesn't count because it creates
> more
> > than one universe, but as I suggested earlier this requires an objective
> > formulation. Which programs count and which ones don't? How can we
> > know whether a program creates a single universe or more than one?
> > We need something more in the theory to solve this problem.
>
> I would be inclined to reject any theory that threw ad hoc
> rationalizations
> for rejecting some universes and accepting others. The whole appeal of
> "everything exists" is its zero information.
>
>
> > Another problem is that the Kolmogorov measure is defined only up to
> > an additive constant. Given a specific, large, program which runs on
> > universal TM "T", we can construct a different UTM T' on which that
> > program is very small. (In essence we hard-wire the program into the
> > T' definition.) This means that I can create a UTM where a magical
> > flying-rabbit universe is more probable than the one we live in.
>
> I agree that this is a hole, and it has bothered me (sort of in the
> back of my mind) for some time. In Juergen's paper, he says
>
> Under different universal priors (based on different universal
> machines), probabilities of a given string differ by no more
> than a constant factor independent of the string size, due to the
> compiler theorem (the constant factor corresponds to the
> probability of guessing a compiler). This justifies the name
> ``universal prior,'' also known as Solomonoff-Levin distribution.
>
> I don't know what this means exactly. It worries me that the
> probabilities are not completely well defined. It seems to me that
> Hal is correct that certain UTM's can be contrived to make certain
> bit strings very likely.
>
> We could borrow the section from Tegmark's paper on making
> predicitions, and apply it to this discussion.
> I'll assume Bruno's computational indeterminism, and restrict the
> discussion to just two consecutive time steps in the computation.
> We want to predict the probability of our seeing Y at time t1,
> given that we now see X at time t0. As Bruno often points out, we
> have no way of knowing exactly who we are or what bit strings we
> are represented by, but we do know that at time t0, we see X.
>
> Let Si be a bit string within a single time snapshot of a universe
> history. Let's also assume that it's possible, in principle, to
> evaluate Si to see if it qualifies as an observer, and if it observes
> X or Y.
>
> Then,
>
> P(Y,t1|X,t0) =
> Sum j { Sum i { P(Y|Sj) P(Sj,t1|Si,t0) P(X|Si) } }
>
> In words, the probability that I'll see Y at t1, given that I see X
> at t0, equals the sum of all probabilities that Sj sees Y, that Sj is
> a computational continuation of Si, and that Si sees X.
>
> Now, being able to determine what a "bit string observes" depends,
> of course, on the TM. The TM "breathes life" into the bit strings.
> Also, the determination of the middle term depends on the TM.
> I'm just thinking out loud, here, really, but it seems to me that
> even a "constant" factor will throw this equation out of whack. We
> should be able to show that this probability is independent of the
> TM.
>
>
> > A related problem is the uncomputability of the Kolmogorov measure.
> > There is no way in general to know what is the shortest program to
> > construct a given string or a given universe. Yet probabilities are
> > real and so apparently someone/something is in effect computing them.
> > In other words, our observations of probability imply that uncomputable
> > values are being computed. This is at least a bit paradoxical.
>
> I don't see this as a problem, on the other hand. Just because
> probabilities exist doesn't mean that anyone or anything is computing
> them.
>
>
> >
> > Hal
>
> --
> Chris Maloney
> http://www.chrismaloney.com
>
> "Donuts are so sweet and tasty."
> -- Homer Simpson
Received on Mon Oct 25 1999 - 02:02:22 PDT