Re: Turing vs math

From: <>
Date: Sun, 24 Oct 1999 18:41:48 -0700

Christopher Maloney, <>, writes:
> wrote:
> >
> > Juergen Schmidhuber,, 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).

To see that it is a hole, you have to know what the argument is that it
is a hole in!

The argument attempts to explain why we don't see flying rabbits or
other magical exceptions to the natural and simple laws of physics.
The reason, according to this argument, is that universes with simple
laws of physics can be described (simulated) with a shorter program
than universes which have complicated laws of physics with all kinds of
exceptions like magical flying rabbits. The argument further assumes that
universes exist with greater probability the shorter their program is.
Since flying-rabbit universes have larger programs than non-flying-rabbit
ones, they are therefore of lower probability. Hence we are unlikely
to be living in a flying-rabbit universe.

That is the argument. The hole is that it does not work if we consider
one of the shortest possible universe programs, the universal dovetailer
(UD). This simple program creates, as part of its output, flying rabbits.
Yet it is an incredibly simple program, hence it is very high probability.
In fact, it is very likely that we do live in the universe created by
this program, and since that universe has flying rabbits in it we have
failed to explain why we don't see flying rabbits.

To resolve this, we have to do one of two things, as I see it. We can
disallow the UD as a legal "universe" simulator, by saying that it doesn't
really create one universe, it creates multiple ones. And if we do that,
we can then restrict our attention to programs which create only single
universes, and then indeed we find that flying-rabbit universes are less
probable than others.

However to take this step we need an objective basis for doing so.
We could say that "one universe" is identified with a single spacetime
manifold, or is some kind of structure that has a certain amount of
connectivity and continuity. Since the UD creates multiple independent
structures with no connection to each other, we could argue that it
objectively creates multiple universes. However this adds considerable
baggage to the theory.

The other possibility, which was proposed by Wei Dai and is the one
which makes sense to me, is to state that the probability of an event
or structure is not just a matter of how probable the universe is which
creates it.

Rather, you have to look at how easy it is to localize that particular
structure within the universe. A simple program which outputs an enormous
universe which has, buried in one tiny place, a copy of my mind, should
not count for much. A more complex program which outputs a smaller
universe in which my mind is a proportionately bigger piece might actually
contribute more, even though the program to create the universe is larger.

Hence, the solution is to say that the contribution to the probability of
structure A in universe X is the size of the program to create universe X,
plus the size of the program which, given universe X, outputs structure A.

There is a very strong precedent for this in Kolmogorov complexity.
We say that the complexity of a string is the size of the smallest program
which outputs (only) that string. We could write a trivial counting
program to output all strings, but that doesn't mean each such string
has a small complexity. If you have two programs, one which outputs many
strings, and the other which takes that output and selects some particular
substring for output, then the sum of the sizes of those two programs
represents the total size of the program to output that substring. It
is this total size which is used to calculate K. complexity.

This is exactly what Wei proposes to do for measuring probabilites in
the context of the multiple universes. It is not enough to know the
probability of a universe which includes the desired structure (my mind,
say) somewhere; you also need to add in a measure of how hard it is to
localize that structure within that universe.

This plugs the hole in the argument above, because even though the UD
outputs a flying-rabbit universe, localizing that universe within the UD
output is going to take at least as large a program as one which creates
it in the first place. Hence the net contribution of the UD to the
probability of any given structure is no larger than for a straightforward
program which implements that structure. The hole is thereby plugged.

Received on Sun Oct 24 1999 - 18:44:38 PDT

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