Re: Turing vs math

From: <hal.domain.name.hidden>
Date: Fri, 22 Oct 1999 09:58:45 -0700

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.

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.

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.

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.

Hal
Received on Fri Oct 22 1999 - 10:01:34 PDT