On Thu, Jan 22, 1998 at 10:31:50AM -0800, Hal Finney wrote:
> > Remember that the prior probability of a region is related to the
> > program length plus the coordinate length. If you have a universe with an
> > exotic coordinate system, the lengths of the coordinates would be long and
> > so the regions in that universe would have small priors.
>
> I'm not so sure. Consider a 1 dimensional universe which is the output
> of a cellular automata program, possibly a very complex and long one.
> Let's suppose though that the output has about 50% 1's and 50% 0's.
>
> Consider a typical state:
>
> 0110001010100101000011110101010110101011010101110
>
> The natural coordinate system is:
>
> 111111111122222222223333333333444444444
> 0123456789012345678901234567890123456789012345678
>
> If I understand your model, a TM which took one of these X coordinates
> (and perhaps a time coordinate T) and output the corresponding bit of
> the state above would in some sense "be" that universe.
>
> But I could write a trivial program which produced as output the LSbit
> of the X coordinate being input. Then I could use a different set of
> coordinates to get the output I needed:
>
> 1 1111112221112 ...
> 013246587092416380245791 ...
>
> These coordinates will not be much larger than the natural ones, but
> the trivial TM can now produce the output of an arbitrarily complicated
> CA program. (The X coordinate system will have to change with each time
> step T, but that is just a complex X,T coordinate system.)

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I don't see how this is a counter-example. In order to specify a region
in this coordinate system, you'd have to give the coordinates of every
point inside the region, so the length of a region that contains
non-trivial content would be much greater in this TM than in a TM with a
simple coordinate system, where you only have to give the coordinates of
the boundaries.