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.)
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.
