On Sun, Jun 24, 2012 at 10:36 PM, Robert Munafo <mrob27@gmail.com> wrote:
Okay, so far so good.
On 6/25/12, Mike Stay <metaweta@gmail.com> wrote:
My ideal measure would still assign 2^{-s|p|} or [...]
|p| looks like "absolute value of p" so I guess that's your shorthand for the measure function. What is s?
p is a binary string laid out in a spiral (say) to form the life pattern and |p| is the length of the binary string. s is a real number, usually taken to be computable and greater than 1, but can be left as a free parameter.
[...] or 2^{-s ln(index(p))}
The sum zeta(s) = sum_p 2^{-s lg(index(p))} is another way of writing the Riemann zeta function. The sum sum_p h(p) 2^{-s lg(index(p))} / zeta(s), where h(p) is 1 if it eventually dies out and 0 otherwise, is the Dirichlet generating function for h(p), also known as "the halting probability at inverse temperature s." It converges at s > 1 and gives a partially random real number at computable s > 1. That partially random real is compressible by a factor of s.
to each pattern p, since then I still get a partially random real out at computable s, but the cells would be numbered in a way that makes computing the measure of equivalent patterns relatively easy. Does numbering them in an order traced out by a space-filling curve like a Hilbert curve or a dragon curve give any benefits over using a spiral?
I numbered my cells for a different purpose, but I had a similar motivation (making it so that equivalent rotations and reflections could be somehow ignored, and having the first N integers lie within a roughly sqrt(N) sized part of the grid). At least I think that's your motivation (-: Anyway, I couldn't think of much that was better than "antidiagonals", which is no better than spirals. I always normalized patterns by sliding them into a corner, so antidiagonals makes more sense. If you normalize by "centering" the pattern on the origin, then a spiral centered on the origin would make sense.
That's good to know, thanks! -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com