From: Mike Stay <metaweta@gmail.com> I'd like to compute the first bits of the "halting probability" of Conway's life; since CAs don't halt, I'd like to compute the probability that it evolves to a state that's easily detectable by Golly, like all empty or a still life or cyclic. (Also, I know the number is uncomputable, but that doesn't preclude a finite computable prefix.) Since any translation or reflection of a pattern has the same outcome as the original, I'd like to choose a measure on the set of cells such that computing the measure of all the translations and reflections of a pattern is easy. Any suggestions?
--but isn't this probability zero? I mean, if you were running life on some finite-size grid-torus, then it would be nonzero but... 1. on an infinite grid, there will with prob=1 be some unboundedly large region of the grid containing the worst possible pattern it could contain, which will last for time>=T before you regard it as settled, where T can be made unboundedly large. 2. Returning to the finite-size grid-torus, since there are only a finite set of states, the thing is certain to be periodic, so then the probability is 1.