Conway's Game of Life

Is there a mathematical way to directly calculate iteration n from the first iteration skipping calculating the iterations in between in Conway's Game of Life? I would assume, if it is possible, it would involve arrays.

Solution 1:

Nope. Life is Turing-complete - meaning that every Life pattern can be simulated by a Turing Machine and vice versa - so short of solving the Halting Problem, the only way of knowing how a given configuration will end up is to run the simulation.

I think that applies to any cellular automaton, actually, not just Life (well, short of rules where every pattern dies or otherwise exhibits predictable/consistent behavior) - although I'm not sure how to prove it for a non-Turing-complete CA.

Solution 2:

There are techniques like HashLife for rapidly computing a long-term state of a starting configuration.

However, at some level this is simply an optimization which uses memorization to takes advantage of the redundancy in the ruleset using quadtrees (a functional data structure used here as a kind of deconstructed representation of the field.)

Solution 3:

Your question is not very clear because "calculating n-th step while skipping intermediate steps" is difficult to formalize. However, Game of life is P-complete meaning that any polynomial time algorithm can be reduced to the problem of calculating the future state of a cell in Game of Life. Therefore, if you have an algorithm to predict Game of life which is fast in the worst case, then you deduce fast algorithms for any polynomial time problems, which is very unlikely even if you allow parallelism (cf. P vs. NC). Hashlife works well on typical or very sparse inputs, not on all inputs.