[handsclean]

https://conwaylife.com/wiki/HashLife is an algorithm for doing basically this in Conway’s Game of Life, which is Turing complete. I remember my first impression being complete confusion: here’s a tick-by-tick simulation too varied and complex to encapsulate in a formula, and you’re telling me I can just skip way into its future?

[RetroTechie]

If I read that page correctly, it does this for areas with empty space between them?

Makes sense. Say you have a pattern (surrounded by empty space) that 'flickers': A-B-A-B-A... etc. Then as long as nothing intrudes, nth generation is the same pattern as in n+1000,000th generation. Similar for patterns that do a 3-cycle, 4-cycle etc.

All you'd need is a) a way to detect repeating patterns, and b) do some kind of collision detection between areas/patterns (there's a thing called 'lightspeed' in Life, that helps).

[handsclean]

I don’t fully understand the algorithm, but no, to my understanding it’s much more general than that. In each tick a cell’s state is solely determined by its immediate neighbors, which means the simulation has a “speed of light” of 1 cell/second: to look N ticks into the future, you need only consider cells within N cells of the area you’re computing, no matter what’s outside that. So, for example, if you want to skip a 10x10 area 100 ticks into the future, you consider a 210x210 area centered on your 10x10, compute it once, then in the future use that 210x210 area as a lookup key for the 10x10 100 ticks into the future. I think HashLife is also somehow doing this on multiple scales at once, and some other tricks.