Theoretical results on infinitely dilute OCA soups?

[wzkchem5]

Hello everyone,

Recently I became interested in the dynamics of infinitely large soups in the limit of zero density. For GoL many theoretical results have already been worked out by Nick Gotts, such as the asymptotic behavior of the densities of a few emergent patterns like switch engines (see https://www.tandfonline.com/doi/abs/10. ... 7200406598 and https://link.springer.com/chapter/10.10 ... 6-217-9_20), but I have not yet found any paper on similar studies on other CAs. I wonder if anyone is aware of studies of this kind? (Note that this is a very different problem from automated soup searches, which always work on soups with finite densities; to rigorously obtain the limiting behavior of infinitely dilute soups one will have to enumerate all possible reactions between the smallest patterns, rather than simply extrapolating a series of finite density simulations, otherwise important but rare patterns such as switch engines will be overlooked).
Specifically I'm mainly interested in CAs that are Turing complete and are simple. They need not be 2D; 1D CAs such as Wolfram's Rule 110 also count.
Thank you all!

[Hunting]

I'm also interested in this kind of theory. But then I got messed up because theoretically everything (Including those we don't yet know) may arise from the "infinite dilute OCA soups" (or OCA "early universes" -- JHC).

[wzkchem5]

I'm also interested in this kind of theory. But then I got messed up because theoretically everything (Including those we don't yet know) may arise from the "infinite dilute OCA soups" (or OCA "early universes" -- JHC).

Yes, but given that so many nontrivial results have already been worked out for GoL, I guess that an analogous project for some relatively well-understood OCA is not unimaginable. If you read Nick Gotts' papers you'll see that his arguments went like: by colliding gliders with a blinker, one can make two gliders travelling in the same direction, and still more gliders may be made by colliding the fleet with still more blinkers or blocks, Other collision sequences can shift the relative positions of the gliders or remove one or more gliders, so eventually any sufficiently spaced fleet of gliders, and therefore many important (if not all) glider-constructible objects, can be made this way. Moreover, for every pattern constructible by such glider fleets (except finitely many exceptions), almost all instances of such pattern will be constructed this way after a sufficiently long period of time. It is then an easy task to estimate the asymptotic density of switch engines, or even quadratic growth patterns, after a given amount of time.
Thus once one realized that (1) the blinker and the block are the oscillator and still live with the smallest precursor respectively, that (2) the glider is the spaceship with the smallest precursor, that (3) glider-blonk reactions can give arbitrary well-spaced glider fleets, and that (4) many important patterns can be constructed by well-spaced gliders fleets, one can already say many things about the asymptotic behavior of infinitely dilute GoL soups. To estimate the asymptotic density of switch engines one additionally needs to know the minimum number of cells required to generate a switch engine (10), and that's all. All of these can be done by a finite systematic search.