My goal is to find all rulesets of life-like cellular automata and initial states that, when simulated in a toroidal grid (although non-toroidal would be interesting as well) of arbitrary finite dimensions, eventually end up in a fixed end state that doesn't oscillate and can be trivial (empty or full).
It seems like there are multiple classes of solutions:
Rulesets that always generate a stationary end state from any arbitrary initial state (e.g. All "no death" rules B0/S012345678, B1/S012345678). However, I'm unsure if there any more rulesets with this fixed end state guarantee.
Non-trivial initial states in rulesets not contained in the first category that end in all still life, or trivial states. This section seems like it would be extremely large and difficult to generate these initial states computationally.
Trivial initial states in any ruleset that don't change, or simply flip to the other state.
For my purposes I am interested in the first class of these solutions mostly, so that I can simply generate random initial states and guarantee it will eventually reach a fixed end state.
And also, some way of randomly generating arbitrary initial states that conform to the second class, but my hunch is it's not easy.