NotLiving wrote:On a related note: is there an upper bound on the depth required to support an arbitrary row of still-life with only dead cells on the other side...?
Hmm. I thought at first this was the harder question about how many rows it might take to complete a still life, or stator, once you have a complete straight row of p1 cells.* I have a vague recollection of a discussion between people who were actually good at this kind of thing, years ago but I don't remember how many. In practice it seemed as if N rows were always needed, but it seemed possible that some yet-unseen weird configuration might need N+1 or N+2... for N somewhere between 3 and 5, if I recall correctly.
I think the answer is similar for an arbitrary row of still life with only emptiness beyond: seems to be single digits, but I don't know of a formal proof. There might possibly be a general method for constructing support, where it can be done -- in which case somebody has very likely solved the problem already, I just don't remember it or don't know about it. There aren't all that many cases to deal with, unless you allow ON cells in groups longer than two... but then to make it a still life you have to match it by an inductor on the other side, which is not "only dead cells". Are you intending to include length-3 and longer inductors in your question?
NotLiving wrote:...assuming there is a way to support said row...
Even with length-1 and length-2 chains of ON cells being all that is allowed, there are some configurations that can't actually be made stable if you're working from one side -- obobobo! and longer chains of alternating ON and OFF, right? Or am I missing a trick?
*
EDIT: Vaguely related but not specifying any particular width: Richard Schroeppel's "Cool Out" Conjecture (from sometime before 1992) --
If a configuration C is locally stable over a rectangle R, then there exists a configuration C* such that (a) C* is locally equal to C over R; and (b) C* is globally stable.
-- which is still a conjecture as far as I know,
EDIT but there's a lot I don't know [see next post].
