danny wrote:How would I go about searching for a still life without parents? Should I try writing a script? Or even, dare I say, a distributed computing project? (the last part of that sentence is purely hypothetical as I cannot program well enough or pay for a website)
I'm afraid that anyone who could answer that question in enough detail, would probably have written the script already.
It's a tricky problem. All still lifes have one parent, so you have to run something like a JLS predecessor search to prove that there aren't any other parents... but you can't just run a search on a small part of it and show that nothing can change in that limited context. Maybe that small part changes, and at the same time the area around it changes, and you might have a predecessor after all.
Maybe something similar to Nicolay Beluchenko's GoE search program? Add cells consistent with stability around the edge of a pattern, always picking the choice such that the number of predecessors increases as little as possible, or even decreases. That would be a painfully slow search -- you'd have to run a full lifesrc-type scan for predecessors, for each possible addition, and the number of possibilities keeps increasing as the object gets bigger.
The unique-father problem might actually be even more painful than the constructibility-by-gliders problem. We could maybe come up with some delicately balanced still life where we can show that such-and-such key cells can't ever be OFF, because that would cause a lightspeed collapse of some hard-to-reach central area. Something like that, anyway. But it's very hard to make it so every
cell is a key cell, and absolutely no alternate predecessors can be found even if they're Gardens of Eden. GoE parents wouldn't be any help in finding a glider construction, but they'd still disqualify a unique-father candidate.
... TL;DR: Just prove omniperiodicity instead, it's much