I'm not sure if I understand the question fully... I think rather than all "patterns that have an infinite sequence of ancestors", we would need to at least restrict it to patterns that do not end as a still-life [1] or oscillator[2]... Let S be the set of patterns that have an infinite sequence of ancestors. Many finite, unchanging patterns (still-lifes) are a member of S, because they can be formed by a collision of two or more gliders[3]. For example, a block[4] can be formed by colliding two gliders (see [5] for details). Each pattern containing two gliders that is an ancestor of the block pattern also belongs to S, and these are all distinct from one another. The boat [6] also belongs to S, and has a glider synthesis, therefore it also has an infinite sequence of distinct ancestors that also belong to S. If the Adam/Eve starting pattern's evolution includes any of the block predecessors, its subsequent evolution must lead to the block. But if it included any of the boat predecessors, its subsequent evolution would lead to the boat. The evolution of the pattern can only end one way, so it cannot contain both the block predecessors and the boat predecessors. The same argument works for oscillators and spaceships[7], so we have to exclude those too. And the same argument would extend to puffer trains [8] because for example a puffer train that leaves a trail of blocks will always be distinct from one that leaves a trail of boats. In fact, I think if you can find any two patterns that have a mutually exclusive forward history, they cannot both be in the forward history of the Adam/Eve pattern. [1] http://www.conwaylife.com/wiki/Still_life [2] http://www.conwaylife.com/wiki/Oscillator [3] http://www.conwaylife.com/wiki/Glider [4] http://www.conwaylife.com/wiki/Block [5] http://www.conwaylife.com/wiki/Glider_synthesis [6] http://www.conwaylife.com/wiki/Boat [7] http://www.conwaylife.com/wiki/Spaceship [8] http://www.conwaylife.com/wiki/Puffer_train On Mon, Dec 5, 2011 at 20:29, Dan Asimov <dasimov@earthlink.net> wrote:
<< FWIW, it feels like a direct Adam/Eve nonexistence proof might be simpler than a direct Garden of Eden existence proof.
Good point. Now let me make a feeble attempt to rescue the Adam & Eve question.
Consider those patterns each having an infinite sequence of ancestors.
Is there a starting pattern for which every one of these eventually appears somewhere among its descendants?
(Alternatively, consider each pattern that, for every N in Z+, has a sequence of N ancestors. Same question.)
--Dan
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