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Any rule that doesn't have gardens of eden?
Posted: December 28th, 2015, 2:07 pm
by drc
Besides B/S012345678?
Re: Any rule that doesn't have gardens of eden?
Posted: December 28th, 2015, 2:15 pm
by BlinkerSpawn
No reversible rule can have gardens of eden.
Re: Any rule that doesn't have gardens of eden?
Posted: December 28th, 2015, 5:42 pm
by Sphenocorona
Conversely, if I'm not mistaken, all non-reversible rules must have gardens of eden, or orphans. This is because for a non-reversible rule, there is always at least one pattern that has more than one immediate predecessor - otherwise, we could run a pattern backwards arbitrarily far, and the rule would be reversible. Since there are two predecessors for some pattern, there must be orphans, patterns with no predecessors.
I'm pretty sure the Replicator rule B1357/S1357 is reversible (though sometimes needing infinitely populated predecessors to patterns with finite population), so it cannot have orphans. The same is true for B1357/S02468.
I'm not sure what other outer-totalistic rules are reversible, if any.
Re: Any rule that doesn't have gardens of eden?
Posted: December 29th, 2015, 1:52 am
by SuperSupermario24
Sphenocorona wrote:I'm pretty sure the Replicator rule B1357/S1357 is reversible (though sometimes needing infinitely populated predecessors to patterns with finite population), so it cannot have orphans. The same is true for B1357/S02468.
But the problem with including infinite population is that there are multiple infinite agars that will just lead to a universe with all cells dead, and therefore it's not reversible even with infinite populations.
(unless I'm missing something here)