It's possible that the rule-string I used in the experiments I half-recall wasn't RL (or equivalently LR) but something slightly more complicated. But there is some short rule-string that yields the behavior I described (provably yielding bilaterally symmetric world-states infinitely often). Jim Propp On Sunday, May 24, 2015, James Propp <jamespropp@gmail.com> wrote:
Thanks for your reply, Steve. But why do you say "very short" instead of "finite"? Perhaps you're saying that there's a dichotomy in the observational data, wherein initial conditions either return quickly or never return at all?
Anyway, I am indeed asking about the model in which a bee travels along the edges of a hexagonal grid and each hexagon carries one bit which (a) tells the bug traveling along the incoming edge which way to turn and (b) flips each time it is used.
Question: If all the bits start out being equal, is there a later time at which the bits are all equal again? That is, is the bee's path bounded for this particular initial state?
Perhaps one of you who is comfortable with CA software could generate some pictures of what one gets after, say, a million steps. (I never went that far, and my old pictures, which I can't locate, were ASCII.) If you choose the stopping time properly, the picture will exhibit bilateral symmetry with respect to the line that extends the edge that the bee started on.
Jim Propp
On Saturday, May 23, 2015, Steve Witham <sw@tiac.net <javascript:_e(%7B%7D,'cvml','sw@tiac.net');>> wrote:
I had a link to an paper proving some basic-sounding theorems about the bee. At the moment, I can't find it among my links about Langton's Ant. The alternative to the path being unbounded is that the bug cycles; there are "initial" states producing very short cycles for the bee. (Scare quotes because such a state is just one of the states in the loop.)
The bee I'm talking about has a hexagonal cell with one bit of state: turn left or right and toggle the state, as with Langton's Ant. I believe Patterson's worm has a lattice where each edge has a bit of state, and once eaten, an edge can't be traveled again (and so it's sometimes possible for the worm to paint itself into a corner and halt).
--Steve
Date: Thu, 21 May 2015 18:41:36 -0400 From: James Propp <jamespropp@gmail.com>
I studied a very simple combination of initial-state and rule-string that demonstrably gives states of the universe with bilateral symmetry infinitely often, and such that the path of the bee appears to be unbounded, but I was never able to prove the latter assertion; I looked in vain for a combinatorially-defined "arrow of time" for this system.
Since a couple of decades have passed, maybe this problem has been solved while I was thinking about others things!
Jim Propp
------------------------------ Date: Fri, 22 May 2015 04:24:01 +0200 From: "Adam P. Goucher" <apgoucher@gmx.com>
The most well-known hexagonal analogues of Langton's ant are Paterson's Worms.
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