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1D rules in a finite universe

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1D rules in a finite universe

Postby 137ben » November 27th, 2010, 3:45 pm

I decided to have a look at a few aspects of these rules. For convenience, I made a rule table that only stores the current generation. This saves processing power, allows us to use features like pop-plot.py and oscar.py, and, allows us to compare two different patterns one on top of the other. Here is the table for W110 (I saved it as R110.table).
n_states:2
neighborhood:vonNeumann
symmetries:none

var a={0,1}
var b={0,1}


# C,N,E,S,W,C'

0,a,0,b,0,0
0,a,0,b,1,0
0,a,1,b,0,1
0,a,1,b,1,1
1,a,0,b,0,1
1,a,0,b,1,1
1,a,1,b,0,1
1,a,1,b,1,0


For W22, simply use B3/S23:T0,1. Of course, this does not allow you to compare multiple patterns side-by-side, so a rule table would still be helpful.
Actually, how hard would it be to write a script to generate a rule tree for 1d elementary CA?

One question is: if you start with a single cell in a circular world, what will the eventual period be?
For R110, in a circular universe of width n, we have period:
1,1,1,2,1,9,14,7,7,25,110,18,351...
For W22 we have:
1,1,1,1,1,1,1,1,4,4,4,4,4,1,1,1,12,12,12,12

Then, there's the question of how long it takes to stabilize. In W22, the time taken for a single cell to stabilize in an n cell circular universe is:
1,0,2,2,5,4,4,4,2,2,2,2,2,8...
137ben
 
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Re: 1D rules in a finite universe

Postby ynotds » January 13th, 2011, 8:16 am

Golly supports all even numbered Wolfram rules presenting successive iterations down the grid and now in 2.2 including bounded or cyclic grids, by setting the rule to something like W22:P35,0 or W110:T36,0.

I'm not aware of any method for using rule tables/trees to produce the desirable down the grid display, but don't see creating a rule tree (at least for even numbered rules) to operate on a single line of cells to be particularly difficult. (Such a process could potentially be converted to a similar but up the grid display using a script, but that might be overkill and preclude using other scripts anyway.)

My own first post to Wolfram's NKS Forum remains the only public record of a few months studying a system I called "Trapper" which turned up multiple times in my studies of "Life in a Tube" and which is equivalent to several 3 colour 1D CA rules. A quick example:
#CXRLE Pos=-36,0
x = 72, y = 290, rule = B3/S23:T72,0
2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b
2o$2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b
2o2b2o3$72o4$72o8$72o12$72o4$72o4$72o4$72o4$72o4$72o4$72o4$72o4$72o4$
72o4$72o8$72o4$72o4$72o4$72o4$72o4$72o4$72o4$72o4$72o8$72o28$72o8$72o
4$72o4$72o8$72o4$72o12$72o8$72o4$72o4$72o12$72o4$72o4$72o4$72o12$72o
24$72o4$72o8$72o4$72o8$72o2$3b2ob2o3b2ob2o3b2ob2o3b2ob2o3b2ob2o3b2ob2o
3b2ob2o3b2ob2o3b2ob2o$4bobo5bobo5bobo5bobo5bobo5bobo5bobo5bobo5bobo$4b
obo5bobo5bobo5bobo5bobo5bobo5bobo5bobo5bobo$3b2ob2o3b2ob2o3b2ob2o3b2ob
2o3b2ob2o3b2ob2o3b2ob2o3b2ob2o3b2ob2o!
As mentioned in my final follow up in that thread, I only later discovered the similarity to Rules 109 and 73 which have their own means of partitioning a larger grid into separate stripes, each of which will run irreversibly through an often very large number of configurations before eventually settling into an also often large loop, one example of which is attached to that follow up.
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