[asymmetric neighborhoods] - A New Kind of Science: The NKS Forum

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asymmetric neighborhoods

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Posted by: Joel Schiff

Has anyone in the CA literature considered asymmetric neighborhoods, say for example in 1-dimension, r=1 on the left and r=2 on the right?


Posted by: Richard Phillips

To my knowledge not as such.

There are three things that one can bear in mind here, which you are probably aware of already, but I'll repeat them for wider benefit anyway.

Firstly, when one considers any neighbourhood, and looks at all the rules possible for it, these rules include all those that in effect ignore certain inputs.

An example of this is for Wolfram's 256 elementary (r=1, k=2) cellular automata, all rules numbered in binary wwxxyyzz, - ie. the 16 rules with successive pairs of bits the same - ignore their rightmost input cell. These could be considered as a set of rules in their own right with an asymmetric neighbourhood with r=1 on the left and r=0 on the right.

For the example you give, the 2 color rules with r=1 on the left and r=2 on the right, we need just look at the symmetric r=2 rules and that contains all those we wanted.

So rules in an asymmetric neighbourhood are always included as a subset of a larger symmetric neighbourhood.

So to answer your original question, because people have looked at the r=2 symmetric neighbourhood they have looked at all the included asymmetric neighbourhoods.

My second point is that that does not mean it is not worth studying the asymmetric neighbourhoods, as you clearly realise! By constructing them explicitely you focus in on them and can ask questions about them. Also one can compare them as a group to say the similar symmetric rules.

Thirdly, if one simply shifts a neighbourhood in one direction (left or right in 1D say) the overall behaviour of the resulting rules does not really change. All that happens is that the pictures of rule evolution get skewed (for finite wraparound boundaries one can think of this as continously rotating the cells). Of course actual cell values along say a vertical line will change, but the point is that globally one has not got anything new. However imposing finite boundaries that are fixed breaks this symmetry.

By the way, the Mathematica built-in CellularAutomaton function has facility for asymetric rules eg.
CellularAutomaton[{n, 2, {{-1},{0},{1},{2}}, {{1},0}, 100] gives r=1 on the left and r=2 on the right.




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