For simplicity we consider only binary CA. The extension to S > 2 is straightforward.
Outer totalistic CA rules depend on only the number of live cells in a neighborhood, with the additional condition that we allow different outcomes if the central cell is alive or dead. For one-dimensional, N = 3 rules the nbhd around the central cell can have 0, 1, or 2 cells alive. Thus S = 2, N = 3 outer totalistic CA rules have six nbhd configurations:
1. Central cell dead, 0 surrounding cells alive
2. Central cell alive, 0 surrounding cells alive
3. Central cell dead, 1 surrounding cell alive
4. Central cell alive, 1 surrounding cell alive
5. Central cell dead, 2 surrounding cells alive
6. Central cell alive, 2 surrounding cells alive
Here is how we tabulate the nbhd configurations. The number corresponding to the configurations is given below each box. | ![]() |
For example, selecting the left box under 1 means if the central cell is alive and exactly one of the two surrounding cells in the nbhd is alive, then the central cell remains alive in the next generation. The corresponding nbhd configurations are colored red. Selecting the right box under 1 means if the central cell is dead and exactly one of the two surrounding cells is alive, then the central cell becomes alive in the next generation. The corresponding nbhd configurations are colored blue. | ![]() |
Outer totalistic rules can be formulated for one-dimensional N = 5, two-dimensional von Neumann, and two-dimensional Moore CA.
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Here is an example of an Outer Totalistic rule for a von Neumann neighborhood.
Checking the left box under 1 means the central cell is alive and exactly one
of the four surrounding cells is alive. The corresponding neighborhoods are shown
in blue.
Checking the right box under 2 means the central cell is dead and exactly two of the four sorrounding cells are alive. The corresponding neighborhoods are shown in red. | ![]() |
Continue to 4. B. Cellular Automata Basics