Totalistic Rules  

A totalistic rule for a cellular automaton determines the fate of a cell by adding up the states of the cells around it ("neighbors"). The sum of neighbors is taken without regard for what direction the neighbors are in. For example, in these three patterns all the central cell A has three "live" neighbors:

. o o . . o o . . o A . . A o . A o . . . . . o o . .

In a totalistic rule, the fate of A would be the same in all three cases. (The fate of the other cells, and therefore of the whole 4-cell pattern, is different in each case).

In totalistic rules typically the previous state of the cell is also considered, allowing for different fates based on the combination of whether the cell itself is live and how many live neighbors it has. A complete description of the rule includes an answer for every possible combination, like this:

cell was alive cell was dead
0 live neighbors DIES no change
1 live neighbors DIES no change
2 live neighbors no change no change
3 live neighbors no change BIRTH
4 live neighbors DIES no change
5 live neighbors DIES no change
6 live neighbors DIES BIRTH
7 live neighbors DIES no change
8 live neighbors DIES no change

For reasons more to do with implementation in a computer program, the rule table is more commonly thought of as being like this:

cell was alive cell was dead
0 live neighbors dies no change
1 live neighbors dies no change
2 live neighbors SURVIVES no change
3 live neighbors SURVIVES BIRTH
4 live neighbors dies no change
5 live neighbors dies no change
6 live neighbors dies BIRTH
7 live neighbors dies no change
8 live neighbors dies no change

Simulating Patterns of Unlimited Size

For this discussion the term horizon refers to the infinite area surrounding the pattern or area of interest. In most cases this infinite area is assumed to be "blank": all cells have value 0.

Inverse and Flashing Rules

If a totalistic rule includes B0 (or lacks Z0) then any large area of 0 cells, including the horizon, will become all 1 cells on the next generation.

If the rule also contains S8, then the horizon remains all 1. This is an "inverse rule".

If a rule has B0 and D8 (i.e. it lacks S8) then any large solid area (including the horizon) will alternate between 0 and 1. In a simulation, the horizon and any solid regions will "flash" or "strobe" on and off.


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