An outer-totalistic cellular automaton is a special case of a general totalistic cellular automaton. Outer-totalistic rules are a proper superset of totalistic rules. In particular, consider the cellular automaton rule

so that the center cell with value changes to value when bordered by cells with values and . The cells with values and are called the outer cells.

In a totalistic cellular automatic, the total value of the cells () is considered, and for each possible value of that total, the rule output is given. So a list of entries, each from 0 to are needed.

In an outer-totalistic cellular automaton, both the center cell value () and the outer total () are considered. Note these are trivially independent quantities. For each combination of the center value and outer-total , the rule output is given. So a matrix with rows and columns is needed with entries each 0 to .

This can be generalized to more outer cells (e.g., two on each side), to two dimensions, and so on.

A -color outer-totalistic cellular automaton can be generated in Mathematica using

  CellularAutomaton[{n, {k, {k, 1, k}}, 1}, init, steps,
    {All, All}]

Similarly, 9-cell two-dimensional outer totalistic rules can be given for a single row through time and the last step, respectively, by

  First /@ CellularAutomaton[{n,
      {k, {{k, k, k}, {k, 1, k}, {k, k, k}}}, {1, 1}
    },
    init, steps, {All, {0}, All}]
  First[CellularAutomaton[{n,
      {k, {{k, k, k}, {k, 1, k}, {k, k, k}}}, {1, 1}
    },
    init, steps, {-1, All, All}]]

This entry contributed by Richard Phillips

CITE THIS AS:

Phillips, Richard. "Outer-Totalistic Cellular Automaton." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Outer-TotalisticCellularAutomaton.html