An outer-totalistic cellular automaton is a generalization of the totalistic cellular automaton. Totalistic rules are a proper superset of outer-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 the Wolfram Language 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}]]

REFERENCES:

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 927-928, 2002.