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}]]
