In one obvious sense, "3-state outer-totalistic" and "2-state isotropic" are two entirely different rulespaces, and hence they give different possibilities.
In another sense...
The full set of possibilities (range-1 Moore neighbourhood, three cellstates, outer totalistic rules):
- the middle cell can be in one of three states (factor 3)
- there are 45 distinguishable possibilities for the bag of neighbour cell states (factor 45):
(the first digit = the number of state-1 neighbours, second digit = the number of state-2 neighbours)
Code: Select all
00 01 02 03 04 05 06 07 08 10 11 12 13 14 15 16 17 20 21 22 23 24 25 26 30 31 32 33 34 35 40 41 42 43 44 50 51 52 53 60 61 62 70 71 80 - for each of 3 x 45 = 135 distinguishable 3x3 conditions, one can separately choose the future state of the middle cell (out of 3 possibilities).
In comparison, "only" 2^102 = 5070602400912917605986812821504 different cellular automata can be described using Hensel notation.
If 3^135 seems too many, one could consider smaller 3-state rulespaces.
For example, make the three transitions 0 -> 2, 1 -> 1, 2 -> 1 forbidden. Then "only" 2^135 = 43556142965880123323311949751266331066368 CA can be defined (where state 1 can be intuitively understood as "newborn cell" and state 2 can be intuitively understood as "old cell").
Further, arrange things so that a cell "knows" its own current state (0, 1, 2), but a cell cannot distinguish state-1 neighbours from state-2 neighbours. (Every cell knows that it is 3-state, but sees the neighbourhood as 2-state.) Then only 2^27 = 134217728 different cellular automata can be defined. (You can apply different survival conditions to "newborn" and "old" cells, but otherwise it's analogous to Life-like cellular automata.)
There are many other possibilities for choosing a sub-rulespace within 3-state R1 Moore OT.
