2D cellular automaton where each cell has its own state transition rules defined by 9+9 randomly chosen cells.
Preabstract. We consider 2D cellular automaton with closed torus-like space and 8-neighbours cells, similar to classical Conway's Game of Life, where 9+9 rules of state transition for each cell are not fixed, but change accordingly to the state of 18 cells that are preselected randomly and individually for this cell. The cells either follow these very rules they define, or are protected to remain in fixed state.
Oscillating patterns are observed...
There are attractors besides completely empty or completely alive field...
In a 2-dimensional cellular automaton (CA), a space, e.g. Euclidean plane $\mathbb{R}^2$, is divided regularly into cells, e.g. unit squares, that change their state accordingly to certain rules. When there are two states, 0 and 1, they are often called “empty/inactive” (0) and “alive/active” (1), and rules specify when state change occurs, which is either birth ($0 \rightarrow 1$) or death ($1 \rightarrow 0$). In totalistic CAs state transition rules depend only on the number of alive cells among 8 neighbours of a given cell (Moore neighbourhood).
Perhaps the most famous such CA is J.H. Conway's Game of Life ([1], [4]), whose rules are B3/D0145678, or, in a more traditional way, if we describe “survival” instead of “death”, B3/S23. There are $2^{18} = 262144$ such rulesets, some of them also provide complex cell behaviour.
CAs of lesser (1D) or higher (3D, 4D, ...) dimensionality, and/or with different cell shapes, states etc. were considered as well. By enumerating discrete cells of space in a spiral-like fashion starting from some origin, any such CA can be represented by a certain 1D CA, with more complicated rule expressions though.
Usually the rules are “external” (whatever happens inside CA, they do not change; put differently, they can be changed only from outside of CA) and “homogeneous” (common for all cells).
In due course, internalization of rules was studied in e.g. [6] by Pavlic et al. for 1D CAs, where rules depend on the whole CA space.
From one point of view, the rules remain external, they just become non-totalistic, since the state of a cell at the next moment now depends not only on the number of its alive neighbours, but on the state of cells that may be “far”. And there is a loss of symmetry.
When “Rules for Empty” are “Birth” and “Rules for Alive” are “Survival”, there are, obviously, at least 2 fixed points of the transition to the next step: completely empty or completely alive fields, and the process often converges to those. Other fixed points or short cycles are observed as well, with almost empty or almost alive fields. Longer non-looped trajectories are of more interest though.
...under eternal construction...