2D cellular automaton, part of which defines its birth/death rules. Hint: press G
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 birth/death rules for the whole space are not fixed, but change accordingly to state of 18 selected cells. These cells either follow the very rules they define, or are protected to be in fixed state to avoid some degenerate behaviours.
Oscillating patterns were observed in some cases... Total extinction is possible as well...
The question arises of rulesets that can be encoded, without protection, by cells behaving accordingly to such ruleset, and those that cannot...
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 “dead/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 birth/death rules depend only on the number of alive cells of a given cell among its 8 neighbours (Moore neighborhood).
Perhaps the most known such CA is J.H. Conway's Game of Life [1], 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, any such CA can be represented by a certain 1D CA, with more complicated rule expressions though.
Usually the rules are “external” in the sense that whatever happens within CA, they do not change; put differently, only someone or something outside CA can change the rules.
In due course, internalization of rules was studied in e.g. [5] 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.
...under eternal construction...