Cellular Automata with Global Control

The "rule selection functions" may be described as follows. The parity rule uses the second rule if the parity of the cellular automaton at a given point in time is even; otherwise it chooses the first rule. The majority rule uses the second rule if the majority of the cellular automaton cells are in state 1; otherwise it uses the first rule. The remaining two functions hint at the additional capabilities of the code behind this Demonstration. The "time parity" selection function exemplifies what has been termed an interactive cellular automaton, in which the cellular automaton can be "programmed" to choose an update rule based on the iteration number; the state of the cellular automaton thus has a spatial and temporal component. Finally, the spacetime parity rule looks at the parity of the sum of (a) the parity of the cellular automaton and (b) the time parity. It thus creates the possibility of spatio-temporal interaction.

The code underlying this Demonstration should accommodate all sorts of cellular automaton rules, including those with more than 2 colors or a radius greater than 1.

The cellular automata used here all have widths fixed by their initial conditions.

Cellular automata with global control may prove quite useful in modeling interactions in which behavior depends on the behavior of neighbors and on global information produced by some sort of aggregating mechanism such as a market.

Snapshot 1: the interaction of Rule 11 and Rule 28 with majority rule function selection starting from a de Bruijn Sequence

Snapshot 2: the interaction of Rule 11 and Rule 28 with time parity rule selection starting from a de Bruijn Sequence

Snapshot 3: the interaction of Rule 30 and Rule 76 with spacetime parity rule selection starting from seed

Snapshot 4: the interaction of Rule 30 and Rule 43 with time parity rule selection starting from randomness