Table 14: Global Properties for Finite Lattices

Global properties of cellular automata on finite lattices.

This table gives some properties of state transition diagrams for finite cellular automata. Each picture shows values plotted as a function of lattice size , varying from 3 to 16. (Periodic boundary conditions are assumed for the cellular automaton evolution.)

The quantities shown are as follows. (In all cases, values for integer sizes are shown joined by lines.)

``Mean transient length'' represents the average number of steps necessary for any particular state to evolve to a cycle. ``Maximum transient length'' gives the maximum number of steps needed.

``Mean cycle length'' gives the average length of the cycle on to which any particular state evolves. (Each cycle is thus weighted in the average with the number of states which evolve to it.) ``Maximum cycle length'' gives the longest cycle for each value of . Some such cycles are shown in table 13.

``Fraction on cycles'' (given in logarithmic form) represents the fraction of all possible states which appear on cycles, and thus can occur after a long time. This quantity is related to the set (topological) entropy for invariant set of the evolution.

The last picture for each rule gives the total number of distinct cycles (in logarithmic form). This can be considered as the number of possible distinct attractors for the evolution.

Table by Holly Peck (Los Alamos National Laboratory). (Original program by S. Wolfram.)

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