wolfram code

Wolfram code

A numbering system for one-dimensional CA rules, introduced by Stephen Wolfram.

For a CA with $S$ states and neighborhood size $N$, each cell’s next state is determined by one of $S^N$ possible neighborhood configurations. A rule maps each configuration to a next state, giving $S^{S^N}$ possible rules total.

The Wolfram code uniquely identifies each rule by encoding its complete transition function as an $S$-ary number:

$$n=\sum _{i=0}^{S^N-1}{s}_i\cdot S^i$$

where $s_i$ is the next state for configuration $i$.

Algorithm

1. List all $S^N$ neighborhood configurations in descending numerical order
2. For each configuration, determine the next state according to the rule
3. Interpret the sequence of next states as an $S$-ary number, convert to decimal

See elementary CA (1D) for the most common case ($S=2$, $N=3$, giving 256 possible rules) with detailed examples and visualizations.