Compound Cellular Automata

Previously, I described the Compound Automata. Here, I decided to show a bit more of the pattern finding workflow.

In this example, I will start with some arbitrary rule, $R_0(\mu)$, with $\mu$ being some parameters. Then, I will combine several instances of $R_0$ with different parameters into more complex rules.

Density

Density, the fraction of “alive” cells is a very important parameter to decide which patterns will be combined together into a more complicated rule.

Below, the outputs of my $R_0$ rule with different parameters, resulting in different densities.

In general, the first step for me to generate a large collection of $R_0$ parameters with different densities, to later combine them.

First level combination

Now, lets take a larger neighborhood. The largest neighborhood used in $R_0$ has range $r=14$, so for $R_1$, the next rule, I will use a neighborhood with range $r=100$. Lets call this neighborhood $N_1$.

Our first compound rule, $R_1$, will be defined like this:

• if $\sum_{N_1} cells < a_1$, we apply $R_0(\mu_1)$, our base rule with some parameters $\mu_1$
• if $\sum_{N_1} cells > b_1$, we apply $R_0(\mu_2)$, our base rule with some parameters $\mu_2$, different from $\mu_1$
• else, nothing changes.

For this first example, I will take the following two paramsets.

$R_0(\mu_1)$ with density 0.64:

$R_0(\mu_2)$ with density 0.24:

There are several values of $a_1$ and $b_1$ that will work. Lower values will lead to dominance of the low density pattern. In the animation below, I set $b_1=a_1+50$, and then slowly increased the values of $a_1$. You can observe how we start with low density pattern, then the high density pattern slowly emerges and then dominates it.