Classification of 1D Cellular Automata

Examples of Class I Behavior:


1
2
3
4
5
6
7
If the bottom of the image is all solid color or if the lines going down are solid, then these are Class I. It means that the CA has approached a fixed point, and that it won't change anymore. The CAs in 5-7 have a short Class IV transient at the top before the CA exhibits Class I behavior.

Examples of Class II Behavior:


1
2
3
4
5
6
7
8
9
10
11
12
In 1-5, the strands going down are periodic since they repeatedly cycle through different states all the way down. In 3 and 4, the strands oscillate between two states. The pattern is a little hard to see in 5, but zooming in should reveal it. The CA in 5 also has a short IV transient at the top. In 6 and 7, the pattern repeats also, but in these cases, it stretches the entire width of the image (the space dimension) as well. There is a short IV transient in 7. The CA in 8 is an example of unbounded periodicity, as explained on page 239 of the textbook. The CA in 9 is similar to that in 1, but there is a IV transient at the top. The CAs in 10-12 are periodic since the pattern repeats. In 11 and 12, there is a Class III transient at the beginning before the pattern repeats, but they are still classified as Class II. The behavior at the bottom of the image is what determines the classification of the CA.

Examples of Class III Behavior:


1
2
3
4
5
6
7
8
9
10
11
Examples 1 and 2 are complete noise, and 3 and 4 are starting to become something interesting, but for the most part, they are noise as well. The CAs in 5-11 would be considered near Class IV, but they are still Class III. They do not exhibit the complex behavior that is present in Class IV CAs. Examples 10 and 11 are even closer to Class IV than examples 5-9.

Examples of Class IV Behavior:


1
2
3
4
5
6
7
These examples have some nice Class IV behavior. They basically look as if holes had been cut out of them. Sometimes these "holes" can be fairly large, as in example 2. Sometimes these CAs will also show strands hanging down inside these "holes", sort of like stalactites in a cave.


8
9
These two examples are more borderline with Class III since the complex patterns are more subtle. The CA in 8 does have some interesting structure, if you look closely. The orange-red "holes" in 9 are representative of Class IV behavior, but most likely, this example is barely a Class IV. It would most likely be a Class IV but near Class III.