Additive cellular automata over ℤp and the bottom of (CA,≤)

Abstract

In a previous work we began to study the question of “how to compare” cellular automata (CA). In that context it was introduced a preorder (CA,≤) admitting a global minimum and it was shown that all the CA satisfying very simple dynamical properties as nilpotency or periodicity are located “on the bottom of (CA,≤)”. Here we prove that also the (algebraically amenable) additive CA over ℤ p are located on the bottom of (CA,≤). This result encourages our conjecture that says that the “distance” from the minimum could represent a measure of “complexity” on CA. We also prove that the additive CA over ℤ p with p prime are pairwise incomparable. This fact improves our understanding of (CA,≤) because it means that the minimum, even in the canonical order compatible with ≤, has infinite outdegree.