Generically Nilpotent Cellular Automata

TL;DR

This paper investigates the asymptotic behavior of one-dimensional cellular automata by characterizing those with singleton generic limit sets and establishing their computational complexity and limitations.

Contribution

It provides a characterization of automata with singleton generic limit sets and proves the class's $\, ext{Sigma}^0_2$-completeness, highlighting computational limitations.

Findings

Automata with singleton generic limit sets are characterized.

The class of such automata is $\, ext{Sigma}^0_2$-complete.

The unique configuration in these automata cannot be algorithmically identified.

Abstract

We study the generic limit sets of one-dimensional cellular automata, which intuitively capture their asymptotic dynamics while discarding transient phenomena. As our main results, we characterize the automata whose generic limit set is a singleton, and show that this class is $\Sigma^0_2$-complete. We also prove that given a CA whose generic limit set is guaranteed to be a singleton, the sole configuration it contains cannot be algorithmically determined.