Generically Nilpotent Cellular Automata

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.