Hard Asymptotic Sets for One-Dimensional Cellular Automata

TL;DR

This paper demonstrates that the language of asymptotic and nonwandering sets in one-dimensional cellular automata can be highly complex, specifically $ ext{SIGMA}^1_1$-hard, indicating deep computational complexity.

Contribution

It establishes the $ ext{SIGMA}^1_1$-hardness of the asymptotic and nonwandering sets for 1D cellular automata, extending understanding of their computational complexity.

Findings

Asymptotic sets can be $ ext{SIGMA}^1_1$-hard.

Nonwandering sets can be $ ext{SIGMA}^1_1$-hard.

Standard constructions are sufficient for these proofs.

Abstract

We prove that the (language of the) asymptotic set (and the nonwandering set) of a one-dimensional cellular automaton can be $\SIGMA^1_1$-hard. We do not go into much detail, since the constructions are relatively standard.