On Gottschalk's surjunctivity conjecture for non-uniform cellular automata
Xuan Kien Phung
TL;DR
This paper extends Gottschalk's surjunctivity conjecture for group $G$ from uniform cellular automata to non-uniform cellular automata (NUCA) with finite memory on surjunctive groups, showing that reversible NUCA are invertible (indeed stably invertible) when they are local perturbations of CA. Building on Gromov–Weiss, it proves that NUCA satisfying stable injectivity over a surjunctive universe are stably invertible, and it extends the dual-surjunctivity results of Capobianco–Kari–Taati to NUCA with uniformly bounded singularity, including certain globally perturbed dynamics. The work also analyzes the relationship between surjectivity, invertibility, and stable invertibility in NUCA, and connects to asynchronous CA through local perturbations with bounded singularity. Overall, the results broaden surjunctivity theory to non-uniform dynamics on group shifts, with implications for linear NUCA and dual-surjunctivity questions in broader group classes.
Abstract
Gottschalk's surjunctivity conjecture for a group $G$ states that it is impossible for cellular automata (CA) over the universe $G$ with finite alphabet to produce strict embeddings of the full shift into itself. A group universe $G$ satisfying Gottschalk's surjunctivity conjecture is called a surjunctive group. The surjunctivity theorem of Gromov and Weiss shows that every sofic group is surjunctive. In this paper, we study the surjunctivity of local perturbations of CA and more generally of non-uniform cellular automata (NUCA) with finite memory and uniformly bounded singularity over surjunctive group universes. In particular, we show that such a NUCA must be invertible whenever it is reversible. We also obtain similar results which extend to the class of NUCA a certain dual-surjunctivity theorem of Capobianco, Kari, and Taati for CA.