On Gottschalk's surjunctivity conjecture for non-uniform cellular automata

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.

💡 Research Summary

The paper investigates the surjunctivity and invertibility properties of non‑uniform cellular automata (NUCA) on groups that are known to satisfy Gottschalk’s surjunctivity conjecture (so‑called surjunctive groups, e.g., all sofic groups). A NUCA is a generalisation of a classical cellular automaton (CA) where each cell may use a different local rule; formally, for a countable group G, a finite alphabet A, and a finite memory set M⊂G, a configuration of local defining maps s∈S^G (with S=A^{A^M}) determines a global map σ_s:A^G→A^G by σ_s(x)(g)=s_g(x|_{gM}). The paper focuses on NUCA with finite memory and uniformly bounded singularity (the set of cells where the local rule deviates from a fixed rule is uniformly bounded in a precise sense).

Key dynamical notions are introduced: pre‑injectivity (injectivity on asymptotic configurations), post‑surjectivity (every configuration asymptotic to an image has a pre‑image asymptotic to the original), and their “stable” versions, which require the property to hold for every local perturbation σ_p with p∈Σ(s). Reversibility (existence of a left‑inverse NUCA) coincides with stable injectivity, while invertibility means bijectivity together with a NUCA inverse; stable invertibility is a stronger condition guaranteeing that every local perturbation has an inverse.

The main contributions are four theorems:

Theorem A (local perturbations on surjunctive groups). Let G be a countable surjunctive group, M a finite subset, A a finite alphabet, and s∈S^G asymptotically constant. If σ_s is stably injective (equivalently reversible) then σ_s is stably invertible. Consequently, any reversible NUCA that is a finite‑support perturbation of a CA on a surjunctive group must be invertible.

Corollary 1 specialises Theorem A using known results: (i) if G is amenable and σ_s is injective, then σ_s is stably invertible; (ii) if G is sofic and σ_s is reversible, then σ_s is stably invertible.

Theorem B (dual‑surjunctivity for post‑injunctive groups). A group G is called post‑injunctive if every post‑surjective CA on G is pre‑injective (Capobianco‑Kari‑Taati proved this for sofic groups). For any countable post‑injunctive group, finite memory M, and asymptotically constant s, if σ_s is stably post‑surjective then σ_s is stably invertible. This extends the dual‑surjunctivity theorem from CA to NUCA.

Theorem C and Theorem D lift Theorems A and B to the broader class of NUCA with uniformly bounded singularity (the set of “singular” cells is uniformly bounded for every finite neighbourhood). The same conclusions—stable invertibility from stable injectivity or stable post‑surjectivity—hold under this weaker hypothesis.

A crucial technical tool is the dual NUCA construction for linear NUCA. For a linear NUCA σ_s (A a finite‑dimensional vector space), the dual map σ_s^∘ is defined via transposition of the local linear maps. Proposition 1 establishes a series of equivalences: σ_s invertible ⇔ σ_s^∘ invertible, σ_s stably injective ⇔ σ_s^∘ stably post‑surjective, and σ_s stably post‑surjective ⇔ σ_s^∘ stably injective. Moreover, uniform bounded singularity and asymptotic constancy are preserved under duality. Using these equivalences, Corollary 2 shows that the two central open questions—whether stable injectivity (Question 1) and stable post‑surjectivity (Question 2) always imply stable invertibility—are equivalent for linear NUCA.

The paper also discusses examples where injective NUCA fail to be surjective, showing that such counter‑examples necessarily violate the uniformly bounded singularity condition, thereby underscoring the sharpness of the hypotheses.

Finally, the authors pose two open problems: (1) does stable injectivity always imply stable invertibility for arbitrary NUCA? (2) does stable post‑surjectivity always imply stable invertibility? They resolve these affirmatively for linear NUCA and for NUCA with uniformly bounded singularity, leaving the general case open.

Overall, the work extends the classical surjunctivity theorem of Gromov‑Weiss and the dual‑surjunctivity theorem of Capobianco‑Kari‑Taati from uniform cellular automata to a broad class of non‑uniform, possibly asynchronous, cellular automata. It provides a robust framework for analysing reversibility and invertibility in systems where local update rules may vary across space, with immediate implications for the theory of computation, symbolic dynamics, and the study of asynchronous models in physics and biology.