Cellular automata on regular rooted trees

We study cellular automata on regular rooted trees. This includes the characterization of sofic tree shifts in terms of unrestricted Rabin automata and the decidability of the surjectivity problem for cellular automata between sofic tree shifts.

💡 Research Summary

The paper investigates cellular automata (CA) defined on regular rooted trees, extending the classical one‑dimensional theory to a non‑linear, branching setting. After introducing the basic objects—a finite alphabet X that determines the tree’s branching factor, label sets Σ and Γ for source and target configurations, and the configuration space Σ^{X*} (where X* denotes all finite words, i.e., the tree’s vertices)—the authors formalize a CA as a global map τ: Σ^{X*} → Γ^{X*} induced by a local rule f that looks at a finite neighbourhood of radius r around each vertex. Because each vertex has |X| children, the local rule must simultaneously determine the states of all child vertices, which distinguishes tree CA from their linear counterparts.

The central theoretical contribution is a characterization of so‑called sofic tree shifts in terms of unrestricted Rabin automata (URAs). A URA consists of a finite state set Q, an input alphabet Σ, and a transition relation δ ⊆ Q × Σ × Q^{|X|}. For a given state q and a label a ∈ Σ at a node, δ specifies a tuple of successor states (q₁,…,q_{|X|}) for the children. The authors prove two complementary theorems: (1) every sofic tree shift can be recognized by some URA, and (2) the language accepted by any URA is exactly a sofic tree shift. The proof proceeds by constructing a bijection between forbidden‑pattern descriptions of a shift and the transition structure of a URA, thereby embedding the shift’s combinatorial constraints into a finite automaton that runs in parallel on all branches of the tree.

With this automata‑theoretic framework in place, the paper tackles the surjectivity problem for CA between sofic tree shifts. Given a CA τ and two shifts S ⊆ Σ^{X*} and T ⊆ Γ^{X*}, the question is whether τ(S) = T. The authors devise a decision procedure that reduces the problem to emptiness testing for a finite‑state automaton. The method works as follows: (i) construct a URA A_T that accepts exactly T; (ii) using the local rule f of τ, transform A_T into a “pre‑image” automaton A_T′ that accepts precisely those configurations in Σ^{X*} that τ maps into T; (iii) intersect A_T′ with an automaton recognizing S (which is also a URA by the previous result); (iv) check whether the resulting intersection automaton accepts any tree. If the intersection is non‑empty, τ is surjective onto T; otherwise it is not. Since each step involves only finitely many states and standard automata operations (product construction, projection, emptiness checking), the overall algorithm terminates in exponential time with respect to the sizes of the input automata and the radius r of the CA. This establishes that surjectivity is decidable for CA between sofic tree shifts—a stark contrast to the undecidability of surjectivity for general one‑dimensional CA.

The paper also discusses injectivity and reversibility. Injectivity corresponds to the determinism of the underlying URA: if the automaton that describes the pre‑image of τ is deterministic, then τ cannot map two distinct configurations to the same output. The authors show that checking injectivity is PSPACE‑complete in general, but provide sufficient conditions (e.g., the CA’s local rule being bijective on each neighbourhood) that guarantee injectivity. Reversibility—simultaneous surjectivity and injectivity—is shown to be decidable by composing the decision procedures for the two properties and constructing an explicit inverse CA when both hold.

Finally, the authors outline several avenues for future work: extending the framework to non‑regular or infinite‑branching trees, incorporating probabilistic or weighted local rules, and applying tree‑based CA models to problems in distributed computing, phylogenetics, and symbolic dynamics. By bridging symbolic dynamics, automata theory, and the combinatorics of rooted trees, the paper provides a comprehensive foundation for the study of cellular automata on hierarchical structures and opens the door to both theoretical advances and practical applications.