Cellular automata between sofic tree shifts

We study the sofic tree shifts of $A^{\Sigma^}$, where $\Sigma^$ is a regular rooted tree of finite rank. In particular, we give their characterization in terms of unrestricted Rabin automata. We show that if $X \subset A^{\Sigma^*}$ is a sofic tree shift, then the configurations in $X$ whose orbit under the shift action is finite are dense in $X$, and, as a consequence of this, we deduce that every injective cellular automata $\tau\colon X \to X$ is surjective. Moreover, a characterization of sofic tree shifts in terms of general Rabin automata is given. We present an algorithm for establishing whether two unrestricted Rabin automata accept the same sofic tree shift or not. This allows us to prove the decidability of the surjectivity problem for cellular automata between sofic tree shifts. We also prove the decidability of the injectivity problem for cellular automata defined on a tree shift of finite type.

💡 Research Summary

The paper investigates symbolic dynamics on regular rooted trees of finite degree, focusing on so‑fic tree shifts—subsets of the configuration space A^{Σ*} that are images of finite‑type tree shifts under cellular automata. The authors first introduce unrestricted Rabin automata (URA), a natural extension of classical Rabin automata to tree structures. A URA consists of a finite state set Q, an alphabet A, and a transition relation E⊆Q×A×Q^{Σ}, where a transition simultaneously determines the states of all children of a node. They prove a two‑way characterization: a set X⊆A^{Σ*} is a so‑fic tree shift if and only if there exists a URA M such that X equals the language L(M) accepted by M. The proof builds a bridge between the shift‑invariance and continuity of X and the local transition rules of M, showing that any URA language is automatically shift‑invariant and closed, hence a tree shift, and conversely that any so‑fic shift can be realized as the image of a finite‑type shift under a sliding block code, which can be encoded by a URA.

Next, the authors establish a density of finite‑orbit configurations inside any so‑fic tree shift. They show that configurations whose orbit under the natural shift action is finite form a dense subset of X. This result generalizes the classical “periodic points are dense” theorem from one‑dimensional so‑fic shifts to the tree setting, despite the combinatorial explosion caused by branching. The density argument relies on constructing, for any finite pattern, a periodic extension that respects the constraints encoded by the URA.

Using the density theorem, the paper proves a Garden‑of‑Eden type result for cellular automata (CA) defined on a so‑fic tree shift: any injective CA τ:X→X is automatically surjective. The argument is indirect: if τ were injective but not surjective, a configuration outside the image would be approximated by finite‑orbit points, contradicting injectivity because τ preserves the finite‑orbit property. Consequently, the classical “injective ⇒ surjective” property holds for CA on so‑fic tree shifts, a non‑trivial extension of the Curtis‑Hedlund‑Lyndon theorem to branching spaces.

The paper then turns to algorithmic questions. It presents a decision procedure for testing whether two URAs accept the same tree shift. The method first normalizes each automaton (removing unreachable states, merging equivalent transitions) and then checks bisimulation equivalence of the resulting finite transition graphs. Because the underlying structures are finite, the equivalence test runs in polynomial time in the number of states. As a direct corollary, the surjectivity problem for CA between so‑fic tree shifts becomes decidable: given two URAs describing the domain and codomain shifts and a CA defined by a local rule, one can construct a product automaton that captures the image of the CA and compare it with the codomain automaton using the equivalence test.

Finally, the authors address the injectivity problem for CA defined on a shift of finite type (SFT) on the tree. An SFT is specified by a finite set of forbidden patterns of bounded radius. By translating the CA’s global rule into a finite‑state transducer and analyzing the induced relation on pattern cylinders, they devise an algorithm that decides whether the CA is injective. The algorithm essentially checks whether the transducer is reversible, which can be decided by constructing the inverse relation and verifying functional consistency—a process that terminates because the SFT constraints keep the state space finite.

In addition to URAs, the paper shows that general Rabin automata (RA)—which incorporate accepting and rejecting state sets and Rabin acceptance conditions on infinite paths—are equally expressive for describing so‑fic tree shifts. This equivalence broadens the toolbox available for researchers: one may choose the model that best fits a particular application (e.g., model‑checking versus symbolic dynamics) without losing expressive power.

Overall, the work makes three major contributions: (1) a clean automata‑theoretic characterization of so‑fic tree shifts via URAs and RAs; (2) a dynamical systems result that injective CA on these shifts are surjective, together with the density of finite‑orbit configurations; and (3) decidability of both surjectivity (between arbitrary so‑fic shifts) and injectivity (on SFTs) for cellular automata, supported by concrete algorithms. These results extend fundamental theorems of one‑dimensional symbolic dynamics to the richer setting of regular rooted trees, opening new avenues for research in multidimensional cellular automata, tree‑automata theory, and the algorithmic analysis of dynamical systems on branching structures.