Cellular automata between sofic tree shifts

TL;DR

This paper characterizes sofic tree shifts using Rabin automata, proves density of finite orbit configurations, and establishes decidability results for cellular automata properties on these shifts.

Contribution

It provides a new characterization of sofic tree shifts via Rabin automata and introduces algorithms for automata equivalence and cellular automata properties.

Findings

Configurations with finite orbits are dense in sofic tree shifts.

Injective cellular automata on sofic tree shifts are surjective.

Decidability of surjectivity and injectivity for cellular automata on these shifts.

Abstract

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.