On Decidability Properties of One-Dimensional Cellular Automata

TL;DR

This paper proves that the first-order theory of the phase-space of one-dimensional cellular automata, extended with counting quantifiers, is decidable by showing these structures are omega-automatic.

Contribution

It extends decidability results to more expressive logics for cellular automata phase-spaces by establishing their omega-automatic structure.

Findings

First-order theory with counting quantifiers is decidable for cellular automata.

Phase-spaces of cellular automata are omega-automatic structures.

New decidable properties and more efficient algorithms for surjective automata.

Abstract

In a recent paper Sutner proved that the first-order theory of the phase-space $\mathcal{S}_\mathcal{A}=(Q^\mathbb{Z}, \longrightarrow)$ of a one-dimensional cellular automaton $\mathcal{A}$ whose configurations are elements of $Q^\mathbb{Z}$, for a finite set of states $Q$, and where $\longrightarrow$ is the "next configuration relation", is decidable. He asked whether this result could be extended to a more expressive logic. We prove in this paper that this is actuallly the case. We first show that, for each one-dimensional cellular automaton $\mathcal{A}$, the phase-space $\mathcal{S}_\mathcal{A}$ is an omega-automatic structure. Then, applying recent results of Kuske and Lohrey on omega-automatic structures, it follows that the first-order theory, extended with some counting and cardinality quantifiers, of the structure $\mathcal{S}_\mathcal{A}$, is decidable. We give some examples of new decidable properties for one-dimensional cellular automata. In the case of surjective cellular automata, some more efficient algorithms can be deduced from results of Kuske and Lohrey on structures of bounded degree. On the other hand we show that the case of cellular automata give new results on automatic graphs.