# On Decidability Properties of One-Dimensional Cellular 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.
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https://hal.archives-ouvertes.fr/hal-00371169
Contributeur : Olivier Finkel <>
Soumis le : lundi 1 juin 2009 - 15:49:45
Dernière modification le : vendredi 4 janvier 2019 - 17:32:32
Document(s) archivé(s) le : mercredi 22 septembre 2010 - 13:16:14

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• HAL Id : hal-00371169, version 2
• ARXIV : 0903.4615

### Citation

Olivier Finkel. On Decidability Properties of One-Dimensional Cellular Automata. Journal of Cellular Automata, Old City Publishing, 2011, 6 (2-3), pp.181-193. ⟨hal-00371169v2⟩

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