FO Logic on Cellular Automata Orbits Equals MSO Logic

Author Guillaume Theyssier

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Guillaume Theyssier
  • I2M, CNRS, Université Aix-Marseille, France


We thank anonymous referees for their feedback and their suggestions to improve the presentation. We also warmly thank N. Pytheas Fogg for their hints about domino problems on regular graphs and numerous stimulating discussions that inspired this work.

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Guillaume Theyssier. FO Logic on Cellular Automata Orbits Equals MSO Logic. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 154:1-154:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We introduce an extension of classical cellular automata (CA) to arbitrary labeled graphs, and show that FO logic on CA orbits is equivalent to MSO logic. We deduce various results from that equivalence, including a characterization of finitely generated groups on which FO model checking for CA orbits is undecidable, and undecidability of satisfiability of a fixed FO property for CA over finite graphs. We also show concrete examples of FO formulas for CA orbits whose model checking problem is equivalent to the domino problem, or its seeded or recurring variants respectively, on any finitely generated group. For the recurring domino problem, we use an extension of the FO signature by a relation found in the well-known Garden of Eden theorem, but we also show a concrete FO formula without the extension and with one quantifier alternation whose model checking problem does not belong to the arithmetical hierarchy on group ℤ².

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Logic
  • Theory of computation → Formal languages and automata theory
  • MSO logic
  • FO logic
  • cellular automata
  • domino problem
  • Cayley graphs


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