FO logic on cellular automata orbits equals MSO logic

TL;DR

This paper establishes an equivalence between FO logic on cellular automata orbits and MSO logic, leading to new undecidability results and concrete examples linking model checking problems to classical computational problems.

Contribution

It introduces an extension of CA to labeled graphs and proves FO logic on CA orbits is equivalent to MSO logic, enabling new undecidability results and examples.

Findings

FO logic on CA orbits is equivalent to MSO logic.

Undecidability of FO model checking for CA over certain groups.

Model checking for CA can be as hard as the domino problem.

Abstract

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 Z^2.