We consider the MSO model-checking problem for simple linear loops, or equivalently discrete-time linear dynamical systems, with semialgebraic predicates (i.e., Boolean combinations of polynomial inequalities on the variables). We place no restrictions on the number of variables, or equivalently the ambient dimension. We establish decidability of the model-checking problem provided that each semialgebraic predicate either has intrinsic dimension 1, or is contained within some three-dimensional subspace. We also note that lifting either of these restrictions and retaining decidability would necessarily require major breakthroughs in number theory.
Submitted, 2021. 21 pages.
© 2021 Toghrul Karimov, Engel Lefaucheux, Joël
Ouaknine, David Purser, Anton Varonka,
Markus A. Whiteland, and James Worrell.