The Pseudo-Skolem Problem is Decidable

Authors Julian D'Costa , Toghrul Karimov , Rupak Majumdar , Joël Ouaknine , Mahmoud Salamati , Sadegh Soudjani , James Worrell

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Julian D'Costa
  • Department of Computer Science, University of Oxford, UK
Toghrul Karimov
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Rupak Majumdar
  • Max Planck Institute for Software Systems, Kaiserslautern, Germany
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Mahmoud Salamati
  • Max Planck Institute for Software Systems, Kaiserslautern, Germany
Sadegh Soudjani
  • Newcastle University, Newcastle upon Tyne, UK
James Worrell
  • Department of Computer Science, University of Oxford, UK

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Julian D'Costa, Toghrul Karimov, Rupak Majumdar, Joël Ouaknine, Mahmoud Salamati, Sadegh Soudjani, and James Worrell. The Pseudo-Skolem Problem is Decidable. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study fundamental decision problems on linear dynamical systems in discrete time. We focus on pseudo-orbits, the collection of trajectories of the dynamical system for which there is an arbitrarily small perturbation at each step. Pseudo-orbits are generalizations of orbits in the topological theory of dynamical systems. We study the pseudo-orbit problem, whether a state belongs to the pseudo-orbit of another state, and the pseudo-Skolem problem, whether a hyperplane is reachable by an ε-pseudo-orbit for every ε. These problems are analogous to the well-studied orbit problem and Skolem problem on unperturbed dynamical systems. Our main results show that the pseudo-orbit problem is decidable in polynomial time and the Skolem problem on pseudo-orbits is decidable. The former extends the seminal result of Kannan and Lipton from orbits to pseudo-orbits. The latter is in contrast to the Skolem problem for linear dynamical systems, which remains open for proper orbits.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Pseudo-orbits
  • Orbit problem
  • Skolem problem
  • linear dynamical systems


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