Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en D'Costa, Julian; Karimov, Toghrul; Majumdar, Rupak; Ouaknine, Joël; Salamati, Mahmoud; Soudjani, Sadegh; Worrell, James https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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The Pseudo-Skolem Problem is Decidable

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Abstract

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.

BibTeX - Entry

@InProceedings{dcosta_et_al:LIPIcs.MFCS.2021.34,
  author =	{D'Costa, Julian and Karimov, Toghrul and Majumdar, Rupak and Ouaknine, Jo\"{e}l and Salamati, Mahmoud and Soudjani, Sadegh and Worrell, James},
  title =	{{The Pseudo-Skolem Problem is Decidable}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{34:1--34:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14474},
  URN =		{urn:nbn:de:0030-drops-144742},
  doi =		{10.4230/LIPIcs.MFCS.2021.34},
  annote =	{Keywords: Pseudo-orbits, Orbit problem, Skolem problem, linear dynamical systems}
}

Keywords: Pseudo-orbits, Orbit problem, Skolem problem, linear dynamical systems
Seminar: 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
Issue date: 2021
Date of publication: 18.08.2021


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