Document Open Access Logo

The Pseudo-Skolem Problem is Decidable

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

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2021.34.pdf
  • Filesize: 0.8 MB
  • 21 pages

Document Identifiers

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.MFCS.2021.34

Author Details

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

Funding

  • D'Costa, Julian: emmy.network foundation under the aegis of the Fondation de Luxembourg.
  • Majumdar, Rupak: DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).
  • Ouaknine, Joël: ERC grant AVS-ISS (648701), and DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science). Joël Ouaknine is also affiliated with Keble College, Oxford as http://emmy.network/ Fellow.
  • Worrell, James: EPSRC Fellowship EP/N008197/1.

References

  1. Dmitri V. Anosov. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. Math., 90, 1967. Google Scholar
  2. Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in Real Algebraic Geometry. Springer, 2006. Google Scholar
  3. Rufus Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, volume 470 of Lecture Notes in Mathematics. Springer-Verlag, 1975. Google Scholar
  4. Jin-Yi Cai. Computing Jordan normal forms exactly for commuting matrices in polynomial time. Int. J. Found. Comput. Sci., 5(3/4):293-302, 1994. Google Scholar
  5. Jin-Yi Cai, Richard J. Lipton, and Yechezkel Zalcstein. The complexity of the A B C problem. SIAM J. Comput., 29(6), 2000. Google Scholar
  6. Henri Cohen. A Course in Computational Algebraic Number Theory. Springer, 1993. Google Scholar
  7. George E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In H. Brakhage, editor, Automata Theory and Formal Languages, pages 134-183, Berlin, Heidelberg, 1975. Springer Berlin Heidelberg. Google Scholar
  8. Charles C. Conley. Isolated invariant sets and the Morse index, volume 25 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, 1978. Google Scholar
  9. Guoqiang Ge. Algorithms Related to Multiplicative Representations of Algebraic Numbers. PhD thesis, U.C. Berkeley, 1993. Google Scholar
  10. Godfrey H. Hardy and Edward M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 1999. Google Scholar
  11. Michael A. Harrison. Lectures on linear sequential machines. Technical report, DTIC Document, 1969. Google Scholar
  12. Ravindran Kannan and Richard J. Lipton. Polynomial-time algorithm for the orbit problem. J. ACM, 33(4):808-821, 1986. Google Scholar
  13. David W. Masser. Linear relations on algebraic groups. In New Advances in Transcendence Theory. Camb. Univ. Press, 1988. Google Scholar
  14. Maurice Mignotte, Tarlok N. Shorey, and Robert Tijdeman. The distance between terms of an algebraic recurrence sequence. J. für die reine und angewandte Math., 349, 1984. Google Scholar
  15. Joël Ouaknine and James Worrell. Positivity problems for low-order linear recurrence sequences. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 366-379, 2014. Google Scholar
  16. Joël Ouaknine and James Worrell. On linear recurrence sequences and loop termination. ACM SIGLOG News, 2(2):4-13, 2015. Google Scholar
  17. Christos H. Papadimitriou and Georgios Piliouras. From nash equilibria to chain recurrent sets: An algorithmic solution concept for game theory. Entropy, 20(10):782, 2018. Google Scholar
  18. James Renegar. On the computational complexity and geometry of the first-order theory of the reals. J. Symb. Comp., 1992. Google Scholar
  19. Eduardo D. Sontag. Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer-Verlag, Berlin, Heidelberg, 1998. Google Scholar
  20. Terence Tao. Structure and randomness: pages from year one of a mathematical blog. American Mathematical Society, 2008. Google Scholar
  21. Nikolai K. Vereshchagin. The problem of appearance of a zero in a linear recurrence sequence (in russian). Mat. Zametki, 38(2), 1985. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact