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Bounding the Escape Time of a Linear Dynamical System over a Compact Semialgebraic Set

Authors Julian D'Costa, Engel Lefaucheux , Eike Neumann, Joël Ouaknine , James Worrell

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Author Details

Julian D'Costa
  • Department of Computer Science, University of Oxford, UK
Engel Lefaucheux
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
  • Université de Lorraine, Inria, LORIA, Nancy, France
Eike Neumann
  • Swansea University, Swansea, UK
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
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.
  • Ouaknine, Joël: Also affiliated with Keble College, Oxford as http://emmy.network/ Fellow, and supported by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).

Cite As Get BibTex

Julian D'Costa, Engel Lefaucheux, Eike Neumann, Joël Ouaknine, and James Worrell. Bounding the Escape Time of a Linear Dynamical System over a Compact Semialgebraic Set. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.MFCS.2022.39

Abstract

We study the Escape Problem for discrete-time linear dynamical systems over compact semialgebraic sets. We establish a uniform upper bound on the number of iterations it takes for every orbit of a rational matrix to escape a compact semialgebraic set defined over rational data. Our bound is doubly exponential in the ambient dimension, singly exponential in the degrees of the polynomials used to define the semialgebraic set, and singly exponential in the bitsize of the coefficients of these polynomials and the bitsize of the matrix entries. We show that our bound is tight by providing a matching lower bound.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
Keywords
  • Discrete linear dynamical systems
  • Program termination
  • Compact semialgebraic sets
  • Uniform termination bounds

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