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Reachability in Dynamical Systems with Rounding

Authors Christel Baier , Florian Funke , Simon Jantsch , Toghrul Karimov , Engel Lefaucheux , Joël Ouaknine , Amaury Pouly , David Purser , Markus A. Whiteland

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Author Details

Christel Baier
  • Technische Universität Dresden, Germany
Florian Funke
  • Technische Universität Dresden, Germany
Simon Jantsch
  • Technische Universität Dresden, Germany
Toghrul Karimov
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Engel Lefaucheux
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
  • Department of Computer Science, Oxford University, UK
Amaury Pouly
  • Université de Paris, CNRS, IRIF, F-75006, Paris, France
David Purser
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Markus A. Whiteland
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany

Funding

This work was funded by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science), the Cluster of Excellence EXC 2050/1 (CeTI, project ID 390696704, as part of Germany’s Excellence Strategy), DFG-projects BA-1679/11-1 and BA-1679/12-1, and the Research Training Group QuantLA (GRK 1763).
  • Ouaknine, Joël: Supported by ERC grant AVS-ISS (648701).
  • Pouly, Amaury: Supported by CODYS project ANR-18-CE40-0007.

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Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel Lefaucheux, Joël Ouaknine, Amaury Pouly, David Purser, and Markus A. Whiteland. Reachability in Dynamical Systems with Rounding. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.FSTTCS.2020.36

Abstract

We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix M ∈ ℚ^{d × d}, an initial vector x ∈ ℚ^{d}, a granularity g ∈ ℚ_+ and a rounding operation [⋅] projecting a vector of ℚ^{d} onto another vector whose every entry is a multiple of g, we are interested in the behaviour of the orbit 𝒪 = ⟨[x], [M[x]],[M[M[x]]],… ⟩, i.e., the trajectory of a linear dynamical system in which the state is rounded after each step. For arbitrary rounding functions with bounded effect, we show that the complexity of deciding point-to-point reachability - whether a given target y ∈ ℚ^{d} belongs to 𝒪 - is PSPACE-complete for hyperbolic systems (when no eigenvalue of M has modulus one). We also establish decidability without any restrictions on eigenvalues for several natural classes of rounding functions.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • dynamical systems
  • rounding
  • reachability

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