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On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems

Authors Toghrul Karimov, Joël Ouaknine, James Worrell

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Author Details

Toghrul Karimov
  • 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, University of Oxford, UK
James Worrell
  • Department of Computer Science, University of Oxford, UK

Funding

  • Ouaknine, Joël: Supported by ERC grant AVS-ISS (648701) and DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).
  • Worrell, James: Supported by EPSRC Fellowship EP/N008197/1.

Cite AsGet BibTex

Toghrul Karimov, Joël Ouaknine, and James Worrell. On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.MFCS.2020.54

Abstract

Consider a discrete dynamical system given by a square matrix M ∈ ℚ^{d × d} and a starting point s ∈ ℚ^d. The orbit of such a system is the infinite trajectory ⟨ s, Ms, M²s, …⟩. Given a collection T₁, T₂, …, T_m ⊆ ℝ^d of semialgebraic sets, we can associate with each T_i an atomic proposition P_i which evaluates to true at time n if, and only if, M^ns ∈ T_i. This gives rise to the LTL Model-Checking Problem for discrete linear dynamical systems: given such a system (M,s) and an LTL formula over such atomic propositions, determine whether the orbit satisfies the formula. The main contribution of the present paper is to show that the LTL Model-Checking Problem for discrete linear dynamical systems is decidable in dimension 3 or less.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
Keywords
  • Linear dynamical systems
  • Orbit Problem
  • LTL model checking

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