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Invariants for Continuous Linear Dynamical Systems

Authors Shaull Almagor , Edon Kelmendi, Joël Ouaknine, James Worrell

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ACM Subject Classification
  • Theory of computation → Logic and verification
  • Computing methodologies → Algebraic algorithms
  • Mathematics of computing → Continuous mathematics
  • Mathematics of computing → Continuous functions
  • Theory of computation → Finite Model Theory
  • Software and its engineering → Formal software verification
Keywords
  • Invariants
  • continuous linear dynamical systems
  • continuous Skolem problem
  • safety
  • o-minimality

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Abstract

Continuous linear dynamical systems are used extensively in mathematics, computer science, physics, and engineering to model the evolution of a system over time. A central technique for certifying safety properties of such systems is by synthesising inductive invariants. This is the task of finding a set of states that is closed under the dynamics of the system and is disjoint from a given set of error states. In this paper we study the problem of synthesising inductive invariants that are definable in o-minimal expansions of the ordered field of real numbers. In particular, assuming Schanuel’s conjecture in transcendental number theory, we establish effective synthesis of o-minimal invariants in the case of semi-algebraic error sets. Without using Schanuel’s conjecture, we give a procedure for synthesizing o-minimal invariants that contain all but a bounded initial segment of the orbit and are disjoint from a given semi-algebraic error set. We further prove that effective synthesis of semi-algebraic invariants that contain the whole orbit, is at least as hard as a certain open problem in transcendental number theory.

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Shaull Almagor, Edon Kelmendi, Joël Ouaknine, and James Worrell. Invariants for Continuous Linear Dynamical Systems. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 107:1-107:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.107

Author Details

Shaull Almagor
  • Department of Computer Science, Technion, Haifa, Israel
Edon Kelmendi
  • Department of Computer Science, Oxford University, UK
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
  • Department of Computer Science, Oxford University, UK
James Worrell
  • Department of Computer Science, Oxford University, UK

Funding

  • Almagor, Shaull: Supported by a European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 837327.
  • 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.

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