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Differential Equation Invariance Axiomatization

Authors:

André Platzer,

Yong Kiam TanAuthors Info & Claims

Journal of the ACM (JACM), Volume 67, Issue 1

Article No.: 6, Pages 1 - 66

https://doi.org/10.1145/3380825

Published: 03 April 2020 Publication History

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Abstract

This article proves the completeness of an axiomatization for differential equation invariants described by Noetherian functions. First, the differential equation axioms of differential dynamic logic are shown to be complete for reasoning about analytic invariants. Completeness crucially exploits differential ghosts, which introduce additional variables that can be chosen to evolve freely along new differential equations. Cleverly chosen differential ghosts are the proof-theoretical counterpart of dark matter. They create a new hypothetical state, whose relationship to the original state variables satisfies invariants that did not exist before. The reflection of these new invariants in the original system then enables its analysis.

An extended axiomatization with existence and uniqueness axioms is complete for all local progress properties, and, with a real induction axiom, is complete for all semianalytic invariants. This parsimonious axiomatization serves as the logical foundation for reasoning about invariants of differential equations. Indeed, it is precisely this logical treatment that enables the generalization of completeness to the Noetherian case.

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Cited By

Platzer A(2025)Hybrid dynamical systems logic and its refinementsScience of Computer Programming10.1016/j.scico.2024.103179239(103179)Online publication date: Jan-2025
https://doi.org/10.1016/j.scico.2024.103179
Prebet EPlatzer A(2024)Uniform Substitution for Differential Refinement LogicAutomated Reasoning10.1007/978-3-031-63501-4_11(196-215)Online publication date: 2-Jul-2024
https://doi.org/10.1007/978-3-031-63501-4_11
Kabra ALaurent JMitsch SPlatzer A(2024)CESAR: Control Envelope Synthesis via Angelic RefinementsTools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-031-57246-3_9(144-164)Online publication date: 4-Apr-2024
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Index Terms

Differential Equation Invariance Axiomatization
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
2. Theory of computation
  1. Logic

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 67, Issue 1

February 2020

265 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3379978

Editor:
Éva Tardos
Cornell University

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Copyright © 2020 Owner/Author.

This work is licensed under a Creative Commons Attribution International 4.0 License.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 03 April 2020

Accepted: 01 January 2020

Revised: 01 December 2019

Received: 01 May 2019

Published in JACM Volume 67, Issue 1

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Cited By

Platzer A(2025)Hybrid dynamical systems logic and its refinementsScience of Computer Programming10.1016/j.scico.2024.103179239(103179)Online publication date: Jan-2025
https://doi.org/10.1016/j.scico.2024.103179
Prebet EPlatzer A(2024)Uniform Substitution for Differential Refinement LogicAutomated Reasoning10.1007/978-3-031-63501-4_11(196-215)Online publication date: 2-Jul-2024
https://doi.org/10.1007/978-3-031-63501-4_11
Kabra ALaurent JMitsch SPlatzer A(2024)CESAR: Control Envelope Synthesis via Angelic RefinementsTools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-031-57246-3_9(144-164)Online publication date: 4-Apr-2024
https://doi.org/10.1007/978-3-031-57246-3_9
Strauss MMitsch S(2023)Slow Down, Move Over: A Case Study in Formal Verification, Refinement, and Testing of the Responsibility-Sensitive Safety Model for Self-Driving CarsTests and Proofs10.1007/978-3-031-38828-6_9(149-167)Online publication date: 18-Jul-2023
https://dl.acm.org/doi/10.1007/978-3-031-38828-6_9
Brieger MMitsch SPlatzer A(2023)Uniform Substitution for Dynamic Logic with Communicating Hybrid ProgramsAutomated Deduction – CADE 2910.1007/978-3-031-38499-8_6(96-115)Online publication date: 2-Sep-2023
https://doi.org/10.1007/978-3-031-38499-8_6
Platzer A(2023)Refinements of Hybrid Dynamical Systems LogicRigorous State-Based Methods10.1007/978-3-031-33163-3_1(3-14)Online publication date: 15-May-2023
https://doi.org/10.1007/978-3-031-33163-3_1
Sheng HBentkamp AZhan B(2023)HHLPy: Practical Verification of Hybrid Systems Using Hoare LogicFormal Methods10.1007/978-3-031-27481-7_11(160-178)Online publication date: 3-Mar-2023
https://doi.org/10.1007/978-3-031-27481-7_11
Tan YMitsch SPlatzer ABartocci EPutot S(2022)Verifying Switched System Stability With LogicProceedings of the 25th ACM International Conference on Hybrid Systems: Computation and Control10.1145/3501710.3519541(1-11)Online publication date: 4-May-2022
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Lin QMitsch SPlatzer ADolan J(2022)Safe and Resilient Practical Waypoint-Following for Autonomous VehiclesIEEE Control Systems Letters10.1109/LCSYS.2021.31257176(1574-1579)Online publication date: 2022
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Ghorbal KSogokon A(2022)Characterizing positively invariant setsJournal of Symbolic Computation10.1016/j.jsc.2022.01.004113:C(1-28)Online publication date: 1-Nov-2022
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