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Constraint-Based Framework for Reasoning with Differential Equations

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Cyber-Physical Systems Security

Abstract

An extension of constraint satisfaction problems with differential equations is proposed. Reasoning with differential equations is mandatory to analyze or verify dynamical systems, such as cyber-physical ones. A constraint-based framework is presented to model a wider class of problems based on logical combination of high-level properties. In addition, the complete correctness is verified using a set-membership approach in this framework. Finally, examples are given to demonstrate the benefits of the presented framework.

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References

  1. F. Benhamou, D. McAllester, P. Van Hentenryck, CLP (intervals) revisited. Technical Report, Providence (1994)

    Google Scholar 

  2. O. Bouissou, A. Chapoutot, S. Mimram, HySon: precise simulation of hybrid systems with imprecise inputs, in IEEE Rapid System Prototyping (2012)

    Google Scholar 

  3. G. Chabert, L. Jaulin, Contractor programming. Artif. Intell. 173(11), 1079–1100 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. X. Chen, E. Ábrahám, S. Sankaranarayanan, Flow*: an analyzer for non-linear hybrid systems, in Proceedings of the International Conference on Computer Aided Verification (Springer, Berlin, 2013), pp. 258–263

    Google Scholar 

  5. X. Chen, S. Sankaranarayanan, E. Ábrahám, Under-approximate flowpipes for non-linear continuous systems, in Proceedings of the Conference on Formal Methods in Computer-Aided Design (FMCAD Inc., Austin, 2014), pp. 59–66

    Google Scholar 

  6. J. Cruz, P. Barahona, Constraint satisfaction differential problems, in Principles and Practice of Constraint Programming. Lecture Notes in Computer Science, vol. 2833 (Springer, Berlin, 2003), pp. 259–273

    Chapter  MATH  Google Scholar 

  7. J. Cruz, P. Barahona, Constraint reasoning over differential equations. Appl. Numer. Anal. Comput. Math. 1(1), 140–154 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. L.H. de Figueiredo, J. Stolfi, Self-validated numerical methods and applications. Brazilian Mathematics Colloquium Monographs. IMPA/CNPq, Rio de Janeiro (1997)

    Google Scholar 

  9. J.A. dit Sandretto, A. Chapoutot, DynIBEX: a differential constraint library for studying dynamical systems, in Conference on Hybrid Systems: Computation and Control (2016). Poster

    Google Scholar 

  10. J.A. dit Sandretto, A. Chapoutot, Validated explicit and implicit Runge-Kutta methods. Reliab. Comput. 22, 56–77 (2016)

    Google Scholar 

  11. J.A. dit Sandretto, A. Chapoutot, Validated simulation of differential algebraic equations with Runge-Kutta methods. Reliab. Comput. 22, 56–77 (2016)

    Google Scholar 

  12. J.A. dit Sandretto, A. Chapoutot, O. Mullier, Tuning PI controller in non-linear uncertain closed-loop systems with interval analysis, in 2nd International Workshop on Synthesis of Complex Parameters, OpenAccess Series in Informatics, vol. 44, pp. 91–102. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2015)

    Google Scholar 

  13. J.A. dit Sandretto, A. Chapoutot, O. Mullier, Formal verification of robotic behaviors in presence of bounded uncertainties. J. Softw. Eng. Robot. 8(1), 78–88 (2017)

    Google Scholar 

  14. D. Eveillard, D. Ropers, H. De Jong, C. Branlant, A. Bockmayr, A multi-scale constraint programming model of alternative splicing regulation. Theor. Comput. Sci. 325(1), 3–24 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Gajda, M. Jankowska, A. Marciniak, B. Szyszka, A survey of interval Runge–Kutta and multistep methods for solving the IVP, in Parallel Processing and Applied Mathematics. Lecture Notes in Computer Science, vol. 4967 (Springer, Berlin, 2008), pp. 1361–1371

    Google Scholar 

  16. A. Goldsztejn, W. Hayes, Rigorous inner approximation of the range of functions, in 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (2006), p. 19

    Google Scholar 

  17. A. Goldsztejn, L. Jaulin, Inner approximation of the range of vector-valued functions. Reliab. Comput. 14, 1–23 (2010)

    MathSciNet  Google Scholar 

  18. A. Goldsztejn, O. Mullier, D. Eveillard, H. Hosobe, Including ODE based constraints in the standard CP framework, in Principles and Practice of Constraint Programming. Lecture Notes in Computer Science, vol. 6308 (Springer, Berlin, 2010), pp. 221–235

    Google Scholar 

  19. E. Goubault, S. Putot, Under-approximations of computations in real numbers based on generalized affine arithmetic, in Proceedings of the Static Analysis Symposium, vol. 4634. Lecture Notes in Computer Science (Springer, Berlin, 2007), pp. 137–152

    Google Scholar 

  20. E. Goubault, S. Putot, Forward inner-approximated reachability of non-linear continuous systems, in Proceedings of the International Conference on Hybrid Systems: Computation and Control (ACM, New York, 2017), pp. 1–10

    MATH  Google Scholar 

  21. E. Goubault, O. Mullier, S. Putot, M. Kieffer, Inner approximated reachability analysis, in Proceedings of the International Conference on Hybrid Systems: Computation and Control (ACM, New York, 2014), pp. 163–172

    MATH  Google Scholar 

  22. M. Janssen, Y. Deville, P. Van Hentenryck, Multistep filtering operators for ordinary differential equations, in Principles and Practice of Constraint Programming. Lecture Notes in Computer Science, vol. 1713 (Springer, Berlin, 1999), pp. 246–260

    Chapter  Google Scholar 

  23. L. Jaulin, E. Walter, Set inversion via interval analysis for nonlinear bounded-error estimation. Automatica 29(4), 1053–1064 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Jaulin, M. Kieffer, O. Didrit, E. Walter, Applied Interval Analysis (Springer, Berlin, 2001)

    Book  MATH  Google Scholar 

  25. E. Kaucher, Interval Analysis in the Extended Interval Space IR (Springer, Berlin, 1980), pp. 33–49

    MATH  Google Scholar 

  26. Y. Lebbah, O. Lhomme, Accelerating filtering techniques for numeric CSPs. J. Artif. Intell. 139(1), 109–132 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. R.J. Lohner, Enclosing the solutions of ordinary initial and boundary value problems, in Computer Arithmetic (B.G. Teubner, Stuttgart, 1987), pp. 255–286

    Google Scholar 

  28. R.E. Moore, Interval Analysis. Series in Automatic Computation (Prentice Hall, Englewood Cliffs, 1966)

    Google Scholar 

  29. N.S. Nedialkov, K. Jackson, G. Corliss, Validated solutions of initial value problems for ordinary differential equations. Appl. Math. Comput. 105(1), 21–68 (1999)

    MathSciNet  MATH  Google Scholar 

  30. A. Rauh, M. Brill, C. GüNther, A novel interval arithmetic approach for solving differential-algebraic equations with ValEncIA-IVP. Int. J. Appl. Math. Comput. Sci. 19(3), 381–397 (2009)

    Article  MATH  Google Scholar 

  31. F. Rossi, P. Van Beek, T. Walsh, Handbook of Constraint Programming (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  32. M. Rueher, Solving continuous constraint systems, in International Conference on Computer Graphics and Artificial Intelligence (2005)

    Google Scholar 

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Acknowledgements

This research benefited from the support of the “Chair Complex Systems Engineering – Ecole Polytechnique, THALES, DGA, FX, Dassault Aviation, DCNS Research, ENSTA ParisTech, Télécom ParisTech, Fondation ParisTech, and FDO ENSTA,” and it is also partially funded by DGA MRIS “Safety for Complex Robotic Systems.”

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Authors and Affiliations

  1. ENSTA ParisTech, Université Paris-Saclay, Palaiseau Cedex, France

    Julien Alexandre dit Sandretto, Alexandre Chapoutot & Olivier Mullier

Authors
  1. Julien Alexandre dit Sandretto
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  2. Alexandre Chapoutot
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  3. Olivier Mullier
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Corresponding author

Correspondence to Julien Alexandre dit Sandretto .

Editor information

Editors and Affiliations

  1. İstinye University, İstanbul, Turkey

    Çetin Kaya Koç

  2. Nanjing University of Aeronautics and Astronautics, Nanjing, China

    Çetin Kaya Koç

  3. University of California Santa Barbara, Santa Barbara, CA, USA

    Çetin Kaya Koç

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Alexandre dit Sandretto, J., Chapoutot, A., Mullier, O. (2018). Constraint-Based Framework for Reasoning with Differential Equations. In: Koç, Ç.K. (eds) Cyber-Physical Systems Security. Springer, Cham. https://doi.org/10.1007/978-3-319-98935-8_2

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  • DOI: https://doi.org/10.1007/978-3-319-98935-8_2

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-98934-1

  • Online ISBN: 978-3-319-98935-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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