Abstract Interpretation of the Physical Inputs of Embedded Programs

  • Olivier Bouissou
  • Matthieu Martel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4905)


We define an abstraction of the continuous variables that serve as inputs to embedded software. In existing static analyzers, these variables are most often abstracted by a constant interval, and this approach has shown its limits. We propose a different method that analyzes in a more precise way the continuous environment. This environment is first expressed as the semantics of a special continuous program, and we define a safe abstract semantics. We introduce the abstract domain of interval valued step functions and show that it safely over-approximates the set of continuous functions. The theory of guaranteed integration is then used to effectively compute an abstract semantics and we prove that this abstract semantics is safe.


Hybrid System Step Function Abstract Interpretation Hybrid Automaton Abstract Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alur, R., et al.: The algorithmic analysis of hybrid systems. Theoretical Computer Science 138(1), 3–34 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bertrane, J.: Static Analysis by Abstract Interpretation of the Quasi-synchronous Composition of Synchronous Programs. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 97–112. Springer, Heidelberg (2005)Google Scholar
  3. 3.
    Bieberbach, L.: On the remainder of the runge-kutta formula. Z.A.M.P. 2, 233–248 (1951)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cousot, P., et al.: Design and Implementation of a Special-Purpose Static Program Analyzer for Safety-Critical Real-Time Embedded Software. In: Mogensen, T.Æ., Schmidt, D.A., Sudborough, I.H. (eds.) The Essence of Computation. LNCS, vol. 2566, pp. 85–108. Springer, Heidelberg (2002)Google Scholar
  5. 5.
    Bouissou, O.: Analyse statique par interpretation abstraite de système hybrides discrets-continus. Technical Report 05-301, CEA-LIST (2005)Google Scholar
  6. 6.
    Bouissou, O., Martel, M.: GRKLib: A guaranteed runge-kutta library. In: International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, IEEE, Los Alamitos (2006)Google Scholar
  7. 7.
    Bourdoncle, F.: Abstract interpretation by dynamic partitioning. Journal of Functional Programming 2(4), 407–423 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: POPL, pp. 238–252. ACM Press, New York (1977)Google Scholar
  9. 9.
    Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: POPL, pp. 84–97. ACM Press, New York (1978)Google Scholar
  10. 10.
    Daumas, M., Lester, D.: Stochastic formal methods: An application to accuracy of numeric software. In: Proceedings of the 40th IEEE Annual Hawaii International Conference on System Sciences (2007)Google Scholar
  11. 11.
    Lieutier, A., Edalat, A., Pattinson, D.: A Computational Model for Multi-variable Differential Calculus. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 505–519. Springer, Heidelberg (2005)Google Scholar
  12. 12.
    Edalat, A., Pattinson, D.: A Domain Theoretic Account of Picard’s Theorem. In: Díaz, J., et al. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 494–505. Springer, Heidelberg (2004)Google Scholar
  13. 13.
    Martel, M., Goubault, É., Putot, S.: Asserting the Precision of Floating-Point Computations: A Simple Abstract Interpreter. In: Le Métayer, D. (ed.) ESOP 2002 and ETAPS 2002. LNCS, vol. 2305, pp. 209–212. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Goubault, E., Martel, M., Putot, S.: Some future challenges in the validation of control systems. In: ERTS (2006)Google Scholar
  15. 15.
    Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations I: nonstiff problems, 2nd revised edn. Springer, Heidelberg (1993)Google Scholar
  16. 16.
    Halbwachs, N., Proy, Y., Raymond, P.: Verification of linear hybrid systems by means of convex approximations. In: LeCharlier, B. (ed.) SAS 1994. LNCS, vol. 864, pp. 223–237. Springer, Heidelberg (1994)Google Scholar
  17. 17.
    Henzinger, T.A.: The theory of hybrid automata. In: Symposium on Logic in Computer Science, pp. 278–292. IEEE Press, Los Alamitos (1996)Google Scholar
  18. 18.
    Henzinger, T.A., et al.: Beyond HYTECH: Hybrid Systems Analysis Using Interval Numerical Methods. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 130–144. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Carr III, J.W.: Error bounds for the runge-kutta single-step integration process. JACM 5(1), 39–44 (1958)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lohner, R.: Einschließung der Lösung gewöhnlicher Anfangsund Randwertaufgaben und Anwendungen. PhD thesis, Universität Karlsruhe (1988)Google Scholar
  21. 21.
    Martel, M.: An overview of semantics for the validation of numerical programs. In VMCAI. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 59–77. Springer, Heidelberg (2005)Google Scholar
  22. 22.
    Martel, M.: Towards an abstraction of the physical environment of embedded systems. In: NSAD (2005)Google Scholar
  23. 23.
    Mosterman, P.J.: An overview of hybrid simulation phenomena and their support by simulation packages. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 165–177. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  24. 24.
    Nedialkov, N.S., Jackson, K.R.: An interval Hermite-Obreschkoff method for computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. In: Developments in Reliable Computing, pp. 289–310. Kluwer, Dordrecht (1999)Google Scholar
  25. 25.
    IEEE Task P754. ANSI/IEEE 754-1985, Standard for Binary Floating-Point Arithmetic. IEEE, New York, August 12 (1985).Google Scholar
  26. 26.
    Rauh, A., Auer, E., Hofer, E.: ValEncIA-IVP: A case study of validated solvers for initial value problems. In: SCAN (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Olivier Bouissou
    • 1
  • Matthieu Martel
    • 2
  1. 1.CEA LISTLaboratoire MeASIGif-sur-YvetteFrance
  2. 2.Laboratoire ELIAUS-DALIUniversité de PerpignanPerpignan

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