Robust Computations with Dynamical Systems

  • Olivier Bournez
  • Daniel S. Graça
  • Emmanuel Hainry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)


In this paper we discuss the computational power of Lipschitz dynamical systems which are robust to infinitesimal perturbations.

Whereas the study in [1] was done only for not-so-natural systems from a classical mathematical point of view (discontinuous differential equation systems, discontinuous piecewise affine maps, or perturbed Turing machines), we prove that the results presented there can be generalized to Lipschitz and computable dynamical systems.

In other words, we prove that the perturbed reachability problem (i.e. the reachability problem for systems which are subjected to infinitesimal perturbations) is co-recursively enumerable for this kind of systems. Using this result we show that if robustness to infinitesimal perturbations is also required, the reachability problem becomes decidable. This result can be interpreted in the following manner: undecidability of verification doesn’t hold for Lipschitz, computable and robust systems.

We also show that the perturbed reachability problem is co-r.e. complete even for C  ∞ -systems.


Verification Model-checking Computable Analysis Analog Computations 


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  1. 1.
    Asarin, E., Bouajjani, A.: Perturbed Turing machines and hybrid systems. In: Proc. 16th Annual IEEE Symposium, Logic in Computer Science, pp. 269–278 (2001)Google Scholar
  2. 2.
    Alur, R., Pappas, G.J. (eds.): HSCC 2004. LNCS, vol. 2993. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  3. 3.
    Alur, R., Dill, D.L.: Automata for modeling real-time systems. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 322–335. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  4. 4.
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? J. Comput. System Sci. 57(1), 94–124 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Asarin, E., Maler, O., Pnueli, A.: Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoret. Comput. Sci. 138, 35–65 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Collins, P.: Continuity and computability of reachable sets. Theor. Comput. Sci. 341, 162–195 (2005)zbMATHCrossRefGoogle Scholar
  7. 7.
    Fränzle, M.: Analysis of hybrid systems: An ounce of realism can save an infinity of states. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 126–140. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. 8.
    Henzinger, T.A., Raskin, J.F.: Robust undecidability of timed and hybrid systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 145–159. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Asarin, E., Collins, P.: Noisy Turing machines. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1031–1042. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, Heidelberg (1983)Google Scholar
  11. 11.
    Hirsch, M.W., Smale, S., Devaney, R.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, London (2004)zbMATHGoogle Scholar
  12. 12.
    Weihrauch, K.: Computable Analysis: an Introduction. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  13. 13.
    Birkhoff, G., Rota, G.C.: Ordinary Differential Equations, 4th edn. John Wiley & Sons, Chichester (1989)Google Scholar
  14. 14.
    Graça, D.S., Campagnolo, M.L., Buescu, J.: Robust simulations of Turing machines with analytic maps and flows. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 169–179. Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Graça, D.S., Campagnolo, M.L., Buescu, J.: Computability with polynomial differential equations. Adv. Appl. Math. 40(3), 330–349 (2008)zbMATHCrossRefGoogle Scholar
  16. 16.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. (Ser. 2-42), 230–265 (1936)Google Scholar
  17. 17.
    Grzegorczyk, A.: Computable functionals. Fund. Math. 42, 168–202 (1955)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Lacombe, D.: Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles III. C. R. Acad. Sci. Paris 241, 151–153 (1955)Google Scholar
  19. 19.
    Ko, K.-I.: Computational Complexity of Real Functions. Birkhäuser, Basel (1991)Google Scholar
  20. 20.
    Puri, A.: Dynamical properties of timed automata. In: Ravn, A.P., Rischel, H. (eds.) FTRTFT 1998. LNCS, vol. 1486, pp. 210–227. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  21. 21.
    Ott, E.: Chaos in Dynamical Systems, 2nd edn. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  22. 22.
    Collins, P., Graça, D.S.: Effective computability of solutions of differential inclusions — the ten thousand monkeys approach. Journal of Universal Computer Science 15(6), 1162–1185 (2009)MathSciNetGoogle Scholar
  23. 23.
    Bournez, O., Cosnard, M.: On the computational power of dynamical systems and hybrid systems. Theoret. Comput. Sci. 168(2), 417–459 (1996)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Olivier Bournez
    • 1
  • Daniel S. Graça
    • 2
    • 3
  • Emmanuel Hainry
    • 4
    • 5
  1. 1.Ecole Polytechnique, LIXPalaiseau CedexFrance
  2. 2.DM/FCTUniversidade do AlgarveFaroPortugal
  3. 3.SQIG/Instituto de TelecomunicaçõesLisbonPortugal
  4. 4.LORIAVandœuvre-lès-Nancy CedexFrance
  5. 5.Nancy Université, Université Henri PoincaréNancyFrance

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