Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Asarin, E., Bouajjani, A.: Perturbed Turing machines and hybrid systems. In: Proc. 16th Annual IEEE Symposium, Logic in Computer Science, pp. 269–278 (2001)
Alur, R., Pappas, G.J. (eds.): HSCC 2004. LNCS, vol. 2993. Springer, Heidelberg (2004)
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)
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)
Asarin, E., Maler, O., Pnueli, A.: Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoret. Comput. Sci. 138, 35–65 (1995)
Collins, P.: Continuity and computability of reachable sets. Theor. Comput. Sci. 341, 162–195 (2005)
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)
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)
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)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, Heidelberg (1983)
Hirsch, M.W., Smale, S., Devaney, R.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, London (2004)
Weihrauch, K.: Computable Analysis: an Introduction. Springer, Heidelberg (2000)
Birkhoff, G., Rota, G.C.: Ordinary Differential Equations, 4th edn. John Wiley & Sons, Chichester (1989)
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)
Graça, D.S., Campagnolo, M.L., Buescu, J.: Computability with polynomial differential equations. Adv. Appl. Math. 40(3), 330–349 (2008)
Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. (Ser. 2-42), 230–265 (1936)
Grzegorczyk, A.: Computable functionals. Fund. Math. 42, 168–202 (1955)
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)
Ko, K.-I.: Computational Complexity of Real Functions. Birkhäuser, Basel (1991)
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)
Ott, E.: Chaos in Dynamical Systems, 2nd edn. Cambridge University Press, Cambridge (2002)
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)
Bournez, O., Cosnard, M.: On the computational power of dynamical systems and hybrid systems. Theoret. Comput. Sci. 168(2), 417–459 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bournez, O., Graça, D.S., Hainry, E. (2010). Robust Computations with Dynamical Systems. In: Hliněný, P., Kučera, A. (eds) Mathematical Foundations of Computer Science 2010. MFCS 2010. Lecture Notes in Computer Science, vol 6281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15155-2_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-15155-2_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15154-5
Online ISBN: 978-3-642-15155-2
eBook Packages: Computer ScienceComputer Science (R0)