Abstract
We revisited decidability of the reachability problem for low dimensional hybrid systems. Even though many attempts have been done to draw the boundary between decidable and undecidable hybrid systems there are still many open problems in between. In this paper we show that the reachability question for some two dimensional hybrid systems are undecidable and that for other 2-dim systems this question remains unanswered, showing that it is as hard as the reachability problem for Piecewise Affine Maps, that is a well known open problem.
Partially supported by CNRS Project MathSTIC “Squash— Analyse Qualitative de Systémes Hybrides”.
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References
R. Alur, C. Courcoubetis, N. Halbwachs, T. A. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. The algorithmic analysis of hybrid systems. TCS, 138:3–34, 1995.
R. Alur, C. Courcoubetis, T. A. Henzinger, and P.-H. Ho. Hybrid automata: An algorithmic approach to the specification and verification of hybrid systems. LNCS 736, p. 209–229. Springer-Verlag, 1993.
R. Alur and D. L. Dill. A theory of timed automata. TCS, 126:183–235, 1994.
R. Alur, T. A. Henzinger, and M. Y. Vardi. Parametric real-time reasoning. In 25th ACM STOC, p. 592–601, 1993.
R. Alur, S. Kannan, and S. La Torre. Polyhedral flows in hybrid automata. In HSCC’99, LNCS 1569, p. 5–18. Springer-Verlag, 1999.
E. Asarin and O. Maler. On some relations between dynamical systems and transition systems. In ICALP’94, LNCS 820. Springer, 1994.
E. Asarin, O. Maler, and A. Pnueli. Reachability analysis of dynamical systems having piecewise-constant derivatives. TCS, 138:35–65, 1995.
E. Asarin, G. Schneider, and S. Yovine. On the decidability of the reachability problem for planar digfierential inclusions. In HSCC’2001, LNCS 2034, p. 89–104.
L. S. Bobrow and M. A. Arbib. Discrete Mathematics. W. B. Saunders, 1974.
V. Blondel and J. Tsitsilkis. Complexity of stability and controllability of elementary hybrid systems. Automatica, 35:479–489, 1999.
O. Bournez. Complexité algorithmique des systèmes dynamiques continus et hybrides. PhD thesis, ENS de Lyon, 1999.
P. Bouyer, C. Dufourd, E. Fleury, and A. Petit. Are timed automata updatable? In CAV’2000, LNCS 1855, p. 464–479.
P. Bouyer, C. Dufourd, E. Fleury, and A. Petit. Expressiveness of updatable timed automata. In MFCS’2000, LNCS 1893, p. 232–242.
M. S. Branicky. Universal computation and other capabilities of hybrid and continuous dynamical systems. TCS, 138(1), 1995.
K. Čerāns. Algorithmic problems in analysis of real-time systems specifications. PhD thesis, Univ. of Latvia, 1992.
K. Čerāns and J. Vïksna. Deciding reachability for planar multi-polynomial systems. In Hybrid Systems III, LNCS 1066. Springer, 1996.
R. L. Devaney. An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Redwood City, 2nd edition, 1989.
J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems and Linear Algebra. Springer-Verlag, New York, 1990.
M. Henle. A combinatorial introduction to topology. Dover publications, Inc., 1979.
T. A. Henzinger, P. W. Kopke, A. Puri, and P. Varaiya. What’s decidable about hybrid automata? In STOC’1995, p. 373–382. ACM Press.
T. A. Henzinger. The theory of hybrid automata. In LICS’96, p. 278–292, 1996.
M. W. Hirsch and S. Smale. Differential Equations, Dynamical Systems and Linear Algebra. Academic Press Inc., 1974.
Y. Kesten, A. Pnueli, J. Sifakis, and S. Yovine. Integration graphs: a class of decidable hybrid systems. In Hybrid Systems, LNCS 736, p. 179–208, 1993.
P. Koiran. My favourite problems. http://www.ens-lyon.fr/~koiran/problems.html.
P. Koiran, M. Cosnard, and M. Garzon. Computability with low-dimensional dynamical systems. TCS, 132(1):113–128, 1994.
P. Koiran and C. Moore. Closed-form analytic maps in one and two dimensions can simulate universal Turing machines. TCS, 210:217–223, 1999.
O. Maler and A. Pnueli. Reachability analysis of planar multi-linear systems. In CAV’93, LNCS 697, p. 194–209. Springer-Verlag, 1993.
M. L. Minsky. Computation: Finite and Infinite Machines. Prentice-Hall, 1967.
C. Moore. Unpredictability and undecidability in dynamical systems. Physical Review Letters, 64(20), 1990.
X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. An approach to the description and analysis of hybrid systems. In Hybrid Systems, LNCS 736, p. 149–178, 1993.
C. H. Papadimitriou. Computational complexity. Addison-Wesley, 1994.
A. Puri and P. Varaiya. Decidability of hybrid systems with rectangular differential inclusions. In CAV’94, LNCS 818, p. 95–104. Springer, 1994.
G. Schneider. Algorithmic Analysis of Polygonal Hybrid Systems. PhD thesis, VERIMAG— UJF, Grenoble, 2002.
K. S. Sibirsky. Introduction to Topological Dynamics. Noordhoff International Publishing, Leyden, 1975.
S. Simić, K. Johansson, S. Sastry, and J. Lygeros. Towards a geometric theory of hybrid systems. In HSCC’00, LNCS 1790. Springer, 2000.
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Asarin, E., Schneider, G. (2002). Widening the Boundary between Decidable and Undecidable Hybrid Systems* . In: Brim, L., Křetínský, M., Kučera, A., Jančar, P. (eds) CONCUR 2002 — Concurrency Theory. CONCUR 2002. Lecture Notes in Computer Science, vol 2421. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45694-5_14
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