Reachability becomes undecidable in hybrid automata (HA) that can simulate a Turing (TM) or Minsky (MM) machine. Asarin and Schneider have shown that, between the decidable 2-dim Piecewise Constant Derivative (PCD) class and the undecidable 3-dim PCD class, there lies the “open” class 2-dim Hierarchical PCD (HPCD). This class was shown to be equivalent to the class of 1-dim Piecewise Affine Maps (PAM). In this paper, we first explore 2-dim HPCD’s proximity to decidability, by showing that they are equivalent to 2-dim PCDs with translational resets, and to HPCDs without resets. A hierarchy of intermediates also equivalent to the HPCD class is presented, revealing semblance to timed and initialized rectangular automata. We then explore the proximity to the undecidability frontier. We show that 2-dim HPCDs with zeno executions or integer-checks can simulate the 2-counter MM. We conclude by retreating HPCDs as PAMs, to derive a simple over-approximating algorithm for reachability. This also defines a decidable subclass 1-dim Onto PAM (oPAM). The novel non-trivial transformation of 2-dim HPCDs into “almost decidable” systems, is likely to pave the way for approximate reachability algorithms, and the characterization of decidable subclasses. It is hoped that these ideas eventually coalesce into a complete understanding of the reachability problem for the class 2-dim HPCD (1-dim PAM).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The Algorithmic Analysis of Hybrid Systems. Theoretical Computer Science 138, 3–34 (1995)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alur, R., Dill, D.L.: A Theory of Timed Automata. TCS 126, 183–235 (1994)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Asarin, E., Maler, O., Pnueli, A.: Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoretical Computer Science 138, 35–65 (1995)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Asarin, E., Schneider, G.: Widening the boundary between decidable and undecidable hybrid systems. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 193–208. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Asarin, E., Schneider, G., Yovine, S.: On the decidability of the reachability problem for planar differential inclusions. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 89–104. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Berard, B., Dufourd, C.: Timed automata and additive clock constraints. Information Processing Letter 75(1-2), 1–7 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Henzinger, T., Kopke, P.W., Puri, A., Varaiya, P.: What’s Decidable about Hybrid Automata. In: Symposium on the Theory of Computing (STOC), pp. 373–382 (1995)Google Scholar
  8. 8.
    Henzinger, T.A., Sastry, S.: Hybrid Systems-Computation and Control. In: Henzinger, T.A., Sastry, S.S. (eds.) HSCC 1998. LNCS, vol. 1386, Springer, Heidelberg (1998)Google Scholar
  9. 9.
    Koiran, P.: My favourte problems (1999), http://perso.ens-lyon.fr/pascal.koiran/problems.html
  10. 10.
    Lafferiere, G., Pappas, G.J., Sastry, S.: O-minimal Hybrid Systems. Mathematics of Control, Signals, and Systems 13(1), 1–21 (2000)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lafferriere, G., Pappas, G.J., Yovine, S.: Symbolic reachability computation for families of linear vector fields. J. Symb. Comput. 32(3), 231–253 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Maler, O., Pnueli, A.: Reachability analysis of planar multi-linear systems. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, Springer, Heidelberg (1993)Google Scholar
  13. 13.
    Minsky, M.L.: Recursive unsolvability of post’s problem of tag and other topics in theory of turing machines. Ann. of Math. 74, 437–455 (1961)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Mysore, V., Mishra, B.: Algorithmic Algebraic Model Checking III: Approximate Methods. Infinity (2005)Google Scholar
  15. 15.
    Piazza, C., Antoniotti, M., Mysore, V., Policriti, A., Winkler, F., Mishra, B.: Algorithmic Algebraic Model Checking I: The Case of Biochemical Systems and their Reachability Analysis. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 5–19. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Puri, A., Varaiya, P.: Decidebility of hybrid systems with rectangular differential inclusions. Computer Aided Verification, pp. 95–104 (1994)Google Scholar
  17. 17.
    Schneider, G.: Algorithmic Analysis of Polygonal Hybrid Systems. Ph.D. thesis. VERIMAG - UJF, Grenoble, France (2002)Google Scholar
  18. 18.
    Tabuada, P., Pappas, G.J.: Model checking ltl over controllable linear systems is decidable. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 498–513. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    Teschl, G.: Ordinary differential equations and dynamical systems (2004), Lecture Notes from http://www.mat.univie.ac.at/gerald/ftp/book-ode/index.html
  20. 20.
    Turing, A.: On computable numbers, with an application to the entscheidungs problem. Proceedings of the London Mathematical Society 2(42), 230–265 (1936)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Venkatesh Mysore
    • 1
  • Amir Pnueli
    • 1
    • 2
  1. 1.Courant Institute of Mathematical SciencesNYUNew YorkU.S.A
  2. 2.The Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations