Refining the Undecidability Frontier of Hybrid Automata
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).
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- 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.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.Koiran, P.: My favourte problems (1999), http://perso.ens-lyon.fr/pascal.koiran/problems.html
- 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
- 14.Mysore, V., Mishra, B.: Algorithmic Algebraic Model Checking III: Approximate Methods. Infinity (2005)Google Scholar
- 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.Puri, A., Varaiya, P.: Decidebility of hybrid systems with rectangular differential inclusions. Computer Aided Verification, pp. 95–104 (1994)Google Scholar
- 17.Schneider, G.: Algorithmic Analysis of Polygonal Hybrid Systems. Ph.D. thesis. VERIMAG - UJF, Grenoble, France (2002)Google Scholar
- 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.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