Abstract
In this paper, we investigate the computability and complexity of reachability problems for two-dimensional hierarchical piecewise constant derivative (HPCD) systems. The main interest in HPCDs stems from the fact that their reachability problem is on the border between decidability and undecidability, since it is equivalent to that of reachability for one-dimensional piecewise affine maps (PAMs) which is a long standing open problem. Understanding the most expressive hybrid system models that retain decidability for reachability has generated a great deal of interest over the past few years. In this paper, we show a restriction of HPCDs (called RHPCDs) which leads to the reachability problem becoming decidable. We then study which additional powers we must add to the RHPCD model to render it 1D PAM-equivalent. Finally, we show NP-hardness of reachability for nondeterministic RHPCDs.
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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(1), 3–34 (1995)
Asarin, E., Maler, O., Pneuli, A.: Reachability analysis of dynamical systems having piecewise constant derivatives. Theoretical Computer Science 138, 35–65 (1995)
Asarin, E., Mysore, V., Pneuli, A., Schneider, G.: Low dimensional hybrid systems - decidable, undecidable, don’t know. Information and Computation 211, 138–159 (2012)
Asarin, E., Schneider, G.: Widening the boundary between decidable and undecidable hybrid systems. In: Brim, L., Jančar, P., Křetínský, M., Kučera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 193–208. Springer, Heidelberg (2002)
Asarin, E., Schneider, G., Yovine, S.: Algorithmic analysis of polygonal hybrid systems, part I: Reachability. Theoretical Computer Science 379, 231–265 (2007)
Balluchi, A., Benvenuti, L., di Benedetto, M., Pinello, C., Sangiovanni-Vincentelli, A.: Automotive engine control and hybrid systems: challenges and opportunities. Proceedings of the IEEE 88(7), 888–912 (2000)
Bauer, N., Kowalewski, S., Sand, G.: A case study: Multi product batch plant for the demonstration of control and scheduling problems. In: ADPM, Dortmund, Germany, pp. 969–974 (2000)
Blondel, V., Tsitsiklis, J.: When is a pair of matrices mortal? Information Processing Letters 63, 283–286 (1997)
Blondel, V., Tsitsiklis, J.N.: A survey of computational complexity results in systems and control. Automatica 36, 1249–1274 (2000)
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)
Kurganskyy, O., Potapov, I., Sancho-Caparrini, F.: Reachability problems in low-dimensional iterative maps. International Journal of Foundations of Computer Science 19(4), 935–951 (2008)
Maler, O., Pnueli, A.: Reachability analysis of planar multi-linear systems. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 194–209. Springer, Heidelberg (1993)
Prabhakar, P., Vladimerou, V., Viswanathan, M., Dullerud, G.E.: A decidable class of planar linear hybrid systems. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 401–414. Springer, Heidelberg (2008)
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Bell, P.C., Chen, S. (2013). Reachability Problems for Hierarchical Piecewise Constant Derivative Systems. In: Abdulla, P.A., Potapov, I. (eds) Reachability Problems. RP 2013. Lecture Notes in Computer Science, vol 8169. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41036-9_6
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DOI: https://doi.org/10.1007/978-3-642-41036-9_6
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