Abstract
It is well known that in an o-minimal hybrid system the continuous and discrete components can be separated, and therefore the problem of finite bisimulation reduces to the same problem for a transition system associated with a continuous dynamical system. It was recently proved by several authors that under certain natural assumptions such finite bisimulation exists. In the paper we consider o-minimal systems defined by Pfaffian functions, either implicitly (via triangular systems of ordinary differential equations) or explicitly (by means of semi-Pfaffian maps). We give explicit upper bounds on the sizes of bisimulations as functions of formats of initial dynamical systems. We also suggest an algorithm with an elementary (doubly-exponential) upper complexity bound for computing finite bisimulations of these 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
Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1997)
Brihaye, T., Michaux, C., Riviere, C., Troestler, C.: On o-minimal hybrid systems. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 219–233. Springer, Heidelberg (2004)
Gabrielov, A., Vorobjov, N.: Complexity of computations with Pfaffian and Noetherian functions. In: Normal Forms, Bifurcations and Finiteness Problems in Differential Equations. NATO Science Series II, vol. 137, pp. 211–250. Kluwer, Dordrecht (2004)
Gabrielov, A., Vorobjov, N.: Complexity of cylindrical decompositions of sub- Pfaffian sets. J. Pure and Appl. Algebra 164, 1–2, 179–197 (2001)
Gabrielov, A., Vorobjov, N.: Betti numbers of semialgebraic sets defined by quantifier-free formulae. In: Discrete and Computational Geometry (2004, To appear)
Grossman, R.L., et al.: HS 1991 and HS 1992. LNCS, vol. 736, p. 474. Springer, Heidelberg (1993)
Henzinger, T.A.: The theory of hybrid automata. In: Proceedings of 11th Ann. Symp. Logic in Computer Sci., pp. 278–292. IEEE Computer Society Press, Los Alamitos (1996)
Lafferriere, G., Pappas, G.J., Sastry, S.: O-minimal hybrid systems. Math. Control Signals Systems 13, 1–21 (2000)
Pericleous, S., Vorobjov, N.: New complexity bounds for cylindrical decompositions of sub-Pfaffian sets. In: Aronov, B., et al. (eds.) Discrete and Computational Geometry. Goodman-Pollack Festschrift, pp. 673–694. Springer, Heidelberg (2003)
Zell, T.: Betti numbers of semi-Pfaffian sets. J. Pure Appl. Algebra 139, 323–338 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Korovina, M., Vorobjov, N. (2004). Pfaffian Hybrid Systems. In: Marcinkowski, J., Tarlecki, A. (eds) Computer Science Logic. CSL 2004. Lecture Notes in Computer Science, vol 3210. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30124-0_33
Download citation
DOI: https://doi.org/10.1007/978-3-540-30124-0_33
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23024-3
Online ISBN: 978-3-540-30124-0
eBook Packages: Springer Book Archive