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Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems

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Logical Approaches to Computational Barriers (CiE 2006)

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Abstract

In this paper we study a class of hybrid systems defined by Pfaffian maps. It is a sub-class of o-minimal hybrid systems which capture rich continuous dynamics and yet can be studied using finite bisimulations. The existence of finite bisimulations for o-minimal dynamical and hybrid systems has been shown by several authors (see e.g. [3,4,13]). The next natural question to investigate is how the sizes of such bisimulations can be bounded. The first step in this direction was done in [10] where a double exponential upper bound was shown for Pfaffian dynamical and hybrid systems. In the present paper we improve this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of systems on which the exponential bound is attained. The bounds provide a basis for designing efficient algorithms for computing bisimulations, solving reachability and motion planning problems.

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References

  1. Alur, R., Dill, D.L.: A Theory of Timed Automata. Theoretical Computer Science 126, 183–235 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)

    Book  MATH  Google Scholar 

  3. Brihaye, T., Michaux, C.: On expressiveness and decidability of o-minimal hybrid systems. J. Complexity 21, 447–478 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Davoren, J.M.: Topologies, continuity and bisimulations. ITA 33(4/5), 357–382 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Gabrielov, A., Vorobjov, N.: Complexity of stratifications of semi-Pfaffian sets. Discrete Comput. Geom. 14, 71–91 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gabrielov, A., Vorobjov, N.: Complexity of computations with Pfaffian and Noetherian functions. In: Ilyashenko, Y., et al. (eds.) Normal Forms, Bifurcations and Finiteness Problems in Differential Equations. NATO Science Series II, vol. 137, pp. 211–250. Kluwer Academic Publishers, Dordrecht (2004)

    Chapter  Google Scholar 

  7. Gabrielov, A., Vorobjov, N., Zell, T.: Betti numbers of semialgebraic and sub-Pfaffian sets. J. London Math. Soc. 69(1), 27–43 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hirsch, M.: Differential Topology. Springer, New York (1976)

    Book  MATH  Google Scholar 

  9. Khovanskii, A.: Fewnomials. Translations of Mathematical Monographs, vol. 88. American Mathematical Society, Providence (1991)

    Google Scholar 

  10. Korovina, M., Vorobjov, N.: Pfaffian hybrid systems. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 430–441. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  11. Korovina, M., Vorobjov, N.: Upper and lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems. Technical Report Nr 05-01, Universitat Siegen, pp. 1–16 (2005), http://www.brics.dk/~korovina/upperbounds.pdf

  12. Kurdyka, K., Orro, P., Symon, S.: Semialgebraic sard theorem for generalized critical values. J. Differential Geometry 56, 67–92 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Lafferriere, G., Pappas, G., Sastry, S.: O-minimal hybrid systems. Math. Control Signals Systems 13, 1–21 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Massey, W.S.: A Basic Course in Algebraic Topology. Springer, New York (1991)

    MATH  Google Scholar 

  15. van den Dries, L.: Tame Topology and O-minimal Structures. London Mathematical Society Lecture Notes Series, vol. 248. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  16. Wilkie, A.J.: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9(4), 1051–1094 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zell, T.: Betti numbers of semi-Pfaffian sets. J. Pure Appl. Algebra 139, 323–338 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Theoretische Informatik, Universität Siegen, D-57068, Germany

    Margarita Korovina

  2. IIS SB RAS, Novosibirsk, Russia

    Margarita Korovina

  3. Department of Computer Science, University of Bath, Bath, BA2 7AY, England

    Nicolai Vorobjov

Authors
  1. Margarita Korovina
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  2. Nicolai Vorobjov
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Swansea University, Singleton Park, SA2 8PP, Swansea, United Kingdom

    Arnold Beckmann

  2. Department of Computer Science, University of Wales Swansea, Singleton Park, SA2 8PP, Swansea, UK

    Ulrich Berger

  3. Universiteit van Amsterdam, Plantage Muidergracht 24, 1018, Amsterdam, TV, The Netherlands

    Benedikt Löwe

  4. School of Physical Sciences, Swansea University, Singleton Park, SA2 8PP, Swansea, Wales, United Kingdom

    John V. Tucker

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© 2006 Springer-Verlag Berlin Heidelberg

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Korovina, M., Vorobjov, N. (2006). Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds) Logical Approaches to Computational Barriers. CiE 2006. Lecture Notes in Computer Science, vol 3988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780342_28

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  • DOI: https://doi.org/10.1007/11780342_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35466-6

  • Online ISBN: 978-3-540-35468-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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