Advertisement

ΣK–constraints for Hybrid Systems

  • Margarita Korovina
  • Oleg Kudinov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5947)

Abstract

In this paper we introduce and study computational aspects of \(\it \Sigma_K\)-constraints which are powerful enough to represent computable continuous data, but also simple enough to be an approach to approximate constraint solving for a large class of quantified continuous constraints. We illustrate how \(\it \Sigma_K\)-constraints can be used for reasoning about hybrid systems.

Keywords

Hybrid System Computable Function Hybrid Automaton Continuous Constraint Reachability Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anai, H., Weispfenning, V.: Reach set computation using real quantifier elimination. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, p. 63. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Brattka, V., Weihrauch, K.: Computability on subsets of euclidean space I: Closed and compact sets. TCS 219, 65–93 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barwise, J.: Admissible sets and Structures. Springer, Berlin (1975)zbMATHGoogle Scholar
  4. 4.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  5. 5.
    Benhamou, F., Goualard, F., Languénou, E., Christie, M.: Interval constraint solving for camera control and motion planning. ACM Trans. Comput. Log. 5(4), 732–767 (2004)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Caviness, B.F., Johnson, J.R. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien (1998)zbMATHGoogle Scholar
  7. 7.
    Collins, G.E.: Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  8. 8.
    Ershov, Y.L.: Definability and computability. Plenum, New York (1996)Google Scholar
  9. 9.
    Henzinger, T.A., Rusu, V.: Reachability Verification for Hybrid Automata. In: Henzinger, T.A., Sastry, S.S. (eds.) HSCC 1998. LNCS, vol. 1386, pp. 190–205. Springer, Heidelberg (1998)Google Scholar
  10. 10.
    Immerman, N.: Descriptive Complexity. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  11. 11.
    Blass, A., Gurevich, Y.: Background, reserve and Gandy machines. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 1–17. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  12. 12.
    Korovina, M.V., Kudinov, O.V.: Towards Computability over Effectively Enumerable Topological Spaces. Electr. Notes Theor. Comput. Sci. 202, 305–313 (2008)CrossRefGoogle Scholar
  13. 13.
    Korovina, M.V., Kudinov, O.V.: Towards computability of higher type continuous data. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 235–241. Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Korovina, M.V.: Computational aspects of Σ-definability over the real numbers without the equality test. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 330–344. Springer, Heidelberg (2003)Google Scholar
  15. 15.
    Korovina, M.V.: Gandy’s theorem for abstract structures without the equality test. In: Vardi, M.Y., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 290–301. Springer, Heidelberg (2003)Google Scholar
  16. 16.
    Korovina, M.V., Kudinov, O.V.: Semantic characterisations of second-order computability over the real numbers. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 160–172. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    Korovina, M.V., Kudinov, O.V.: Formalisation of Computability of Operators and Real-Valued Functionals via Domain Theory. In: Blank, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 146–168. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Korovina, M.V., Kudinov, O.V.: Generalised Computability and Applications to Hybrid Systems. In: Bjørner, D., Broy, M., Zamulin, A.V. (eds.) PSI 2001. LNCS, vol. 2244, pp. 494–499. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. 19.
    Korovina, M.V., Kudinov, O.V.: Characteristic properties of majorant-computability over the reals. In: Gottlob, G., Grandjean, E., Seyr, K. (eds.) CSL 1998. LNCS, vol. 1584, pp. 188–203. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  20. 20.
    Dahlhaus, E., Makowsky, J.A.: Query languages for hierarchic databases. Information and Computation 101, 1–32 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Nerode, A., Kohn, W.: Models for Hybrid Systems: Automata, Topologies, Controllability, Observability. In: Grossman, R.L., Ravn, A.P., Rischel, H., Nerode, A. (eds.) HS 1991 and HS 1992. LNCS, vol. 736, pp. 317–357. Springer, Heidelberg (1993)Google Scholar
  22. 22.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1988)Google Scholar
  23. 23.
    Ratschan, S., She, Z.: Constraints for Continuous Reachability in the Verification of Hybrid Systems. In: Calmet, J., Ida, T., Wang, D. (eds.) AISC 2006. LNCS (LNAI), vol. 4120, pp. 196–210. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Tarski, A.: A Decidion Method in Algebra and Geometry. University of California Press, Berkeley (1951)Google Scholar
  25. 25.
    Tiwari, A.: Abstractions for hybrid systems. Formal Methods in System Design 32(1), 57–83 (2008)zbMATHCrossRefGoogle Scholar
  26. 26.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Margarita Korovina
    • 1
  • Oleg Kudinov
    • 2
  1. 1.Centre for Interdisciplinary Computational and Dynamical AnalysisThe University of Manchester and IIS SB RAS Novosibirsk 
  2. 2.Sobolev Institute of Mathematics, Novosibirsk 

Personalised recommendations