Towards an Algebra of Hybrid Systems

  • Peter Höfner
  • Bernhard Möller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3929)


We present a trajectory-based model for describing hybrid systems. For this we use left quantales and left semirings, thus providing a new application for these algebraic structures. Furthermore, we sketch a connection between game theory and hybrid systems.


Hybrid System Boolean Algebra Safety Requirement Hybrid Automaton Galois Connection 
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.


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  1. 1.
    Backhouse, R., Michaelis, D.: Fixed-Point Characterisation of Winning Strategies in Impartial Games. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2003. LNCS, vol. 3051, pp. 34–47. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Cohen, E.: Separation and Reduction. In: Backhouse, R., Oliveira, J.N. (eds.) MPC 2000. LNCS, vol. 1837, pp. 45–59. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Conway, J.H.: Regular Algebra and Finite Machines. Chapman & Hall, Boca Raton (1971)zbMATHGoogle Scholar
  4. 4.
    Davoren, J.M., Nerode, A.: Logics for Hybrid Systems. Proc. IEEE 88, 985–1010 (2000)CrossRefGoogle Scholar
  5. 5.
    Desharnais, J., Möller, B., Struth, G.: Kleene Algebra with Domain. ACM Trans. Computational Logic (to appear, 2006); Preliminary version: Universität Augsburg, Institut für Informatik, Report No. 2003-07 (June 2003) Google Scholar
  6. 6.
    Desharnais, J., Möller, B., Struth, G.: Modal Kleene Algebra and Applications – A Survey. J. Relational Methods in Computer Science 1, 93–131 (2004), zbMATHGoogle Scholar
  7. 7.
    Henzinger, T.: The Theory of Hybrid Automata. In: Proc. 11th Annual IEEE Symposium on Logic in Computer Science, New Brunswick, New Jersey, pp. 278–292 (1996)Google Scholar
  8. 8.
    Höfner, P.: From Sequential Algebra to Kleene Algebra: Interval Modalities and Duration Calculus. Technical Report 2005-5, Institut für Informatik, Universität Augsburg (2005)Google Scholar
  9. 9.
    Höfner, P.: An Algebraic Semantics for Duration Calculus. In: 17th European Summer School in Logic, Language and Information (ESSLLI), Proc. 10th ESSLLI Student Session, Heriot-Watt University Edinburgh, Scotland, August 2005, pp. 99–111 (2005)Google Scholar
  10. 10.
    Isaacs, R.: Differential Games. Wiley, Chichester (1965) Republished: Dover (1999)Google Scholar
  11. 11.
    von Karger, B.: Temporal Algebra. Habilitation thesis, University of Kiel (1997)Google Scholar
  12. 12.
    Kozen, D.: Kleene Algebra with Tests. ACM Trans. Programming Languages and Systems 19, 427–443 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lynch, N.A., Segala, R., Vaandrager, F.W.: Hybrid I/O Automata. Information and Computation 185, 105–157 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Möller, B.: Complete Tests do not Guarantee Domain. Technical Report 2005-6, Institut für Informatik, Universität Augsburg (2005)Google Scholar
  15. 15.
    Möller, B.: Lazy Kleene Algebra. In: Kozen, D. (ed.) MPC 2004. LNCS, vol. 3125, pp. 252–273. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Sintzoff, M.: Iterative Synthesis of Control Guards Ensuring Invariance and Inevitability in Discrete-Decision Games. In: Owe, O., Krogdahl, S., Lyche, T. (eds.) From Object-Orientation to Formal Methods. LNCS, vol. 2635, pp. 272–301. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Peter Höfner
    • 1
  • Bernhard Möller
    • 1
  1. 1.Institut für InformatikUniversität AugsburgAugsburgGermany

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