Playing Hybrid Games with KeYmaera

  • Jan-David Quesel
  • André Platzer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7364)

Abstract

We propose a new logic, called differential dynamic game logic (\({\sf dDG}{\mathcal{L}}\)), that adds several game constructs on top of differential dynamic logic (\({\sf d}\mathcal{L}\)) so that it can be used for hybrid games. The logic \({\sf dDG}{\mathcal{L}}\) is a conservative extension of \({\sf d}\mathcal{L}\), which we exploit for our implementation of \({\sf dDG}{\mathcal{L}}\) in the theorem prover KeYmaera. We provide rules for extending the \({\sf d}\mathcal{L}\) sequent proof calculus to handle the \({\sf dDG}{\mathcal{L}}\) constructs by identifying analogs to operators of \({\sf d}\mathcal{L}\). We have implemented \({\sf dDG}{\mathcal{L}}\) in an extension of KeYmaera and verified a case study in which a robot satisfies a joint safety and liveness objective in a factory automation scenario, in which the factory may perform interfering actions independently.

Keywords

differential dynamic logic hybrid games sequent calculus theorem proving logics for hybrid systems factory automation 

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References

  1. 1.
    Bouyer, P., Brihaye, T., Chevalier, F.: O-minimal hybrid reachability games. Logical Methods in Computer Science 6(1) (2009)Google Scholar
  2. 2.
    Gao, Y., Lygeros, J., Quincampoix, M.: On the Reachability Problem for Uncertain Hybrid Systems. IEEE Transactions on Automatic Control 52(9) (September 2007)Google Scholar
  3. 3.
    Harel, D.: First-Order Dynamic Logic. LNCS, vol. 68. Springer, Heidelberg (1979)MATHCrossRefGoogle Scholar
  4. 4.
    Henzinger, T.A., Horowitz, B., Majumdar, R.: Rectangular Hybrid Games. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 320–335. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Maler, O., Pnueli, A., Sifakis, J.: On the Synthesis of Discrete Controllers for Timed Systems (An Extended Abstract). In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 229–242. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  6. 6.
    Parikh, R.: The logic of games and its applications. In: Annals of Discrete Mathematics, pp. 111–140. Elsevier (1985)Google Scholar
  7. 7.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reas. 41(2), 143–189 (2008)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Platzer, A.: Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics. Springer, Heidelberg (2010)MATHCrossRefGoogle Scholar
  9. 9.
    Platzer, A.: Stochastic Differential Dynamic Logic for Stochastic Hybrid Programs. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 446–460. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Platzer, A.: Differential game logic for hybrid games. Tech. Rep. CMU-CS-12-105, School of Computer Science, Carnegie Mellon University, Pittsburgh (March 2012)Google Scholar
  11. 11.
    Platzer, A., Quesel, J.-D.: KeYmaera: A Hybrid Theorem Prover for Hybrid Systems (System Description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 171–178. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Quesel, J.-D., Fränzle, M., Damm, W.: Crossing the Bridge between Similar Games. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 160–176. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Quesel, J.D., Platzer, A.: Playing Hybrid Games with KeYmaera. Tech. Rep. 84, SFB/TR 14 AVACS (April 2012), http://www.avacs.org ISSN: 1860–9821
  14. 14.
    Tomlin, C., Lygeros, J., Sastry, S.: A Game Theoretic Approach to Controller Design for Hybrid Systems. Proceedings of IEEE 88, 949–969 (2000)CrossRefGoogle Scholar
  15. 15.
    Vladimerou, V., Prabhakar, P., Viswanathan, M., Dullerud, G.: Specifications for decidable hybrid games. Theoretical Computer Science 412(48), 6770–6785 (2011)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jan-David Quesel
    • 1
  • André Platzer
    • 2
  1. 1.Department of Computing ScienceUniversity of OldenburgGermany
  2. 2.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

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