An Improved HHL Prover: An Interactive Theorem Prover for Hybrid Systems

  • Shuling Wang
  • Naijun Zhan
  • Liang Zou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9407)


Hybrid systems are integrations of discrete computation and continuous physical evolution. To guarantee the correctness of hybrid systems, formal techniques on modelling and verification of hybrid systems have been proposed. Hybrid CSP (HCSP) is an extension of CSP with differential equations and some forms of interruptions for modelling hybrid systems, and Hybrid Hoare logic (HHL) is an extension of Hoare logic for specifying and verifying hybrid systems that are modelled using HCSP. In this paper, we report an improved HHL prover, which is an interactive theorem prover based on Isabelle/HOL for verifying HCSP models. Compared with the prototypical release in [22], the new HHL prover realises the proof system of HHL as a shallow embedding in Isabelle/HOL, rather than deep embedding in [22]. In order to contrast the new HHL prover in shallow embedding and the old one in deep embedding, we demonstrate the use of both variants on the safety verification of a lunar lander case study.


Model Check Hybrid System Inference Rule Verification Condition Proof Assistant 
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.



This paper is supported partly by “973 Program” under grant No. 2014CB340701, by NSFC under grants 91118007 and 91418204, by CDZ project CAP (GZ 1023), and by the CAS/SAFEA International Partnership Program for Creative Research Teams.


  1. 1.
    Alur, R.: Formal verification of hybrid systems. In: EMSOFT 2011, pp. 273–278 (2011)Google Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.-H.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Grossman, R.L., Ravn, A.P., Rischel, H., Nerode, A. (eds.) HS 1991 and HS 1992. LNCS, vol. 736, pp. 209–229. Springer, Heidelberg (1993) CrossRefGoogle Scholar
  3. 3.
    Alur, R., Dang, T., Ivančić, F.: Predicate abstraction for reachability analysis of hybrid systems. ACM Trans. Embed. Comput. Syst. 5(1), 152–199 (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Asarin, E., Bournez, O., Dang, T., Maler, O.: Approximate reachability analysis of piecewise-linear dynamical systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, p. 20. Springer, Heidelberg (2000) CrossRefGoogle Scholar
  5. 5.
    Asarin, E., Dang, T., Maler, O.: The d/dt tool for verification of hybrid systems. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, p. 365. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  6. 6.
    Chen, X., Ábrahám, E., Sankaranarayanan, S.: Flow*: an analyzer for non-linear hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 258–263. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  7. 7.
    Eggers, A., Ramdani, N., Nedialkov, N., Fränzle, M.: Improving SAT modulo ODE for hybrid systems analysis by combining different enclosure methods. In: Barthe, G., Pardo, A., Schneider, G. (eds.) SEFM 2011. LNCS, vol. 7041, pp. 172–187. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  8. 8.
    Frehse, G.: PHAVer: algorithmic verification of hybrid systems past hytech. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 258–273. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  9. 9.
    He, J. : From CSP to hybrid systems. In: A Classical Mind, Essays in Honour of C.A.R. Hoare, pp. 171–189. Prentice Hall International (UK) Ltd. (1994)Google Scholar
  10. 10.
    Henzinger, T.A.: The theory of hybrid automata. In: Inan, M.K., Kurshan, R.P. (eds.) LICS’1996. NATO ASI Series, vol. 170, pp. 278–292. Springer, Heidelberg (1996) Google Scholar
  11. 11.
    Liu, J., Lv, J., Quan, Z., Zhan, N., Zhao, H., Zhou, C., Zou, L.: A calculus for hybrid CSP. In: Ueda, K. (ed.) APLAS 2010. LNCS, vol. 6461, pp. 1–15. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  12. 12.
    Liu, J., Zhan, N., Zhao, H.: Computing semi-algebraic invariants for polynomial dynamical systems. In: EMSOFT 2011, pp. 97–106 (2011)Google Scholar
  13. 13.
    Manna, Z., Pnueli, A.: Verifying hybrid systems. In: Grossman, R.L., Ravn, A.P., Rischel, H., Nerode, A. (eds.) HS 1991 and HS 1992. LNCS, vol. 736, pp. 4–35. Springer, Heidelberg (1993) CrossRefGoogle Scholar
  14. 14.
    Platzer, A., Clarke, E.M.: Computing differential invariants of hybrid systems as fixedpoints. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 176–189. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Wildmoser, M., Nipkow, T.: Certifying machine code safety: shallow versus deep embedding. In: Slind, K., Bunker, A., Gopalakrishnan, G.C. (eds.) TPHOLs 2004. LNCS, vol. 3223, pp. 305–320. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  17. 17.
    Zhan, N., Wang, S., Zhao, H.: Formal modelling, analysis and verification of hybrid systems. In: Liu, Z., Woodcock, J., Zhu, H. (eds.) Unifying Theories of Programming and Formal Engineering Methods. LNCS, vol. 8050, pp. 207–281. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  18. 18.
    Zhao, H., Yang, M., Zhan, N., Gu, B., Zou, L., Chen, Y.: Formal verification of a descent guidance control program of a lunar lander. In: Jones, C., Pihlajasaari, P., Sun, J. (eds.) FM 2014. LNCS, vol. 8442, pp. 733–748. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  19. 19.
    Zhou, C., Hansen, M.R.: Duration Calculus – A Formal Approach to Real-Time Systems. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2004) zbMATHGoogle Scholar
  20. 20.
    Zhou, C., Hoare, C.A.R., Ravn, A.P.: A calculus of durations. Inf. Process. Lett. 40(5), 269–276 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chaochen, Z., Ji, W., Ravn, A.P.: A formal description of hybrid systems. In: Alur, R., Sontag, E.D., Henzinger, T.A. (eds.) HS 1995. LNCS, vol. 1066, pp. 511–530. Springer, Heidelberg (1996) CrossRefGoogle Scholar
  22. 22.
    Zou, L., Lv, J., Wang, S., Zhan, N., Tang, T., Yuan, L., Liu, Y.: Verifying Chinese train control system under a combined scenario by theorem proving. In: Cohen, E., Rybalchenko, A. (eds.) VSTTE 2013. LNCS, vol. 8164, pp. 262–280. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  23. 23.
    Zou, L., Zhan, N., Wang, S., Fränzle, M.: Formal verification of simulink/stateflow diagrams. In: ATVA 2015 (2015) (to appear)Google Scholar
  24. 24.
    Zou, L., Zhan, N., Wang, S., Fränzle, M., Qin, S.: Verifying simulink diagrams via a hybrid hoare logic prover. In: EMSOFT 2013, pp. 1–10 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.State Key Laboratory of Computer ScienceInstitute of Software, Chinese Academy of SciencesBeijingChina

Personalised recommendations