A Calculus for Hybrid CSP

  • Jiang Liu
  • Jidong Lv
  • Zhao Quan
  • Naijun Zhan
  • Hengjun Zhao
  • Chaochen Zhou
  • Liang Zou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6461)


Hybrid Communicating Sequential Processes (HCSP) is an extension of CSP allowing continuous dynamics. We are interested in applying HCSP to model and verify hybrid systems. This paper is to present a calculus for a subset of HCSP as a part of our efforts in modelling and verifying hybrid systems. The calculus consists of two parts. To deal with continuous dynamics, the calculus adopts differential invariants. A brief introduction to a complete algorithm for generating polynomial differential invariants is presented, which applies DISCOVERER, a symbolic computation tool for semi-algebraic systems. The other part of the calculus is a logic to reason about HCSP process, which involves communication, parallelism, real-time as well as continuous dynamics. This logic is named as Hybrid Hoare Logic. Its assertions consist of traditional pre- and post-conditions, and also Duration Calculus formulas to record execution history of HCSP process.


Chinese Train Control System Differential Invariant DISCOVERER Duration Calculus Hoare Logic Hybrid CSP Hybrid Logic 


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  1. 1.
    Apt, K., de Boer, F., Olderog, E.-R.: Verfication of Sequential and Concurrent Programs. Springer, Heidelberg (2009) ISBN 978-1-184882-744-8CrossRefzbMATHGoogle Scholar
  2. 2.
    Collins, G.E.: 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
  3. 3.
    Gulwani, S., Tiwari, A.: Constraint-based approach for analysis of hybrid systems. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    : From CSP to hybrid systems. In: The proc. of A Classical Mind: Essays in Honour of C. A. R. Hoare. International Series In Computer Science, pp. 171–189. Prentice-Hall, Englewood Cliffs (1994), ISBN:0-13-294844-3Google Scholar
  5. 5.
    Liu, J., Zhan, N., Zhao, H.: A complete method for generating polynomial differential invariants. Technical Report of State Key Lab. of Comp. Sci., ISCAS-LCS-10-15 (2010)Google Scholar
  6. 6.
    Manna, Z., Pnueli, A.: Models of reactitivity. Acta Informatica 30(7), 609–678 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Olderog, R.-R., Dierks, H.: Real-Time Systems: Formal Secification and Automatic Verification. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  8. 8.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. of Logic and Computation 20(1), 309–352 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. of Automated Reasoning 41, 143–189 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Prajna, S., Jadbabaie, A.: Safety verification of hybrid systems using barrier certificates. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 477–492. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constructing invariants for hybrid systems. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 539–554. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Xia, B.: DISCOVERER: A tool for solving semi-algebraic systems. In: Software Demo at ISSAC 2007, Waterloo, July 30 (2007); Also: ACM SIGSAM Bulletin 41(3),102–103 (September 2007)Google Scholar
  14. 14.
    Yang, L.: Recent advances on determining the number of real roots of parametric polynomials. J. Symbolic Computation 28, 225–242 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yang, L., Hou, X., Zeng, Z.: A complete discrimination system for polynomials. Science in China (Ser. E) 39, 628–646 (1996)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Zhou, C., Wang, J., Ravn, A.: A formal description of hybrid systems. In: Alur, R., Sontag, E.D., Henzinger, T.A. (eds.) HS 1995. LNCS, vol. 1066, Springer, Heidelberg (1996)Google Scholar
  17. 17.
    Zhang, S.: The General Technical Solutions to Chinese Train Control System at Level 3 (CTCS-3). China Railway Publisher (2008)Google Scholar
  18. 18.
    Zhou, C., Hansen, M.: Duration Calculus: A Formal Approach to Real-Time Systems. Springer, Heidelberg (2004), ISBN 3-540-40823-1zbMATHGoogle Scholar
  19. 19.
    Zhou, C., Hoare, C.A.R., Ravn, A.: A calculus of durations. Information Processing Letters 40(5), 269–276 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jiang Liu
    • 1
  • Jidong Lv
    • 2
  • Zhao Quan
    • 1
  • Naijun Zhan
    • 1
  • Hengjun Zhao
    • 1
  • Chaochen Zhou
    • 1
  • Liang Zou
    • 1
  1. 1.State Key Lab. of Computer ScienceInstitute of Software, CASChina
  2. 2.State Key Lab. of Rail Traffic Control and SafetyBeijing Jiaotong UniversityChina

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