Skip to main content

Advertisement

SpringerLink
Go to cart
  1. Home
  2. Journal of Automated Reasoning
  3. Article
Differential Dynamic Logic for Hybrid Systems
Download PDF
Your article has downloaded

Similar articles being viewed by others

Slider with three articles shown per slide. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide.

Predicate Transformer Semantics for Hybrid Systems

31 October 2021

Jonathan Julián Huerta y Munive & Georg Struth

Some Problems of Analysis of Hybrid Automata

26 July 2018

V. V. Skobelev & V. G. Skobelev

An axiomatic approach to existence and liveness for differential equations

12 April 2021

Yong Kiam Tan & André Platzer

Verifying Safety and Persistence in Hybrid Systems Using Flowpipes and Continuous Invariants

24 November 2018

Andrew Sogokon, Paul B. Jackson & Taylor T. Johnson

Logic-based specification and verification of homogeneous dynamic multi-agent systems

28 April 2020

Riccardo De Masellis & Valentin Goranko

Refinements of behavioural abstractions for the supervisory control of hybrid systems

22 April 2020

Jung-Min Yang, Thomas Moor & Jörg Raisch

A Simple Logic of Functional Dependence

24 March 2021

Alexandru Baltag & Johan van Benthem

A Poincaré–Bendixson theorem for hybrid dynamical systems on directed graphs

16 November 2019

William Clark & Anthony Bloch

The Refinement Calculus of Reactive Systems Toolset

07 April 2020

Iulia Dragomir, Viorel Preoteasa & Stavros Tripakis

Download PDF
  • Open Access
  • Published: 20 August 2008

Differential Dynamic Logic for Hybrid Systems

  • André Platzer1 

Journal of Automated Reasoning volume 41, pages 143–189 (2008)Cite this article

  • 997 Accesses

  • 238 Citations

  • Metrics details

A Correction to this article was published on 15 November 2021

This article has been updated

Abstract

Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, we introduce a dynamic logic for hybrid programs, which is a program notation for hybrid systems. As a verification technique that is suitable for automation, we introduce a free variable proof calculus with a novel combination of real-valued free variables and Skolemisation for lifting quantifier elimination for real arithmetic to dynamic logic. The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts. Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations. In a case study with cooperating traffic agents of the European Train Control System, we further show that our calculus is well-suited for verifying realistic hybrid systems with parametric system dynamics.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Change history

  • 15 November 2021

    A Correction to this paper has been published: https://doi.org/10.1007/s10817-021-09608-w

References

  1. Ábrahám-Mumm, E., Steffen, M., Hannemann, U.: Verification of hybrid systems: formalization and proof rules in PVS. In: ICECCS, pp. 48–57. IEEE Computer Society, Los Alamitos (2001). doi:10.1109/ICECCS.2001.930163

    Google Scholar 

  2. Alur, R., Courcoubetis, C., Dill, D.L.: Model-checking for real-time systems. In: LICS, pp. 414–425. IEEE Computer Society, Los Alamitos (1990)

    Google Scholar 

  3. Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of hybrid systems. Theor. Comput. Sci. 138(1), 3–34 (1995). doi:10.1016/0304-3975(94)00202-T

    Article  MathSciNet  MATH  Google Scholar 

  4. Anai, H., Weispfenning, V.: Reach set computations using real quantifier elimination. In: Benedetto, M.D.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC, LNCS, vol. 2034, pp. 63–76. Springer, Berlin (2001). doi:10.1007/3-540-45351-2_9

    Google Scholar 

  5. Asarin, E., Dang, T., Girard, A.: Reachability analysis of nonlinear systems using conservative approximation. In: Maler, O., Pnueli, A. (eds.) Hybrid Systems: Computation and Control, 6th International Workshop, HSCC 2003 Prague, Czech Republic, April 3–5, 2003, Proceedings, LNCS, vol. 2623, pp. 20–35. Springer, Berlin (2003). doi:10.1007/3-540-36580-X_5

    Chapter  Google Scholar 

  6. Beckert, B.: Equality and other theories. In: D’Agostino, M., Gabbay, D., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods. Kluwer, Deventer (1999)

    Google Scholar 

  7. Beckert, B., Hähnle, R., Schmitt, P.H. (eds.): Verification of Object-Oriented Software: The KeY Approach, LNCS, vol. 4334. Springer, Berlin (2007)

    Google Scholar 

  8. Beckert, B., Platzer, A.: Dynamic logic with non-rigid functions: a basis for object-oriented program verification. In: Furbach, U., Shankar, N. (eds.) IJCAR, LNCS, vol. 4130, pp. 266–280. Springer, Berlin (2006)

    Google Scholar 

  9. Branicky, M.S.: Studies in hybrid systems: modeling, analysis, and control. Ph.D. thesis, Dept. Elec. Eng. and Computer Sci., Massachusetts Inst. Technol., Cambridge, MA (1995)

  10. Branicky, M.S.: Universal computation and other capabilities of hybrid and continuous dynamical systems. Theor. Comput. Sci. 138(1), 67–100 (1995). doi:10.1016/0304-3975(94)00147-B

    Article  MathSciNet  MATH  Google Scholar 

  11. Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Automat. Contr. 43(1), 31–45 (1998). doi:10.1109/9.654885

    Article  MathSciNet  MATH  Google Scholar 

  12. Chaochen, Z., Ji, W., Ravn, A.P.: A formal description of hybrid systems. In: Alur, R., Henzinger, T.A., Sontag, E.D. (eds.) Hybrid Systems, LNCS, vol. 1066, pp. 511–530. Springer, Berlin (1995)

    Google Scholar 

  13. Chutinan, A., Krogh, B.H.: Computational techniques for hybrid system verification. IEEE Trans. Automat. Contr. 48(1), 64–75 (2003). doi:10.1109/TAC.2002.806655

    Article  MathSciNet  MATH  Google Scholar 

  14. Clarke, E.M., Fehnker, A., Han, Z., Krogh, B.H., Ouaknine, J., Stursberg, O., Theobald, M.: Abstraction and counterexample-guided refinement in model checking of hybrid systems. Int. J. Found. Comput. Sci. 14(4), 583–604 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT, Cambridge (1999)

    MATH  Google Scholar 

  16. Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12(3), 299–328 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cook, S.A.: Soundness and completeness of an axiom system for program verification. SIAM J. Comput. 7(1), 70–90 (1978). doi:10.1137/0207005

    Article  MathSciNet  MATH  Google Scholar 

  18. Damm, W., Hungar, H., Olderog, E.R.: Verification of cooperating travel agents. Int. J. Control 79(5), 395–421 (2006)

    Article  MATH  Google Scholar 

  19. Damm, W., Mikschl, A., Oehlerking, J., Olderog, E.R., Pang, J., Platzer, A., Segelken, M., Wirtz, B.: Automating verification of cooperation, control, and design in traffic applications. In: Jones, C.B., Liu, Z., Woodcock, J. (eds.) Formal Methods and Hybrid Real-Time Systems, LNCS, vol. 4700, pp. 115–169. Springer, Berlin (2007)

    Chapter  MATH  Google Scholar 

  20. Davoren, J.M.: On hybrid systems and the modal μ-calculus. In: Antsaklis, P.J., Kohn, W., Lemmon, M.D., Nerode, A., Sastry, S. (eds.) Hybrid Systems, LNCS, vol. 1567, pp. 38–69. Springer, Berlin (1997). doi:10.1007/3-540-49163-5_3

    Google Scholar 

  21. Davoren, J.M., Nerode, A.: Logics for hybrid systems. Proc. IEEE 88(7), 985–1010 (2000). doi:10.1109/5.871305

    Article  Google Scholar 

  22. Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Commun. ACM 22(8), 465–476 (1979). doi:10.1145/359138.359142

    Article  MathSciNet  MATH  Google Scholar 

  23. Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. J. Autom. Reason. 31(1), 33–72 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Emerson, E.A., Clarke, E.M.: Using branching time temporal logic to synthesize synchronization skeletons. Sci. Comput. Program. 2(3), 241–266 (1982)

    Article  MATH  Google Scholar 

  25. Emerson, E.A., Halpern, J.Y.: “Sometimes” and “Not Never” revisited: on branching versus linear time temporal logic. J. Assoc. Comput. Mach. 33(1), 151–178 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fitting, M.: First-Order Logic and Automated Theorem Proving, 2nd edn. Springer, New York (1996)

    Book  MATH  Google Scholar 

  27. Fitting, M., Mendelsohn, R.L.: First-Order Modal Logic. Kluwer, Norwell (1999)

    MATH  Google Scholar 

  28. Fränzle, M.: Analysis of hybrid systems: an ounce of realism can save an infinity of states. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL, LNCS, vol. 1683, pp. 126–140. Springer, Berlin (1999)

    Google Scholar 

  29. Frehse, G.: PHAVer: algorithmic verification of hybrid systems past HyTech. In: Morari, M., Thiele, L. (eds.) HSCC, LNCS, vol. 3414, pp. 258–273. Springer, Berlin (2005). doi:10.1007/b106766

    Google Scholar 

  30. Giese, M.: Incremental closure of free variable tableaux. In: Goré, R., Leitsch, A., Nipkow, T. (eds.) IJCAR, LNCS, vol. 2083, pp. 545–560. Springer, Berlin (2001). doi:10.1007/3-540-45744-5_46

    Google Scholar 

  31. Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Mon.hefte Math. Phys. 38, 173–198 (1931). doi:10.1007/BF01700692

    MATH  Google Scholar 

  32. Graça, D.S., Campagnolo, M.L., Buescu, J.: Computability with polynomial differential equations. Adv. Appl. Math. 40, 330–349 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hähnle, R., Schmitt, P.H.: The liberalized δ-rule in free variable semantic tableaux. J. Autom. Reason. 13(2), 211–221 (1994). doi:10.1007/BF00881956

    Article  MathSciNet  MATH  Google Scholar 

  34. Harel, D.: First-Order Dynamic Logic. Springer, New York (1979)

    Book  MATH  Google Scholar 

  35. Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT, Cambridge (2000)

    Book  MATH  Google Scholar 

  36. Henzinger, T.A.: The theory of hybrid automata. In: LICS, pp. 278–292. IEEE Computer Society, Los Alamitos (1996)

    Google Scholar 

  37. Henzinger, T.A., Nicollin, X., Sifakis, J., Yovine, S.: Symbolic model checking for real-time systems. In: LICS, pp. 394–406. IEEE Computer Society, Los Alamitos (1992)

    Google Scholar 

  38. Hutter, D., Langenstein, B., Sengler, C., Siekmann, J.H., Stephan, W., Wolpers, A.: Deduction in the verification support environment (VSE). In: Gaudel, M.C., Woodcock, J. (eds.) FME, LNCS, vol. 1051, pp. 268–286. Springer, Berlin (1996)

    Google Scholar 

  39. Jifeng, H.: From CSP to hybrid systems. In: Roscoe, A.W. (ed.) A Classical Mind: Essays in Honour of C. A. R. Hoare, pp. 171–189. Prentice Hall, Hertfordshire (1994)

    Google Scholar 

  40. Kesten, Y., Manna, Z., Pnueli, A.: Verification of clocked and hybrid systems. Acta Inf. 36(11), 837–912 (2000). doi:10.1007/s002360050177

    Article  MathSciNet  MATH  Google Scholar 

  41. Lafferriere, G., Pappas, G.J., Yovine, S.: A new class of decidable hybrid systems. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC, LNCS, vol. 1569, pp. 137–151. Springer, Berlin (1999)

    Google Scholar 

  42. Manna, Z., Sipma, H.: Deductive verification of hybrid systems using STeP. In: Henzinger, T.A., Sastry, S. (eds.) HSCC, LNCS, vol. 1386, pp. 305–318. Springer, Berlin (1998). doi:10.1007/3-540-64358-3_47

    Google Scholar 

  43. Morayne, M.: On differentiability of Peano type functions. Colloq. Math. LIII, 129–132 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mysore, V., Piazza, C., Mishra, B.: Algorithmic algebraic model checking II: Decidability of semi-algebraic model checking and its applications to systems biology. In: Peled, D., Tsay, Y.K. (eds.) ATVA, LNCS, vol. 3707, pp. 217–233. Springer, Berlin (2005)

    Google Scholar 

  45. Perko, L.: Differential equations and dynamical systems. Springer, New York (1991)

    Book  MATH  Google Scholar 

  46. Platzer, A.: Combining deduction and algebraic constraints for hybrid system analysis. In: Beckert, B. (ed.) VERIFY’07 at CADE, Bremen, Germany, CEUR Workshop Proceedings, vol. 259, pp. 164–178. CEUR-WS.org (2007)

  47. Platzer, A.: Differential dynamic logic for verifying parametric hybrid systems. In: Olivetti, N. (ed.) TABLEAUX, LNCS, vol. 4548, pp. 216–232. Springer, Berlin (2007)

    Google Scholar 

  48. Platzer, A.: A temporal dynamic logic for verifying hybrid system invariants. In: Artëmov, S.N., Nerode, A. (eds.) LFCS, LNCS, vol. 4514, pp. 457–471. Springer, Berlin (2007)

    Google Scholar 

  49. Platzer, A., Clarke, E.M.: The image computation problem in hybrid systems model checking. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC, LNCS, vol. 4416, pp. 473–486. Springer, Berlin (2007)

    Google Scholar 

  50. Pnueli, A.: The temporal logic of programs. In: FOCS, pp. 46–57. IEEE, Piscataway (1977)

    Google Scholar 

  51. Pratt, V.R.: Semantical considerations on Floyd-Hoare logic. In: FOCS, pp. 109–121. IEEE, Piscataway (1976)

    Google Scholar 

  52. Rönkkö, M., Ravn, A.P., Sere, K.: Hybrid action systems. Theor. Comput. Sci. 290(1), 937–973 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sibirsky, K.S.: Introduction to Topological Dynamics. Noordhoff, Leyden (1975)

    MATH  Google Scholar 

  54. Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)

    Book  MATH  Google Scholar 

  55. Tavernini, L.: Differential automata and their discrete simulators. Nonlinear Anal. 11(6), 665–683 (1987). doi:10.1016/0362-546X(87)90034-4

    Article  MathSciNet  MATH  Google Scholar 

  56. Tinelli, C.: Cooperation of background reasoners in theory reasoning by residue sharing. J. Autom. Reason. 30(1), 1–31 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  57. Tiwari, A.: Approximate reachability for linear systems. In: Maler, O., Pnueli, A. (eds.) Hybrid Systems: Computation and Control, 6th International Workshop, HSCC 2003 Prague, Czech Republic, April 3–5, 2003, Proceedings, LNCS, vol 2623, pp. 514–525. Springer, Berlin (2003). doi:10.1007/3-540-36580-X_37

    Chapter  Google Scholar 

  58. Walter, W.: Ordinary Differential Equations. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  59. Zhou, C., Ravn, A.P., Hansen, M.R.: An extended duration calculus for hybrid real-time systems. In: Grossman, R.L., Nerode, A., Ravn, A.P., Rischel, H. (eds.) Hybrid Systems, LNCS, vol. 736, pp. 36–59. Springer, Berlin (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computing Science, University of Oldenburg, 26111, Oldenburg, Germany

    André Platzer

Authors
  1. André Platzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to André Platzer.

Additional information

The original online version of this article was revised due to retrospective open access order

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Cite this article

Platzer, A. Differential Dynamic Logic for Hybrid Systems. J Autom Reasoning 41, 143–189 (2008). https://doi.org/10.1007/s10817-008-9103-8

Download citation

  • Received: 23 August 2007

  • Accepted: 27 June 2008

  • Published: 20 August 2008

  • Issue Date: August 2008

  • DOI: https://doi.org/10.1007/s10817-008-9103-8

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Dynamic logic
  • Differential equations
  • Sequent calculus
  • Axiomatisation
  • Automated theorem proving
  • Verification of hybrid systems
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Over 10 million scientific documents at your fingertips

Switch Edition
  • Academic Edition
  • Corporate Edition
  • Home
  • Impressum
  • Legal information
  • Privacy statement
  • Your US state privacy rights
  • How we use cookies
  • Your privacy choices/Manage cookies
  • Accessibility
  • FAQ
  • Contact us
  • Affiliate program

Not affiliated

Springer Nature

© 2023 Springer Nature Switzerland AG. Part of Springer Nature.