Skip to main content
SpringerLink
Log in

Differential Dynamic Logic dℒ

  • Chapter
  • First Online:
Logical Analysis of Hybrid Systems

Synopsis

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 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    Google Scholar 

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

    Google Scholar 

  3. van den Dries, L., Miller, C.: On the real exponential field with restricted analytic functions. Israel J. Math. 85(1-3), 19–56 (1994). DOI 10.1007/BF02758635

    Article  MATH  MathSciNet  Google Scholar 

  4. Blackburn, P.: Internalizing labelled deduction. J. Log. Comput. 10(1), 137–168 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Scott, D., Strachey, C.: Toward a mathematical semantics for computer languages? Tech. Rep. PRG-6, Oxford Programming Research Group (1971)

    Google Scholar 

  6. Graça, D.S., Campagnolo, M.L., Buescu, J.: Computability with polynomial differential equations. Advances in Applied Mathematics (2007)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. 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: C.B. Jones, Z. Liu, J. Woodcock (eds.) Formal Methods and Hybrid Real-Time Systems, LNCS, vol. 4700, pp. 115–169. Springer (2007). DOI 10.1007/978-3-540-75221-9_6

    Google Scholar 

  9. 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

    Article  Google Scholar 

  10. Gentzen, G.: Untersuchungen über das logische Schließen. Math. Zeit. 39, 405–431 (1935). DOI 10.1007/BF01201363

    Article  MathSciNet  Google Scholar 

  11. Beckert, B., Platzer, A.: Dynamic logic with non-rigid functions: A basis for object-oriented program verification. In: U. Furbach, N. Shankar (eds.) IJCAR, LNCS, vol. 4130, pp. 266– 280. Springer (2006). DOI 10.1007/11814771_23

    Google Scholar 

  12. Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12(10), 576–580 (1969). DOI 10.1145/363235.363259

    Article  MATH  Google Scholar 

  13. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Bianconi, R.: Undefinability results in o-minimal expansions of the real numbers. Ann. Pure Appl. Logic 134(1), 43–51 (2005). DOI 10.1016/j.apal.2004.06.010

    Article  MATH  MathSciNet  Google Scholar 

  16. Perko, L.: Differential equations and dynamical systems, 3 edn. Springer, New York, NY, USA (2006)

    Google Scholar 

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

    MATH  Google Scholar 

  18. Tarski, A.: Sur les ensembles définissables de nombres réels I. Fundam. Math. 17, 210–239 (1931)

    MATH  Google Scholar 

  19. Fitting, M., Mendelsohn, R.L.: First-OrderModal Logic. Kluwer, Norwell, MA, USA (1999)

    Google Scholar 

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

    Google Scholar 

  21. Harel, D., Kozen, D., Tiuryn, J.: Dynamic logic. MIT Press, Cambridge (2000)

    MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  24. 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 and Tsay [226], pp. 217–233. DOI 10.1007/11562948_18

    Google Scholar 

  25. Damm, W., Hungar, H., Olderog, E.R.: Verification of cooperating traffic agents. International Journal of Control 79(5), 395–421 (2006). DOI 10.1080/00207170600587531

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  27. Morayne, M.: On differentiability of Peano type functions. Colloquium Mathematicum LIII, 129–132 (1987)

    Google Scholar 

  28. Platzer, A., Clarke, E.M.: The image computation problem in hybrid systems model checking. In: Bemporad et al. [41], pp. 473–486. DOI 10.1007/978-3-540-71493-4_37

    Google Scholar 

  29. 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  MATH  Google Scholar 

  30. 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  MATH  MathSciNet  Google Scholar 

  31. Lafferriere, G., Pappas, G.J., Yovine, S.: A new class of decidable hybrid systems. In: F.W. Vaandrager, J.H. van Schuppen (eds.) HSCC, LNCS, vol. 1569, pp. 137–151. Springer (1999). DOI 10.1007/3-540-48983-5_15

    Google Scholar 

  32. Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12(3), 299–328 (1991). DOI 10.1016/S0747-7171(08)80152-6

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  34. Tinelli, C.: Cooperation of background reasoners in theory reasoning by residue sharing. J. Autom. Reasoning 30(1), 1–31 (2003). DOI 10.1023/A:1022587501759

    Article  MATH  MathSciNet  Google Scholar 

  35. van den Dries, L., Speissegger, P.: The real field with convergent generalized power series. Trans. Amer. Math. Soc. 350, 4377–4421 (1998). DOI 10.1090/S0002-9947-98-02105-9

    Article  MATH  MathSciNet  Google Scholar 

  36. Giese, M.: Incremental closure of free variable tableaux. In: Goré et al. [139], pp. 545–560. DOI 10.1007/3-540-45744-5_46

    Google Scholar 

  37. Frehse, G.: PHAVer: Algorithmic verification of hybrid systems past HyTech. In: Morari and Thiele [212], pp. 258–273. DOI 10.1007/b106766

    Google Scholar 

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

    MATH  Google Scholar 

  39. Achatz, M., McCallum, S., Weispfenning, V.: Deciding polynomial-exponential problems. In: J.R. Sendra, L. González-Vega (eds.) ISSAC, pp. 215–222. ACM (2008). DOI 10.1145/ 1390768.1390799

    Google Scholar 

  40. Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: Model and optimal control theory. IEEE T. Automat. Contr. 43(1), 31–45 (1998). DOI 10.1109/9. 654885

    Article  MATH  MathSciNet  Google Scholar 

  41. Platzer, A.: Differential dynamic logic for verifying parametric hybrid systems. In: N. Olivetti (ed.) TABLEAUX, LNCS, vol. 4548, pp. 216–232. Springer (2007). DOI 10.1007/978-3-540-73099-6_17

    Google Scholar 

  42. Egerstedt, M., Johansson, K.H., Sastry, S., Lygeros, J.: On the regularization of Zeno hybrid automata. Systems and Control Letters 38, 141–150 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  44. Miller, C.L.: Expansions of the real field with power functions. Ann. Pure Appl. Logic 68(1), 79–94 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  45. Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. J. Autom. Reasoning 31(1), 33–72 (2003). DOI 10.1023/A:1027357912519

    Article  MATH  MathSciNet  Google Scholar 

  46. Asarin, E., Dang, T., Girard, A.: Reachability analysis of nonlinear systems using conservative approximation. In: Maler and Pnueli [200], pp. 20–35. DOI 10.1007/3-540-36580-X_5

    Google Scholar 

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

    Google Scholar 

  48. Bianconi, R.: Nondefinability results for expansions of the field of real numbers by the exponential function and by the restricted sine functions. J. Symb. Log. 62(4), 1173–1178 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  49. Beckert, B., Hähnle, R., Schmitt, P.H. (eds.): Verification of Object-Oriented Software: The KeY Approach, LNCS, vol. 4334. Springer (2007). DOI 10.1007/978-3-540-69061-0

    Google Scholar 

  50. Kozen, D.: Kleene algebra with tests. ACM Trans. Program. Lang. Syst. 19(3), 427–443 (1997). DOI 10.1145/256167.256195

    Article  Google Scholar 

  51. 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)

    Google Scholar 

  52. Kripke, S.A.: Semantical considerations on modal logic. Acta Philosophica Fennica 16, 83– 94 (1963)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA, 15213, USA

    Dr. André Platzer

Authors
  1. Dr. André Platzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to André Platzer .

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Platzer, A. (2010). Differential Dynamic Logic dℒ. In: Logical Analysis of Hybrid Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14509-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-14509-4_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14508-7

  • Online ISBN: 978-3-642-14509-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation