Skip to main content
SpringerLink
Log in

Logic & Proofs for Cyber-Physical Systems

  • Conference paper
  • First Online:
Automated Reasoning (IJCAR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9706))

Included in the following conference series:

Abstract

Cyber-physical systems (CPS) combine cyber aspects such as communication and computer control with physical aspects such as movement in space, which arise frequently in many safety-critical application domains, including aviation, automotive, railway, and robotics. But how can we ensure that these systems are guaranteed to meet their design goals, e.g., that an aircraft will not crash into another one?

This paper highlights some of the most fascinating aspects of cyber-physical systems and their dynamical systems models, such as hybrid systems that combine discrete transitions and continuous evolution along differential equations. Because of the impact that they can have on the real world, CPSs deserve proof as safety evidence.

Multi-dynamical systems understand complex systems as a combination of multiple elementary dynamical aspects, which makes them natural mathematical models for CPS, since they tame their complexity by compositionality. The family of differential dynamic logics achieves this compositionality by providing compositional logics, programming languages, and reasoning principles for CPS. Differential dynamic logics, as implemented in the theorem prover KeYmaera X, have been instrumental in verifying many applications, including the Airborne Collision Avoidance System ACAS X, the European Train Control System ETCS, automotive systems, mobile robot navigation, and a surgical robot system for skull-base surgery. This combination of strong theoretical foundations with practical theorem proving challenges and relevant applications makes Logic for CPS an ideal area for compelling and rewarding research.

This paper focuses on illustrating important principles of cyber-physical systems here. Technical surveys can be found in the literature, e.g., [2, 7, 8, 12, 20, 32, 41, 42]. This material is based upon work supported by the National Science Foundation under NSF CAREER Award CNS-1054246.

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 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Notes

  1. 1.

    http://www.keymaeraX.org/.

  2. 2.

    The students’ self-defined 3-week course projects and their presentations to a panel of experts from industry in the CPS V&V Grand Prix are available from the course web pages http://lfcps.org/course/fcps.html.

References

  1. Alur, R.: Formal verification of hybrid systems. In: Chakraborty, S., Jerraya, A., Baruah, S.K., Fischmeister, S. (eds.) EMSOFT, pp. 273–278. ACM (2011)

    Google Scholar 

  2. Alur, R.: Principles of Cyber-Physical Systems. MIT Press, Cambridge (2015)

    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. Theoret. Comput. Sci. 138(1), 3–34 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alur, R., Henzinger, T., Lafferriere, G., Pappas, G.J.: Discrete abstractions of hybrid systems. Proc. IEEE 88(7), 971–984 (2000)

    Article  Google Scholar 

  5. Branicky, M.S.: General hybrid dynamical systems: modeling, analysis, and control. In: Alur, R., Sontag, E.D., Henzinger, T.A. (eds.) HS 1995. LNCS, vol. 1066, pp. 186–200. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  6. Clarke, E.M., Emerson, E.A., Sifakis, J.: Model checking: algorithmic verification and debugging. Commun. ACM 52(11), 74–84 (2009)

    Article  Google Scholar 

  7. Davoren, J.M., Nerode, A.: Logics for hybrid systems. IEEE 88(7), 985–1010 (2000)

    Article  Google Scholar 

  8. Doyen, L., Frehse, G., Pappas, G.J., Platzer, A.: Verification of hybrid systems. In: Clarke, E.M., Henzinger, T.A., Veith, H. (eds.) Handbook of Model Checking, Chap. 28. Springer, Heidelberg (2017)

    Google Scholar 

  9. Frege, G.: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Verlag von Louis Nebert, Halle (1879)

    Google Scholar 

  10. Gentzen, G.: Untersuchungen über das logische Schließen. I. Math. Zeit. 39(2), 176–210 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  11. Henzinger, T.A., Sifakis, J.: The discipline of embedded systems design. Computer 40(10), 32–40 (2007)

    Article  Google Scholar 

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

    Google Scholar 

  13. Hilbert, D.: Die Grundlagen der Mathematik. Abhandlungen aus dem Seminar der Hamburgischen Universität 6(1), 65–85 (1928)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  15. Jeannin, J.-B., Ghorbal, K., Kouskoulas, Y., Gardner, R., Schmidt, A., Zawadzki, E., Platzer, A.: A formally verified hybrid system for the next-generation airborne collision avoidance system. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 21–36. Springer, Heidelberg (2015)

    Google Scholar 

  16. Kapteyn, J.C.: First attempt at a theory of the arrangement and motion of the sidereal system. Astrophys. J. 55, 302 (1922)

    Article  Google Scholar 

  17. Larsen, K.G.: Verification and performance analysis for embedded systems. In: Chin, W., Qin, S. (eds.) TASE 2009, Third IEEE International Symposium on Theoretical Aspects of Software Engineering, 29–31 July 2009, pp. 3–4. IEEE Computer Society, Tianjin, China (2009)

    Google Scholar 

  18. Lee, E.A., Seshia, S.A.: Introduction to Embedded Systems - A Cyber-Physical Systems Approach. Lulu Press, Raleigh (2013). Lulu.com

    Google Scholar 

  19. Lie, S.: Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen. Teubner, Leipzig (1893)

    Book  MATH  Google Scholar 

  20. Lunze, J., Lamnabhi-Lagarrigue, F. (eds.): Handbook of Hybrid Systems Control: Theory, Tools, Applications. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  21. Maler, O.: Control from computer science. Ann. Rev. Control 26(2), 175–187 (2002)

    Article  MathSciNet  Google Scholar 

  22. Mitsch, S., Platzer, A.: ModelPlex: verified runtime validation of verified cyber-physical system models. In: Bonakdarpour, B., Smolka, S.A. (eds.) RV 2014. LNCS, vol. 8734, pp. 199–214. Springer, Heidelberg (2014)

    Google Scholar 

  23. Nerode, A.: Logic and control. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) CiE 2007. LNCS, vol. 4497, pp. 585–597. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  24. Nerode, A., Kohn, W.: Models for hybrid systems: automata, topologies, controllability, observability. In: Grossman, R.L., Ravn, A.P., Rischel, H., Nerode, A. (eds.) HS 1991 and HS 1992. LNCS, vol. 736, pp. 317–356. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  25. NITRD CPS Senior Steering Group: CPS vision statement. NITRD (2012)

    Google Scholar 

  26. Pappas, G.J.: Wireless control networks: modeling, synthesis, robustness, security. In: Caccamo, M., Frazzoli, E., Grosu, R. (eds.) Proceedings of the 14th ACM International Conference on Hybrid Systems: Computation and Control, HSCC 2011, April 12–14, 2011, pp. 1–2. ACM, Chicago (2011)

    Google Scholar 

  27. Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reas. 41(2), 143–189 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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)

    Chapter  Google Scholar 

  30. Platzer, A.: Quantified differential dynamic logic for distributed hybrid systems. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 469–483. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  31. Platzer, A.: The complete proof theory of hybrid systems. In: LICS, pp. 541–550. IEEE (2012)

    Google Scholar 

  32. Platzer, A.: Logics of dynamical systems. In: LICS, pp. 13–24. IEEE (2012)

    Google Scholar 

  33. Platzer, A.: The structure of differential invariants and differential cut elimination. Log. Meth. Comput. Sci. 8(4), 1–38 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Platzer, A.: Differential game logic. ACM Trans. Comput. Log. 17(1), 1: 1–1: 51 (2015)

    Article  MathSciNet  Google Scholar 

  35. Platzer, A.: A uniform substitution calculus for differential dynamic logic. In: Felty, A., Middeldorp, A. (eds.) CADE. LNCS, vol. 9195, pp. 467–481. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  36. Platzer, A., Clarke, E.M.: Formal verification of curved flight collision avoidance maneuvers: a case study. In: Cavalcanti, A., Dams, D.R. (eds.) FM 2009. LNCS, vol. 5850, pp. 547–562. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

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

    Google Scholar 

  38. President’s Council of Advisors on Science and Technology: Leadership under challenge: Information technology R&D in a competitive world. An Assessment of the Federal Networking and Information Technology R&D, Program, August 2007

    Google Scholar 

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

    Google Scholar 

  40. Smullyan, R.M.: First-Order Logic. Dover, Mineola (1968)

    Book  MATH  Google Scholar 

  41. Tabuada, P.: Verification and Control of Hybrid Systems: A Symbolic Approach. Springer, New York (2009)

    Book  MATH  Google Scholar 

  42. Tiwari, A.: Abstractions for hybrid systems. Form. Meth. Syst. Des. 32(1), 57–83 (2008)

    Article  MATH  Google Scholar 

  43. Tiwari, A.: Logic in software, dynamical and biological systems. In: LICS, pp. 9–10. IEEE Computer Society (2011)

    Google Scholar 

  44. Wing, J.M.: Five deep questions in computing. Commun. ACM 51(1), 58–60 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Carnegie Mellon University, Pittsburgh, USA

    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 .

Editor information

Editors and Affiliations

  1. Aix-Marseille University, Marseille, France

    Nicola Olivetti

  2. SRI International, Menlo Park, California, USA

    Ashish Tiwari

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Platzer, A. (2016). Logic & Proofs for Cyber-Physical Systems. In: Olivetti, N., Tiwari, A. (eds) Automated Reasoning. IJCAR 2016. Lecture Notes in Computer Science(), vol 9706. Springer, Cham. https://doi.org/10.1007/978-3-319-40229-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-40229-1_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40228-4

  • Online ISBN: 978-3-319-40229-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation