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Hybrid Systems Verification with Isabelle/HOL: Simpler Syntax, Better Models, Faster Proofs

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Formal Methods (FM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 13047))

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Abstract

We extend a semantic verification framework for hybrid systems with the Isabelle/HOL proof assistant by an algebraic model for hybrid program stores, a shallow expression model for hybrid programs and their correctness specifications, and domain-specific deductive and calculational support. The new store model yields clean separations and dynamic local views of variables, e.g. discrete/continuous, mutable/immutable, program/logical, and enhanced ways of manipulating them using combinators, projections and framing. This leads to more local inference rules, procedures and tactics for reasoning with invariant sets, certifying solutions of hybrid specifications or calculating derivatives with increased proof automation and scalability. The new expression model provides more user-friendly syntax, better control of name spaces and interfaces connecting the framework with real-world modelling languages.

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Notes

  1. 1.

    github.com/isabelle-utp/Hybrid-Verification, also by clicking our icons.

References

  1. Anand, A., Knepper, R.: ROSCoq: robots powered by constructive reals. In: Urban, C., Zhang, X. (eds.) ITP 2015. LNCS, vol. 9236, pp. 34–50. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-22102-1_3

    Chapter  Google Scholar 

  2. Ricketts, D., Malecha, G., Alvarez, M.M., Gowda, V., Lerner, S.: Towards verification of hybrid systems in a foundational proof assistant. In: MEMOCODE, pp. 248–257. IEEE (2015)

    Google Scholar 

  3. Wang, S., Zhan, N., Zou, L.: An improved HHL prover: an interactive theorem prover for hybrid systems. In: Butler, M., Conchon, S., Zaïdi, F. (eds.) ICFEM 2015. LNCS, vol. 9407, pp. 382–399. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25423-4_25

    Chapter  Google Scholar 

  4. Huerta y Munive, J.J., Struth, G.: Verifying hybrid systems with modal Kleene algebra. In: Desharnais, J., Guttmann, W., Joosten, S. (eds.) RAMiCS 2018. LNCS, vol. 11194, pp. 225–243. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-02149-8_14

    Chapter  Google Scholar 

  5. Huerta y Munive, J.J., Struth, G.: Predicate transformer semantics for hybrid systems: verification components for Isabelle/HOL. arXiv:1909.05618 [cs.LO] (2019)

  6. Huerta y Munive, J.J.: Verification components for hybrid systems. Archive of Formal Proofs (2019)

    Google Scholar 

  7. Foster, S.: Hybrid relations in Isabelle/UTP. In: Ribeiro, P., Sampaio, A. (eds.) UTP 2019. LNCS, vol. 11885, pp. 130–153. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-31038-7_7

    Chapter  Google Scholar 

  8. Foster, S., Huerta y Munive, J.J., Struth, G.: Differential Hoare logics and refinement calculi for hybrid systems with Isabelle/HOL. In: Fahrenberg, U., Jipsen, P., Winter, M. (eds.) RAMiCS 2020. LNCS, vol. 12062, pp. 169–186. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-43520-2_11

    Chapter  Google Scholar 

  9. Immler, F., Traut, C.: The flow of ODEs: Formalization of variational equation and Poincaré map. J. Autom. Reasoning 62(2), 215–236 (2019)

    Google Scholar 

  10. Foster, S., Baxter, J., Cavalcanti, A., Woodcock, J., Zeyda, F.: Unifying semantic foundations for automated verification tools in Isabelle/UTP. Science of Computer Programming, vol. 197, October 2020

    Google Scholar 

  11. Platzer, A.: Logical Foundations of Cyber-Physical Systems. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-63588-0

  12. Boulton, R.J., Gordon, A.D., Gordon, M.J.C., Harrison, J., Herbert, J., Tassel, J.V.: Experience with embedding hardware description languages in HOL. In: IFIP Transactions, vol. A-10, pp. 129–156, North-Holland (1992)

    Google Scholar 

  13. Wildmoser, M., Nipkow, T.: Certifying machine code safety: shallow versus deep embedding. In: Slind, K., Bunker, A., Gopalakrishnan, G. (eds.) TPHOLs 2004. LNCS, vol. 3223, pp. 305–320. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30142-4_22

    Chapter  Google Scholar 

  14. Huerta y Munive, J.J.: Affine systems of ODEs in Isabelle/HOL for hybrid-program verification. In: de Boer, F., Cerone, A. (eds.) SEFM 2020. LNCS, vol. 12310, pp. 77–92. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58768-0_5

    Chapter  Google Scholar 

  15. Huerta y Munive, J.J.: Matrices for ODEs. Archive of Formal Proofs (2020)

    Google Scholar 

  16. Hickman, T., Laursen, C.P., Foster, S., Huerta y Munive, J.J.: Certifying differential equation solutions from computer algebra systems in Isabelle/HOL. arXiv:2102.02679 [cs.LO], February 2021

  17. Teschl, G.: Ordinary Differential Equations and Dynamical Systems. AMS (2012)

    Google Scholar 

  18. Mitsch, S., Huerta y Munive, J.J., Jin, X., Zhan, B., Wang, S., Zhan, N.: ARCH-COMP20 category report: Hybrid systems theorem proving. ARCH 20, 141–161 (2019)

    Google Scholar 

  19. Foster, S., Zeyda, F.: Optics. Archive of Formal Proofs, May 2017

    Google Scholar 

  20. Oles, F.: A Category-theoretic approach to the semantics of programming languages. Ph.D. thesis, Syracuse University (1982)

    Google Scholar 

  21. Back, R., von Wright, J.: Refinement Calculus–A Systematic Introduction. Springer, New York (1998). https://doi.org/10.1007/978-1-4612-1674-2

  22. Foster, J.: Bidirectional programming languages. Ph.D. thesis, University of Pennsylvania (2009)

    Google Scholar 

  23. Foster, S., Baxter, J.: Automated algebraic reasoning for collections and local variables with lenses. In: Fahrenberg, U., Jipsen, P., Winter, M. (eds.) RAMiCS 2020. LNCS, vol. 12062, pp. 100–116. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-43520-2_7

    Chapter  Google Scholar 

  24. Foster, S., Gleirscher, M., Calinescu, R.: Towards deductive verification of control algorithms for autonomous marine vehicles. In: ICECCS. IEEE, October 2020

    Google Scholar 

  25. Immler, F., Hölzl, J.: Numerical analysis of ordinary differential equations in Isabelle/HOL. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 377–392. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32347-8_26

    Chapter  Google Scholar 

  26. Hölzl, J., Immler, F., Huffman, B.: Type classes and filters for mathematical analysis in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 279–294. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39634-2_21

    Chapter  Google Scholar 

  27. Foster, S., Nemouchi, Y., Gleirscher, M., Wei, R., Kelly, T.: Integration of formal proof into unified assurance cases with Isabelle/SACM. Formal Aspects of Computing (2021)

    Google Scholar 

  28. Kuncar, O., Popescu, A.: A consistent foundation for Isabelle/HOL. J. Autom. Reasoning 62, 531–555 (2019)

    Google Scholar 

  29. Cheney, E.W.: Analysis for Applied Mathematics. Springer, New York (2001). https://doi.org/10.1007/978-1-4757-3559-8

  30. Platzer, A.: The structure of differential invariants and differential cut elimination. Log. Meth. Comput. Sci. 8(4), 1–38 (2012). https://doi.org/10.2168/LMCS-8(4:16)2012

  31. Platzer, A., Tan, Y.K.: Differential equation axiomatization: the impressive power of differential ghosts. In: LICS, pp. 819–828. ACM (2018)

    Google Scholar 

  32. Matichuk, D., Murray, T.C., Wenzel, M.: Eisbach: a proof method language for Isabelle. J. Autom. Reasoning 56(3), 261–282 (2016)

    Google Scholar 

  33. Hölzl, J.: Proving inequalities over reals with computation in Isabelle/HOL. In: PLMMS, pp. 38–45. ACM (2009)

    Google Scholar 

  34. Blanchette, J.C., Kaliszyk, C., Paulson, L.C., Urban, J.: Hammering towards QED. J. Formalized Reasoning 9(1), 101–148 (2016)

    Google Scholar 

  35. Li, W., Passmore, G., Paulson, L.: Deciding univariate polynomial problems using untrusted certificates in Isabelle/HOL. J. Autom. Reasoning 62, 29–91 (2019)

    Google Scholar 

  36. Cordwell, K., Yong, K.T., Platzer, A.: A verified decision procedure for univariate real arithmetic with the BKR algorithm. In Cohen, L., Kaliszyk, C. (eds.) ITP. Volume 193 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 14:1–14:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

    Google Scholar 

  37. Fulton, N., Mitsch, S., Quesel, J.-D., Völp, M., Platzer, A.: KeYmaera X: an axiomatic tactical theorem prover for hybrid systems. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS (LNAI), vol. 9195, pp. 527–538. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21401-6_36

    Chapter  Google Scholar 

  38. Slagel, J.T., White, L., Dutle, A.: Formal verification of semi-algebraic sets and real analytic functions. In: CPP, pp. 278–290. ACM (2021)

    Google Scholar 

  39. Bohrer, B., Rahli, V., Vukotic, I., Völp, M., Platzer, A.: Formally verified differential dynamic logic. In: CPP, pp. 208–221. ACM (2017)

    Google Scholar 

  40. Platzer, A.: Differential game logic. Archive of Formal Proofs (2019)

    Google Scholar 

  41. 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). https://doi.org/10.1007/978-3-642-17164-2_1

    Chapter  Google Scholar 

  42. Preoteasa, V., Dragomir, I., Tripakis, S.: Refinement calculus of reactive systems. In: International Conference on Embedded Systems (EMSOFT). IEEE, October 2014

    Google Scholar 

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Acknowledgements

This work is funded by UKRI-EPSRC project CyPhyAssure (grant reference EP/S001190/1), Novo Nordisk Fonden Start Package Grant (NNF20OC0063462), the Assuring Autonomy International Programme (AAIP; grant CSI:Cobot), a partnership between Lloyd’s Register Foundation and the University of York, and Labex DigiCosme through an invited professorship of the fourth author at the Laboratoire d’informatique de l’École polytechnique.

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Authors and Affiliations

  1. University of York, York, UK

    Simon Foster

  2. University of Copenhagen, Copenhagen, Denmark

    Jonathan Julián Huerta y Munive

  3. University of Bremen, Bremen, Germany

    Mario Gleirscher

  4. University of Sheffield, Sheffield, UK

    Georg Struth

Authors
  1. Simon Foster
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  2. Jonathan Julián Huerta y Munive
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  3. Mario Gleirscher
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  4. Georg Struth
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Corresponding author

Correspondence to Simon Foster .

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Editors and Affiliations

  1. University of Twente, Enschede, The Netherlands

    Marieke Huisman

  2. NASA Ames Research Center, Moffett Field, CA, USA

    Corina Păsăreanu

  3. Institute of Software, Chinese Academy of Sciences, Beijing, China

    Naijun Zhan

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Foster, S., Huerta y Munive, J.J., Gleirscher, M., Struth, G. (2021). Hybrid Systems Verification with Isabelle/HOL: Simpler Syntax, Better Models, Faster Proofs. In: Huisman, M., Păsăreanu, C., Zhan, N. (eds) Formal Methods. FM 2021. Lecture Notes in Computer Science(), vol 13047. Springer, Cham. https://doi.org/10.1007/978-3-030-90870-6_20

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  • DOI: https://doi.org/10.1007/978-3-030-90870-6_20

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