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Separating Separation Logic – Modular Verification of Red-Black Trees

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Verified Software. Theories, Tools and Experiments. (VSTTE 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13800))

Abstract

Interactive theorem provers typically use abstract algebraic data structures to focus on algorithmic correctness. Verification of programs in real programming languages also has to deal with pointer structures, aliasing and, in the case of C, memory management. While progress has been made by using Separation Logic, direct verification of code still has to deal with both aspects at once. In this paper, we show a refinement-based approach that separates the two issues by using a suitable modular structure.

We exemplify the approach with a correctness proof for red-black trees, demonstrating that our approach can generate efficient C code that uses parent pointers and avoids recursion. The proof is split into a large part almost identical to high-level algebraic proofs and a separate small part that uses Separation Logic to verify primitive operations on pointer structures.

Partly supported by the Deutsche Forschungsgemeinschaft (DFG), “Verifikation von Flash-Dateisystemen” (grants RE828/13-1 and RE828/13-2).

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References

  1. Abrial, J.R., Hoare, A., Chapron, P.: The B-Book: Assigning Programs to Meanings. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  2. Affeldt, R., Garrigue, J., Qi, X., Tanaka, K.: Proving tree algorithms for succinct data structures. In: 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), vol. 141, pp. 5:1–5:19 (2019)

    Google Scholar 

  3. Appel, A.: Efficient Verified Red-Black Trees (2011)

    Google Scholar 

  4. Armborst, L., Huisman, M.: Permission-based verification of red-black trees and their merging. In: Proceedings of FormaliSE, vol. 21, pp. 111–123 (2021)

    Google Scholar 

  5. Bodenmüller, S., Schellhorn, G., Bitterlich, M., Reif, W.: Flashix: modular verification of a concurrent and crash-safe flash file system. In: Raschke, A., Riccobene, E., Schewe, K.-D. (eds.) Logic, Computation and Rigorous Methods. LNCS, vol. 12750, pp. 239–265. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-76020-5_14

    Chapter  MATH  Google Scholar 

  6. Börger, E.: The ASM refinement method. Formal Aspects Comput. 15(1–2), 237–257 (2003)

    Article  MATH  Google Scholar 

  7. Charguéraud, A.: Program verification through characteristic formulae. In: Proceedings of ACM SIGPLAN International Conference on Functional Programming (ICFP), pp. 321–332. Association for Computing Machinery (2010)

    Google Scholar 

  8. Charguéraud, A.: Higher-order representation predicates in separation logic. In: Proceedings of ACM SIGPLAN Conference on Certified Programs and Proofs (CPP), pp. 3–14. Association for Computing Machinery (2016)

    Google Scholar 

  9. Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)

    MATH  Google Scholar 

  10. Derrick, J., Boiten, E.: Refinement in Z and in Object-Z: Foundations and Advanced Applications. FACIT. Springer, Cham (2001). Second, Revised Edition (2014)

    Google Scholar 

  11. Dross, C., Moy, Y.: Auto-active proof of red-black trees in SPARK. In: Barrett, C., Davies, M., Kahsai, T. (eds.) NFM 2017. LNCS, vol. 10227, pp. 68–83. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57288-8_5

    Chapter  Google Scholar 

  12. Elgaard, J., Møller, A., Schwartzbach, M.I.: Compile-time debugging of C programs working on trees. In: Smolka, G. (ed.) ESOP 2000. LNCS, vol. 1782, pp. 119–134. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-46425-5_8

    Chapter  Google Scholar 

  13. Ernst, G., Pfähler, J., Schellhorn, G., Reif, W.: Modular, crash-safe refinement for ASMs with submachines. Sci. Comput. Program. 131, 3–21 (2016). Abstract State Machines, Alloy, B, TLA, VDM and Z (ABZ 2014)

    Google Scholar 

  14. Ernst, G., Schellhorn, G., Reif, W.: Verification of B\(^{+}\) trees: an experiment combining shape analysis and interactive theorem proving. In: Barthe, G., Pardo, A., Schneider, G. (eds.) SEFM 2011. LNCS, vol. 7041, pp. 188–203. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24690-6_14

    Chapter  Google Scholar 

  15. Faella, M., Parlato, G.: Reasoning about data trees using CHCs. In: Shoham, S., Vizel, Y. (eds.) CAV 2022. LNCS, vol. 13372, pp. 249–271. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-13188-2_13

    Chapter  Google Scholar 

  16. Filliâtre, J.-C., Letouzey, P.: Functors for proofs and programs. In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 370–384. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24725-8_26

    Chapter  MATH  Google Scholar 

  17. Guibas, L.J., Sedgewick, R.: A dichromatic framework for balanced trees. In: Proceedings of the 19th Symposium on Foundations of Computer Science (SFCS), pp. 8–21. IEEE (1978)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  19. Havasi, F.: An improved B+ tree for flash file systems. In: Černá, I., et al. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 297–307. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-18381-2_25

    Chapter  Google Scholar 

  20. Jacobs, B., Smans, J., Philippaerts, P., Vogels, F., Penninckx, W., Piessens, F.: VeriFast: a powerful, sound, predictable, fast verifier for C and Java. In: Bobaru, M., Havelund, K., Holzmann, G.J., Joshi, R. (eds.) NFM 2011. LNCS, vol. 6617, pp. 41–55. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20398-5_4

    Chapter  Google Scholar 

  21. Sanders, P., Mehlhorn, K.: Algorithms and Data Structures - The Basic Toolbox. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  22. Kahrs, S.: Red-black trees with types. J. Funct. Program. 11(4), 182–196 (2001)

    Article  MATH  Google Scholar 

  23. KIV Proofs for the Correctness of Red-Black Trees (2022). https://kiv.isse.de/projects/RBtree.html

  24. Lammich, P.: Refinement to imperative/HOL. In: Urban, C., Zhang, X. (eds.) ITP 2015. LNCS, vol. 9236, pp. 253–269. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-22102-1_17

    Chapter  Google Scholar 

  25. Lammich, P.: Efficient verified implementation of Introsort and Pdqsort. In: Peltier, N., Sofronie-Stokkermans, V. (eds.) IJCAR 2020. LNCS (LNAI), vol. 12167, pp. 307–323. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-51054-1_18

    Chapter  Google Scholar 

  26. Müller, P., Schwerhoff, M., Summers, A.J.: Viper: a verification infrastructure for permission-based reasoning. In: Jobstmann, B., Leino, K.R.M. (eds.) VMCAI 2016. LNCS, vol. 9583, pp. 41–62. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49122-5_2

    Chapter  MATH  Google Scholar 

  27. Nipkow, T.: Automatic functional correctness proofs for functional search trees. In: Blanchette, J.C., Merz, S. (eds.) ITP 2016. LNCS, vol. 9807, pp. 307–322. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43144-4_19

    Chapter  Google Scholar 

  28. O’Connor, L., et al.: Cogent: uniqueness types and certifying compilation. J. Funct. Program. 31, 25 (2021)

    Article  MATH  Google Scholar 

  29. Peña, R.: An assertional proof of red–black trees using Dafny. J. Autom. Reason. 64(4), 767–791 (2019). https://doi.org/10.1007/s10817-019-09534-y

    Article  MATH  Google Scholar 

  30. Polikarpova, N., Tschannen, J., Furia, C.A.: A fully verified container library. In: Bjørner, N., de Boer, F. (eds.) FM 2015. LNCS, vol. 9109, pp. 414–434. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19249-9_26

    Chapter  Google Scholar 

  31. Reynolds, A., Iosif, R., Serban, C., King, T.: A decision procedure for separation logic in SMT. In: Artho, C., Legay, A., Peled, D. (eds.) ATVA 2016. LNCS, vol. 9938, pp. 244–261. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-46520-3_16

    Chapter  MATH  Google Scholar 

  32. Reynolds, J.C.: Separation logic: a logic for shared mutable data structures. In: Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science, pp. 55–74. IEEE (2002)

    Google Scholar 

  33. Schellhorn, G., Bodenmüller, S., Pfähler, J., Reif, W.: Adding concurrency to a sequential refinement tower. In: Raschke, A., Méry, D., Houdek, F. (eds.) ABZ 2020. LNCS, vol. 12071, pp. 6–23. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48077-6_2

    Chapter  Google Scholar 

  34. Schellhorn, G., Bodenmüller, S., Bitterlich, M., Reif, W.: Software & system verification with KIV. In: Ahrendt, W., Beckert, B., Bubel, R., Johnsen, E.B. (eds.) The Logic of Software. A Tasting Menu of Formal Methods. LNCS, vol. 13360, pp. 408–436. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-08166-8_20

    Chapter  Google Scholar 

  35. Sedgewick, R., Wayne, K.: Algorithms, 4th edn. Addison-Wesley (2011)

    Google Scholar 

  36. Zhan, B.: Efficient verification of imperative programs using auto2. In: Beyer, D., Huisman, M. (eds.) TACAS 2018. LNCS, vol. 10805, pp. 23–40. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89960-2_2

    Chapter  Google Scholar 

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Acknowledgement

We would like to thank our students Nikolai Glaab and Felix Pribyl, who have done large parts of the verification of red-black trees.

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Authors and Affiliations

  1. Institute for Software and Systems Engineering, University of Augsburg, Augsburg, Germany

    Gerhard Schellhorn, Stefan Bodenmüller, Martin Bitterlich & Wolfgang Reif

Authors
  1. Gerhard Schellhorn
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  2. Stefan Bodenmüller
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  3. Martin Bitterlich
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  4. Wolfgang Reif
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Corresponding author

Correspondence to Stefan Bodenmüller .

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Editors and Affiliations

  1. Microsoft Research, Karnataka, India

    Akash Lal

  2. Fondazione Bruno Kessler, Trento, Italy

    Stefano Tonetta

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Schellhorn, G., Bodenmüller, S., Bitterlich, M., Reif, W. (2023). Separating Separation Logic – Modular Verification of Red-Black Trees. In: Lal, A., Tonetta, S. (eds) Verified Software. Theories, Tools and Experiments.. VSTTE 2022. Lecture Notes in Computer Science, vol 13800. Springer, Cham. https://doi.org/10.1007/978-3-031-25803-9_8

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  • DOI: https://doi.org/10.1007/978-3-031-25803-9_8

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