Skip to main content
SpringerLink
Log in

A Semi-automatic Proof of Strong Connectivity

  • Conference paper
  • First Online:
Verified Software. Theories, Tools, and Experiments (VSTTE 2017)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 10712))

Abstract

We present a formal proof of the classical Tarjan-1972 algorithm for finding strongly connected components in directed graphs. We use the Why3 system to express these proofs and fully check them by computer. The Why3-logic is a simple multi-sorted first-order logic augmented by inductive predicates. Furthermore it provides useful libraries for lists and sets. The Why3 system allows the description of programs in a Why3-ML programming language (a first-order programming language with ML syntax) and provides interfaces to various state-of-the-art automatic provers and to manual interactive proof-checkers (we use mainly Coq). We do not claim that this proof is new, although we could not find a formal proof of that algorithm in the literature. But one important point of our article is that our proof is here completely presented and human readable.

R. Chen—Partly supported by ANR-13-LAB3-0007, http://www.spark-2014.org/proofinuse and National Natural Science Foundation of China (Grant No. 61672504).

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Similar content being viewed by others

References

  1. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Boston (1974)

    MATH  Google Scholar 

  2. Appel, A.W.: Verified Functional Algorithms, August 2016. www.cs.princeton.edu/~appel/vfa/

  3. Bobot, F., Filliâtre, J.C., Marché, C., Melquiond, G., Paskevich, A.: The Why3 platform, version 0.86.1. LRI, CNRS and Univ. Paris-Sud and INRIA Saclay, version 0.86.1 edn., May 2015. why3.lri.fr/download/manual-0.86.1.pdf

  4. Bobot, F., Filliâtre, J.-C., Marché, C., Melquiond, G., Paskevich, A.: Preserving user proofs across specification changes. In: Cohen, E., Rybalchenko, A. (eds.) VSTTE 2013. LNCS, vol. 8164, pp. 191–201. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54108-7_10. hal.inria.fr/hal-00875395

    Chapter  Google Scholar 

  5. Bobot, F., Filliâtre, J.C., Marché, C., Paskevich, A.: Let’s verify this with Why3. Softw. Tools Technol. Transf. (STTT) 17(6), 709–727 (2015). hal.inria.fr/hal-00967132

  6. Charguéraud, A.: Program verification through characteristic formulae. In: Hudak, P., Weirich, S. (eds.) Proceeding of the 15th ACM SIGPLAN International Conference on Functional Programming (ICFP), pp. 321–332. ACM (2010). arthur.chargueraud.org/research/2010/cfml

  7. Charguéraud, A.: Higher-order representation predicates in separation logic. In: Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, pp. 3–14, CPP 2016. ACM, New York, January 2016

    Google Scholar 

  8. Charguéraud, A., Pottier, F.: Machine-checked verification of the correctness and amortized complexity of an efficient union-find implementation. In: Proceedings of the 6th International Conference on Interactive Theorem Proving (ITP), August 2015

    Google Scholar 

  9. Chen, R., Lévy, J.J.: Full script of Tarjan SCC Why3 proof. Technical report, Iscas and Inria (2017). jeanjacqueslevy.net/why3/graph/abs/scct/2/scc.html

  10. Chen, R., Lévy, J.J.: Une preuve formelle de l’algorithme de Tarjan-1972 pour trouver les composantes fortement connexes dans un graphe. In: JFLA (2017)

    Google Scholar 

  11. Clochard, M.: Preuves taillées en biseau. In: vingt-huitièmes Journées Francophones des Langages Applicatifs (JFLA). Gourette, France, January 2017. hal.inria.fr/hal-01404935

  12. Cohen, C., Théry, L.: Full script of Tarjan SCC Coq/ssreflect proof. Technical report, Inria (2017). github.com/CohenCyril/tarjan

  13. Coq Development Team: the coq 8.5 standard library. Technical report, Inria (2015). coq.inria.fr/distrib/current/stdlib

  14. Filliâtre, J.-C., Paskevich, A.: Why3 — where programs meet provers. In: Felleisen, M., Gardner, P. (eds.) ESOP 2013. LNCS, vol. 7792, pp. 125–128. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-37036-6_8

    Chapter  Google Scholar 

  15. Gonthier, G., Mahboubi, A., Tassi, E.: A small scale reflection extension for the Coq system. Rapport de recherche RR-6455, INRIA (2008). hal.inria.fr/inria-00258384

  16. Gonthier, G., et al.: Finite graphs in mathematical components (2012). ssr.msr-inria.inria.fr/~jenkins/current/Ssreflect.fingraph.html. The full library is available at www.msr-inria.fr/projects/mathematical-components-2

  17. Hobor, A., Villard, J.: The ramifications of sharing in data structures. In: Proceedings of the 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 523–536, POPL 2013. ACM, New York (2013). doi.acm.org/10.1145/2429069.2429131

  18. Lammich, P., Neumann, R.: A framework for verifying depth-first search algorithms. In: Proceedings of the 2015 Conference on Certified Programs and Proofs, pp. 137–146, CPP 2015. ACM, New York (2015). doi.acm.org/10.1145/2676724.2693165

  19. Lévy, J.J.: Essays for the Luca Cardelli Fest. In: Simple Proofs of Simple Programs in Why3. Microsoft Research Cambridge, MSR-TR-2014-104 (2014)

    Google Scholar 

  20. Mehta, F., Nipkow, T.: Proving pointer programs in higher-order logic. In: CADE (2003)

    Google Scholar 

  21. Poskitt, C.M., Plump, D.: Hoare logic for graph programs. In: VSTTE (2010)

    Google Scholar 

  22. Pottier, F.: Depth-first search and strong connectivity in Coq. In: Journées Francophones des Langages Applicatifs (JFLA 2015), January 2015

    Google Scholar 

  23. Raad, A., Hobor, A., Villard, J., Gardner, P.: Verifying concurrent graph algorithms. In: Igarashi, A. (ed.) APLAS 2016. LNCS, vol. 10017, pp. 314–334. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-47958-3_17

    Chapter  Google Scholar 

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

    Google Scholar 

  25. Sergey, I., Nanevski, A., Banerjee, A.: Mechanized verification of fine-grained concurrent programs. In: Proceedings of the 36th ACM SIGPLAN Conference on Programming Language Design and Implementation, pp. 77–87, PLDI 2015. ACM, New York (2015). doi.acm.org/10.1145/2737924.2737964

  26. Tarjan, R.: Depth first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  27. Théry, L.: Formally-proven Kosaraju’s algorithm (2015). Inria report, Hal-01095533

    Google Scholar 

  28. Wengener, I.: A simplified correctness proof for a well-known algorithm computing strongly connected components. Inf. Process. Lett. 83(1), 17–19 (2002)

    Article  MathSciNet  Google Scholar 

  29. Why3 Development Team: Why3 gallery of programs. Technical report, CNRS and Inria (2016). toccata.lri.fr/gallery

Download references

Acknowledgments

Thanks to the Why3 group at Inria-Saclay/LRI-Orsay for very valuable advices, to Cyril Cohen and Laurent Théry for their fantastic expertise in Coq proofs, to Claude Marché and the reviewers for many corrections.

Author information

Authors and Affiliations

  1. Inria Saclay, Saclay, France

    Ran Chen

  2. ISCAS Beijing, Beijing, China

    Ran Chen

  3. Inria Paris, Paris, France

    Jean-Jacques Lévy

Authors
  1. Ran Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Jacques Lévy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Jacques Lévy .

Editor information

Editors and Affiliations

  1. Paris-Sud University, Orsay, France

    Andrei Paskevich

  2. New York University, New York, New York, USA

    Thomas Wies

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chen, R., Lévy, JJ. (2017). A Semi-automatic Proof of Strong Connectivity. In: Paskevich, A., Wies, T. (eds) Verified Software. Theories, Tools, and Experiments. VSTTE 2017. Lecture Notes in Computer Science(), vol 10712. Springer, Cham. https://doi.org/10.1007/978-3-319-72308-2_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-72308-2_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-72307-5

  • Online ISBN: 978-3-319-72308-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation