Advertisement

Sharing HOL4 and HOL Light Proof Knowledge

Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9450)

Abstract

New proof assistant developments often involve concepts similar to already formalized ones. When proving their properties, a human can often take inspiration from the existing formalized proofs available in other provers or libraries. In this paper we propose and evaluate a number of methods, which strengthen proof automation by learning from proof libraries of different provers. Certain conjectures can be proved directly from the dependencies induced by similar proofs in the other library. Even if exact correspondences are not found, learning-reasoning systems can make use of the association between proved theorems and their characteristics to predict the relevant premises. Such external help can be further combined with internal advice. We evaluate the proposed knowledge-sharing methods by reproving the HOL Light and HOL4 standard libraries. The learning-reasoning system HOL(y)Hammer, whose single best strategy could automatically find proofs for 30 % of the HOL Light problems, can prove 40 % with the knowledge from HOL4.

Keywords

Proof Assistant Standard Library Type Constructor Concept Match Interactive Theorem Prover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

This work has been supported by the Austrian Science Fund (FWF): P26201.

References

  1. 1.
    Adams, M.: The common HOL platform. In: Kaliszyk, C., Paskevich, A., (eds.) Fourth International Workshop on Proof Exchange for Theorem Proving, PxTP 2015, Berlin, Germany, 2–3 August 2015. to appear in EPTCS (2015)Google Scholar
  2. 2.
    Asperti, A., Ricciotti, W., Coen, C.S.: Matita tutorial. J. Formaliz. Reason. 7(2), 91–199 (2014)MathSciNetGoogle Scholar
  3. 3.
    Autexier, S., Hutter, D.: Structure formation in large theories. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS, vol. 9150, pp. 155–170. Springer, Heidelberg (2015) CrossRefGoogle Scholar
  4. 4.
    Blanchette, J.C., Haslbeck, M., Matichuk, D., Nipkow, T.: Mining the archive of formal proofs. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS, vol. 9150, pp. 3–17. Springer, Heidelberg (2015) CrossRefGoogle Scholar
  5. 5.
    Bortin, M., Johnsen, E.B., Lüth, C.: Structured formal development in Isabelle. Nordic J. Comput. 13, 1–20 (2006)MathSciNetGoogle Scholar
  6. 6.
    Gauthier, T., Kaliszyk, C.: Matching concepts across HOL libraries. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) CICM 2014. LNCS, vol. 8543, pp. 267–281. Springer, Heidelberg (2014) Google Scholar
  7. 7.
    Gauthier, T., Kaliszyk, C.: Premise selection and external provers for HOL4. In: Leroy, X., Tiu, A., (eds.) Proceedings of the 4th ACM-SIGPLAN Conference on Certified Programs and Proofs, pp. 49–57 (2015)Google Scholar
  8. 8.
    Harrison, J.: Optimizing proof search in model elimination. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104. Springer, Heidelberg (1996) Google Scholar
  9. 9.
    Harrison, J.: HOL Light: an overview. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 60–66. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  10. 10.
    Huet, G., Herbelin, H.: 30 years of research and development around Coq. In: Jagannathan, S., Sewell, P., (eds.) The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2014, San Diego, CA, USA, 20–21 January 2014, pp. 249–250. ACM (2014)Google Scholar
  11. 11.
    Hurd, J.: First-order proof tactics in higher-order logic theorem provers. In: Archer, M., Di Vito, B., Muñoz, C., (eds.) Design and Application of Strategies/Tactics in Higher Order Logics (STRATA 2003), number NASA/CP-2003-212448 in NASA Technical reports, pp. 56–68, September 2003Google Scholar
  12. 12.
    Hurd, J.: The OpenTheory standard theory library. In: Bobaru, M., Havelund, K., Holzmann, G.J., Joshi, R. (eds.) NFM 2011. LNCS, vol. 6617, pp. 177–191. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  13. 13.
    Kaliszyk, C., Krauss, A.: Scalable LCF-Style proof translation. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 51–66. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  14. 14.
    Kaliszyk, C., Rabe, F.: Towards knowledge management for HOL Light. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) CICM 2014. LNCS, vol. 8543, pp. 357–372. Springer, Heidelberg (2014) Google Scholar
  15. 15.
    Kaliszyk, C., Urban, J.: Learning-assisted automated reasoning with Flyspeck. J. Autom. Reason. 53(2), 173–213 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kaliszyk, C., Urban, J.: HOL(y)Hammer: online ATP service for HOL Light. Math. Comput. Sci. 9(1), 5–22 (2015)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kaliszyk, C., Urban, J., Vyskočil, J.: Efficient semantic features for automated reasoning over large theories. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence, IJCAI 2015 (2015). (to appear)Google Scholar
  18. 18.
    Keller, C., Werner, B.: Importing HOL Light into Coq. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 307–322. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  19. 19.
    The Mizar Mathematical Library. http://mizar.org/
  20. 20.
    Obua, S., Skalberg, S.: Importing HOL into isabelle/HOL. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 298–302. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  21. 21.
    Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, a practical link between automated and interactive theorem provers. In: 8th IWIL (2010). Invited talkGoogle Scholar
  22. 22.
    Rabe, F.: The MMT API: a generic MKM system. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS, vol. 7961, pp. 339–343. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  23. 23.
    Slind, K., Norrish, M.: A brief overview of HOL4. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 28–32. Springer, Heidelberg (2008) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.University of InnsbruckInnsbruckAustria

Personalised recommendations