Skip to main content

Lemma Mining over HOL Light

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8312)

Abstract

Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and re-used in later proofs by formal mathematicians. In this work, we suggest and implement criteria defining the estimated usefulness of the HOL Light lemmas for proving further theorems. We use these criteria to mine the large inference graph of all lemmas in the core HOL Light library, adding thousands of the best lemmas to the pool of named statements that can be re-used in later proofs. The usefulness of the new lemmas is then evaluated by comparing the performance of automated proving of the core HOL Light theorems with and without such added lemmas.

Keywords

  • Automate Reasoning
  • Eigenvector Centrality
  • Large Theory
  • Automate Prove
  • Inference Graph

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, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-45221-5_34
  • Chapter length: 15 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-45221-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alama, J., Heskes, T., Kühlwein, D., Tsivtsivadze, E., Urban, J.: Premise selection for mathematics by corpus analysis and kernel methods. Journal of Automated Reasoning (2013), http://dx.doi.org/10.1007/s10817-013-9286-5

  2. Alama, J., Kühlwein, D., Urban, J.: Automated and Human Proofs in General Mathematics: An Initial Comparison. In: Bjørner, N., Voronkov, A. (eds.) LPAR-18 2012. LNCS, vol. 7180, pp. 37–45. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  3. Avrachenkov, K., Dobrynin, V., Nemirovsky, D., Pham, S.K., Smirnova, E.: Pagerank based clustering of hypertext document collections. In: Myaeng, S.-H., Oard, D.W., Sebastiani, F., Chua, T.-S., Leong, M.-K. (eds.) SIGIR, pp. 873–874. ACM (2008)

    Google Scholar 

  4. Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.): ITP 2013. LNCS, vol. 7998. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  5. Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. Journal of Formalized Reasoning 3(2), 153–245 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Hales, T.C.: Introduction to the Flyspeck project. In: Coquand, T., Lombardi, H., Roy, M.-F. (eds.) Dagstuhl Seminar Proceedings, vol. 05021. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl (2005)

    Google Scholar 

  7. Harrison, J.: HOL Light: A tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)

    CrossRef  Google Scholar 

  8. Hoder, K., Voronkov, A.: Sine qua non for large theory reasoning. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 299–314. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  9. Kaliszyk, C., Krauss, A.: Scalable LCF-style proof translation. In: Blazy, et al. (eds.) [4], pp. 51–66

    Google Scholar 

  10. Kaliszyk, C., Urban, J.: Learning-assisted automated reasoning with Flyspeck. CoRR, abs/1211.7012 (2012)

    Google Scholar 

  11. Kaliszyk, C., Urban, J.: Automated reasoning service for HOL Light. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS (LNAI), vol. 7961, pp. 120–135. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  12. Kaliszyk, C., Urban, J.: Stronger automation for Flyspeck by feature weighting and strategy evolution. In: Blanchette, J.C., Urban, J. (eds.) PxTP 2013. EPiC Series, vol. 14, pp. 87–95. EasyChair (2013)

    Google Scholar 

  13. Kühlwein, D., Blanchette, J.C., Kaliszyk, C., Urban, J.: MaSh: Machine learning for Sledgehammer. In: Blazy, et al. (eds.) [4], pp. 35–50

    Google Scholar 

  14. Kühlwein, D., Schulz, S., Urban, J.: E-MaLeS 1.1. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 407–413. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  15. Kühlwein, D., van Laarhoven, T., Tsivtsivadze, E., Urban, J., Heskes, T.: Overview and evaluation of premise selection techniques for large theory mathematics. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 378–392. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  16. McCune, W.: Prover9 and Mace4. 2005–2010, http://www.cs.unm.edu/~mccune/prover9/

  17. Meng, J., Paulson, L.C.: Translating higher-order clauses to first-order clauses. J. Autom. Reasoning 40(1), 35–60 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  18. Page, L., Brin, S., Motwani, R., Winograd, T.: The PageRank citation ranking: Bringing order to the Web. Technical report, Stanford Digital Library Technologies Project (1998)

    Google Scholar 

  19. Pudlák, P.: Search for faster and shorter proofs using machine generated lemmas. In: Sutcliffe, G., Schmidt, R., Schulz, S. (eds.) 3rd International Joint Conference on Automated Reasoning, Proceedings of the FLoC 2006 Workshop on Empirically Successful Computerized Reasoning. CEUR Workshop Proceedings, vol. 192, pp. 34–52 (2006)

    Google Scholar 

  20. Puzis, Y., Gao, Y., Sutcliffe, G.: Automated generation of interesting theorems. In: Sutcliffe, G., Goebel, R. (eds.) FLAIRS Conference, pp. 49–54. AAAI Press (2006)

    Google Scholar 

  21. Riazanov, A., Voronkov, A.: The design and implementation of VAMPIRE. AI Commun. 15(2-3), 91–110 (2002)

    MATH  Google Scholar 

  22. Schulz, S.: Learning search control knowledge for equational deduction. DISKI, vol. 230. Infix Akademische Verlagsgesellschaft (2000)

    Google Scholar 

  23. Schulz, S.: E - A Brainiac Theorem Prover. AI Commun. 15(2-3), 111–126 (2002)

    MATH  Google Scholar 

  24. Sutcliffe, G.: The Design and Implementation of a Compositional Competition-Cooperation Parallel ATP System. In: de Nivelle, H., Schulz, S. (eds.) Proceedings of the 2nd International Workshop on the Implementation of Logics, pp. 92–102. MPI-I-2001-2-006 in Max-Planck-Institut für Informatik. Research Report (2001)

    Google Scholar 

  25. Sutcliffe, G., Puzis, Y.: SRASS - a semantic relevance axiom selection system. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 295–310. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  26. Urban, J.: MPTP - Motivation, Implementation, First Experiments. Journal of Automated Reasoning 33(3-4), 319–339 (2004)

    MathSciNet  CrossRef  MATH  Google Scholar 

  27. Urban, J.: MoMM - fast interreduction and retrieval in large libraries of formalized mathematics. Int. J. on Artificial Intelligence Tools 15(1), 109–130 (2006)

    CrossRef  Google Scholar 

  28. Urban, J.: BliStr: The Blind Strategymaker. CoRR, abs/1301.2683 (2013)

    Google Scholar 

  29. Urban, J., Sutcliffe, G., Pudlák, P., Vyskočil, J.: MaLARea SG1 - Machine Learner for Automated Reasoning with Semantic Guidance. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 441–456. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  30. Urban, J., Vyskočil, J.: Theorem proving in large formal mathematics as an emerging AI field. In: Bonacina, M.P., Stickel, M.E. (eds.) McCune Festschrift. LNCS (LNAI), vol. 7788, pp. 240–257. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  31. Urban, J., Vyskočil, J., Štěpánek, P.: MaLeCoP: Machine learning connection prover. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS (LNAI), vol. 6793, pp. 263–277. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  32. Veroff, R.: Using hints to increase the effectiveness of an automated reasoning program: Case studies. J. Autom. Reasoning 16(3), 223–239 (1996)

    MathSciNet  CrossRef  MATH  Google Scholar 

  33. Wenzel, M., Paulson, L.C., Nipkow, T.: The Isabelle framework. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 33–38. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  34. Wos, L., Overbeek, R., Lusk, E.L., Boyle, J.: Automated Reasoning: Introduction and Applications. Prentice-Hall (1984)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kaliszyk, C., Urban, J. (2013). Lemma Mining over HOL Light . In: McMillan, K., Middeldorp, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2013. Lecture Notes in Computer Science, vol 8312. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45221-5_34

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-45221-5_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45220-8

  • Online ISBN: 978-3-642-45221-5

  • eBook Packages: Computer ScienceComputer Science (R0)