Skip to main content

Part of the book series: Lecture Notes in Computer Science ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

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)

    Chapter  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)

    Chapter  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)

    Chapter  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)

    Chapter  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)

    Chapter  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)

    Chapter  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)

    Article  MathSciNet  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)

    Chapter  Google Scholar 

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

    Article  MathSciNet  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)

    Article  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)

    Chapter  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)

    Chapter  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)

    Chapter  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)

    Article  MathSciNet  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)

    Chapter  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)

Publish with us

Policies and ethics