Skip to main content

MaSh: Machine Learning for Sledgehammer

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7998)

Abstract

Sledgehammer integrates automatic theorem provers in the proof assistant Isabelle/HOL. A key component, the relevance filter, heuristically ranks the thousands of facts available and selects a subset, based on syntactic similarity to the current goal. We introduce MaSh, an alternative that learns from successful proofs. New challenges arose from our “zero-click” vision: MaSh should integrate seamlessly with the users’ workflow, so that they benefit from machine learning without having to install software, set up servers, or guide the learning. The underlying machinery draws on recent research in the context of Mizar and HOL Light, with a number of enhancements. MaSh outperforms the old relevance filter on large formalizations, and a particularly strong filter is obtained by combining the two filters.

Keywords

  • Visibility Graph
  • Proof Assistant
  • High Order Logic
  • Persistent State
  • Proof Term

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: 10.1007/978-3-642-39634-2_6
  • Chapter length: 16 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   64.99
Price excludes VAT (USA)
  • ISBN: 978-3-642-39634-2
  • 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   84.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. The Mizar Mathematical Library, http://mizar.org/

  2. Alama, J., Heskes, T., Kühlwein, D., Tsivtsivadze, E., Urban, J.: Premise selection for mathematics by corpus analysis and kernel methods. J. Autom. Reasoning, http://arxiv.org/abs/1108.3446 (April 2013)

  3. 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 

  4. Barrett, C., Tinelli, C.: CVC3. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 298–302. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  5. Benzmüller, C., Paulson, L.C., Theiss, F., Fietzke, A.: LEO-II—A cooperative automatic theorem prover for higher-order logic (System description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 162–170. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  6. Berghofer, S., Nipkow, T.: Proof terms for simply typed higher order logic. In: Aagaard, M., Harrison, J. (eds.) TPHOLs 2000. LNCS, vol. 1869, pp. 38–52. Springer, Heidelberg (2000)

    CrossRef  Google Scholar 

  7. Blanchette, J.C., Böhme, S., Paulson, L.C.: Extending Sledgehammer with SMT solvers. J. Autom. Reasoning 51(1), 109–128 (2013), http://www21.in.tum.de/~blanchet/jar-smt.pdf

    CrossRef  Google Scholar 

  8. Blanchette, J.C., Böhme, S., Popescu, A., Smallbone, N.: Encoding monomorphic and polymorphic types. In: Piterman, N., Smolka, S. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 493–507. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  9. Blanchette, J.C., Popescu, A., Wand, D., Weidenbach, C.: More SPASS with Isabelle—Superposition with hard sorts and configurable simplification. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 345–360. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  10. Böhme, S., Nipkow, T.: Sledgehammer: Judgement Day. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS (LNAI), vol. 6173, pp. 107–121. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  11. Brown, C.E.: Satallax: An automatic higher-order prover. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 111–117. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  12. Carlson, A.J., Cumby, C.M., Rosen, J.L., Roth, D.: SNoW user guide. Tech. rep., C.S. Dept., University of Illinois at Urbana-Champaign (1999), http://cogcomp.cs.illinois.edu/papers/CCRR99.pdf

  13. Dutertre, B., de Moura, L.: The Yices SMT solver (2006), http://yices.csl.sri.com/tool-paper.pdf

  14. 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 

  15. Hölzl, J., Heller, A.: Three chapters of measure theory in Isabelle/HOL. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) ITP 2011. LNCS, vol. 6898, pp. 135–151. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  16. 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, pp. 56–68. NASA Tech. Reports (2003)

    Google Scholar 

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

  18. Klein, G., Nipkow, T.: Jinja is not Java. In: Klein, G., Nipkow, T., Paulson, L. (eds.) Archive of Formal Proofs (2005), http://afp.sf.net/entries/Jinja.shtml

  19. Klein, G., Nipkow, T., Paulson, L. (eds.): Archive of Formal Proofs, http://afp.sf.net/

  20. 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, vol. 7364, pp. 378–392. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  21. Kühlwein, D., Urban, J.: Learning from multiple proofs: First experiments. In: Fontaine, P., Schmidt, R., Schulz, S. (eds.) PAAR-2012, pp. 82–94 (2012)

    Google Scholar 

  22. Matuszewski, R., Rudnicki, P.: Mizar: The first 30 years. Mechanized Mathematics and Its Applications 4(1), 3–24 (2005)

    Google Scholar 

  23. Meng, J., Paulson, L.C.: Lightweight relevance filtering for machine-generated resolution problems. J. Applied Logic 7(1), 41–57 (2009)

    MathSciNet  MATH  CrossRef  Google Scholar 

  24. de Moura, L., Bjørner, N.: Z3: An efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  25. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)

    Google Scholar 

  26. Paulson, L.C.: The inductive approach to verifying cryptographic protocols. J. Comput. Secur. 6(1-2), 85–128 (1998)

    Google Scholar 

  27. Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, A practical link between automatic and interactive theorem provers. In: Sutcliffe, G., Ternovska, E., Schulz, S. (eds.) IWIL-2010 (2010)

    Google Scholar 

  28. Riazanov, A., Voronkov, A.: The design and implementation of Vampire. AI Comm. 15(2-3), 91–110 (2002)

    MATH  Google Scholar 

  29. Schulz, S.: System description: E 0.81. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 223–228. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  30. Schulz, S.: First-order deduction for large knowledge bases. Presentation at Deduction at Scale 2011 Seminar, Ringberg Castle (2011), http://www.mpi-inf.mpg.de/departments/rg1/conferences/deduction10/slides/stephan-schulz.pdf

  31. Sutcliffe, G.: The 6th IJCAR automated theorem proving system competition—CASC-J6. AI Commun. 26(2), 211–223 (2013)

    Google Scholar 

  32. Urban, J.: MoMM—Fast interreduction and retrieval in large libraries of formalized mathematics. Int. J. AI Tools 15(1), 109–130 (2006)

    CrossRef  Google Scholar 

  33. Urban, J.: MPTP 0.2: Design, implementation, and initial experiments. J. Autom. Reasoning 37(1-2), 21–43 (2006)

    MATH  CrossRef  Google Scholar 

  34. Urban, J.: MaLARea: A metasystem for automated reasoning in large theories. In: Sutcliffe, G., Urban, J., Schulz, S. (eds.) ESARLT 2007. CEUR Workshop Proceedings, vol. 257. CEUR-WS.org (2007)

    Google Scholar 

  35. Urban, J.: An overview of methods for large-theory automated theorem proving. In: Höfner, P., McIver, A., Struth, G. (eds.) ATE-2011. CEUR Workshop Proceedings, vol. 760, pp. 3–8. CEUR-WS.org (2011)

    Google Scholar 

  36. Urban, J.: BliStr: The blind strategymaker. CoRR abs/1301.2683 (2013), http://arxiv.org/abs/1301.2683

  37. Urban, J., Rudnicki, P., Sutcliffe, G.: ATP and presentation service for Mizar formalizations. J. Autom. Reasoning 50(2), 229–241 (2013)

    MathSciNet  MATH  CrossRef  Google Scholar 

  38. 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 

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

    CrossRef  Google Scholar 

  40. Wenzel, M.: Isabelle/Isar—A generic framework for human-readable proof documents. In: Matuszewski, R., Zalewska, A. (eds.) From Insight to Proof—Festschrift in Honour of Andrzej Trybulec. Studies in Logic, Grammar, and Rhetoric, vol. 10(23). Uniwersytet w Białymstoku (2007)

    Google Scholar 

  41. Wenzel, M.: Parallel proof checking in Isabelle/Isar. In: Dos Reis, G., Théry, L. (eds.) PLMMS 2009, pp. 13–29. ACM Digital Library (2009)

    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

Kühlwein, D., Blanchette, J.C., Kaliszyk, C., Urban, J. (2013). MaSh: Machine Learning for Sledgehammer. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds) Interactive Theorem Proving. ITP 2013. Lecture Notes in Computer Science, vol 7998. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39634-2_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-39634-2_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39633-5

  • Online ISBN: 978-3-642-39634-2

  • eBook Packages: Computer ScienceComputer Science (R0)