Advertisement

Journal of Automated Reasoning

, Volume 55, Issue 3, pp 245–256 | Cite as

MizAR 40 for Mizar 40

  • Cezary Kaliszyk
  • Josef Urban
Open Access
Article

Abstract

As a present to Mizar on its 40th anniversary, we develop an AI/ATP system that in 30 seconds of real time on a 14-CPU machine automatically proves 40 % of the theorems in the latest official version of the Mizar Mathematical Library (MML). This is a considerable improvement over previous performance of large-theory AI/ATP methods measured on the whole MML. To achieve that, a large suite of AI/ATP methods is employed and further developed. We implement the most useful methods efficiently, to scale them to the 150000 formulas in MML. This reduces the training times over the corpus to 1–3 seconds, allowing a simple practical deployment of the methods in the online automated reasoning service for the Mizar users (Miz \(\mathbb {A}\mathbb {R}\)).

Keywords

Automated reasoning Formal mathematics Mizar Large theories Machine learning Artificial intelligence Premise selection 

References

  1. 1.
    Alama, J., Heskes, T., Kühlwein, D., Tsivtsivadze, E., Urban, J.: Premise selection for mathematics by corpus analysis and kernel methods. J. Autom. Reason. 52(2), 191–213 (2014)CrossRefGoogle Scholar
  2. 2.
    Alama, J., Kühlwein, D., Urban, J.: Automated and human proofs in general mathematics: an initial comparison. In: Bjørner, N., Voronkov, A. (eds.) LPAR of LNCS, vol. 7180, pp. 37–45. Springer (2012)Google Scholar
  3. 3.
    de Moura, L.M., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS of LNCS, vol. 4963, pp. 337–340. Springer (2008)Google Scholar
  4. 4.
    Deerwester, S.C., Dumais, S.T., Landauer, T.K., Furnas, G.W., Harshman, R.A.: Indexing by latent semantic analysis. JASIS 41(6), 391–407 (1990)CrossRefGoogle Scholar
  5. 5.
    Dudani, S.A.: The distance-weighted k-nearest-neighbor rule. IEEE Trans. Syst. Man Cybern. SMC-6(4), 325–327 (1976)CrossRefGoogle Scholar
  6. 6.
    Grabowski, A., Kornilowicz, A., Naumowicz, A.: Mizar in a nutshell. Journal of Formalized Reasoning 3(2), 153–245 (2010)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Hales, T.: Dense Sphere Packings: A Blueprint for Formal Proofs of London Mathematical Society Lecture Note Series, vol. 400. Cambridge University Press (2012)Google Scholar
  8. 8.
    Hales, T.C., Adams, M., Bauer, G., Dang, D.T., Harrison, J., Hoang, T.L., Kaliszyk, C., Magron, V., McLaughlin, S., Nguyen, T.T., Nguyen, T.Q., Nipkow, T., Obua, S., Pleso, J., Rute, J., Solovyev, A., Ta, A.H.T., Tran, T.N., Trieu, D.T., Urban, J., Vu, K.K., Zumkeller, R.: A formal proof of the Kepler conjecture. CoRR (2015) arXiv:1501.02155
  9. 9.
    Harrison, J.: HOL light: a tutorial introduction. In: Srivas, M.K., Camilleri, A.J. (eds.) FMCAD of LNCS, vol. 1166, pp. 265–269. Springer (1996)Google Scholar
  10. 10.
    Hoder, K., Voronkov, A.: Sine qua non for large theory reasoning. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE of LNCS, vol. 6803, pp. 299–314. Springer (2011)Google Scholar
  11. 11.
    Jones, K.S.: A statistical interpretation of term specificity and its application in retrieval. J. Doc. 28, 11–21 (1972)CrossRefGoogle Scholar
  12. 12.
    Kaliszyk, C., Urban, J.: Automated reasoning service for HOL light. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) MKM/Calculemus/DML of Lecture Notes in Computer Science, vol. 7961, pp. 120–135. Springer (2013)Google Scholar
  13. 13.
    Kaliszyk, C., Urban, J.: PRocH: proof reconstruction for HOL Light. In: Bonacina, M.P. (ed.) CADE of Lecture Notes in Computer Science, vol. 7898, pp. 267–274. Springer (2013)Google Scholar
  14. 14.
    Kaliszyk, C., Urban, J.: Stronger automation for Flyspeck by feature weighting and strategy evolution. In: Blanchette, J.C., Urban, J. (eds.) PxTP 2013 of EPiC Series. EasyChair, vol. 14, pp. 87–95 (2013)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., Josef, Urban.: HOL(y)Hammer: online ATP service for HOL Light. Math. Comput. Sci. 9(1), 5–22 (2015)zbMATHCrossRefGoogle Scholar
  17. 17.
    Kaliszyk, C., Urban, J., Vyskocil, J.: Machine learner for automated reasoning 0.4 and 0.5. CoRR (2014). Accepted to PAAR’14 arXiv:1402. 2359
  18. 18.
    Kaliszyk, C., Urban, J., Vyskocil, J.: Efficient semantic features for automated reasoning over large theories. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IICAI’15). to appear (2015)Google Scholar
  19. 19.
    Kovács, L., Voronkov, A.: First-order theorem proving and vampire. In: Sharygina, N., Veith, H. (eds.) CAV of Lecture Notes in Computer Science, vol. 8044, pp. 1–35. Springer (2013)Google Scholar
  20. 20.
    Kuehlwein, D., Urban, J.: Learning from multiple proofs: first experiments. In: Fontaine, P., Schmidt, R. A., Schulz, S. (eds.) PAAR-2012 of EPiC Series, vol. 21, pp. 82–94. EasyChair (2013)Google Scholar
  21. 21.
    Kühlwein, D., Blanchette, J.C., Kaliszyk, C., Urban, J.: MaSh: machine learning for Sledgehammer. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) Proceeding of the 4th international conference on interactive theorem proving (ITP’13) of LNCS, vol. 7998, pp. 35–50. Springer (2013)Google Scholar
  22. 22.
    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 of LNCS, vol. 7364, pp. 378–392. Springer (2012)Google Scholar
  23. 23.
    Rehurek, R., Sojka, P.: Software framework for topic modelling with large corpora. In: Proceedings of the LREC 2010 workshop on new challenges for NLP frameworks, pp. 45–50. ELRA, Valletta, Malta (2010)Google Scholar
  24. 24.
    Schapire, R.E.: The strength of weak learnability. Mach. Learn. 5, 197–227 (1990)Google Scholar
  25. 25.
    Schulz, S.: E - A brainiac theorem prover. AI Commun. 15(2-3), 111–126 (2002)zbMATHGoogle Scholar
  26. 26.
    Shawe-Taylor, J., Cristianini, N.: Kernel methods for pattern analysis. Cambridge University Press, New York (2004)CrossRefGoogle Scholar
  27. 27.
    Smolka, S.J., Blanchette, J.C.: Robust, semi-intelligible Isabelle proofs from ATP proofs. In: Blanchette, J. C., Urban, J. (eds.) PxTP 2013 of EPiC Series, vol. 14, pp. 117–132. EasyChair (2013)Google Scholar
  28. 28.
    Josef, U.: Translating Mizar for first order theorem provers. In: MKM of LNCS, vol. 2594, pp. 203–215. Springer (2003)Google Scholar
  29. 29.
    Urban, J.: MPTP - motivation, implementation, first experiments. J. Autom. Reason. 33(3-4), 319–339 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Urban, J.: MPTP 0.2: design, implementation, and initial experiments. J. Autom. Reason. 37(1-2), 21–43 (2006)zbMATHCrossRefGoogle Scholar
  31. 31.
    Urban, J.: MaLARea: a metasystem for automated reasoning in large theories. In: Sutcliffe, G., Urban, J., Schulz, S. (eds.) ESARLT of CEUR Workshop Proceedings, vol. 257. CEUR-WS.org (2007)Google Scholar
  32. 32.
    Urban, J.: An overview of methods for large-theory automated theorem proving (Invited Paper). In: Höfner, P., McIver, A., Struth, G. (eds.) ATE Workshop, volume 760 of CEUR Workshop Proceedings, pp. 3–8. CEUR-WS.org (2011)Google Scholar
  33. 33.
    Urban, J.: BliStr: the blind strategymaker, CoRR. arXiv:1301.2683. Accepted to PAAR’14 (2014)
  34. 34.
    Urban, J., Rudnicki, P., Sutcliffe, G.: ATP and presentation service for Mizar formalizations. J. Autom. Reason. 50, 229–241 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Urban, J., Sutcliffe, G., Pudlák, P., Vyskocil, J.: MaLARea SG1 - machine learner for automated reasoning with semantic guidance. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR of LNCS, vol. 5195, pp. 441–456. Springer (2008)Google Scholar
  36. 36.
    Urban, J., Vyskocil, J.: Theorem proving in large formal mathematics as an emerging AI field. In: Bonacina, M. P., Stickel, M. E. (eds.) Automated reasoning and mathematics: essays in memory of william McCune of LNAI, vol. 7788, pp. 240–257. Springer (2013)Google Scholar

Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.University of InnsbruckInnsbruckAustria
  2. 2.Radboud UniversityNijmegenNetherlands

Personalised recommendations