Automated and Human Proofs in General Mathematics: An Initial Comparison

  • Jesse Alama
  • Daniel Kühlwein
  • Josef Urban
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7180)


First-order translations of large mathematical repositories allow discovery of new proofs by automated reasoning systems. Large amounts of available mathematical knowledge can be re-used by combined AI/ATP systems, possibly in unexpected ways. But automated systems can be also more easily misled by irrelevant knowledge in this setting, and finding deeper proofs is typically more difficult. Both large-theory AI/ATP methods, and translation and data-mining techniques of large formal corpora, have significantly developed recently, providing enough data for an initial comparison of the proofs written by mathematicians and the proofs found automatically. This paper describes such an initial experiment and comparison conducted over the 50000 mathematical theorems from the Mizar Mathematical Library.


Automate Theorem Prove Initial Comparison General Mathematic Mizar Mathematical Library Proof Length 
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.


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  1. 1.
    Alama, J., Kühlwein, D., Tsivtsivadze, E., Urban, J., Heskes, T.: Premise selection for mathematics by corpus analysis and kernel methods. CoRR, abs/1108.3446Google Scholar
  2. 2.
    Alama, J., Mamane, L., Urban, J.: Dependencies in formal mathematics. CoRR, abs/1109.3687 (2011),
  3. 3.
    Dahn, I.: Robbins algebras are Boolean: A revision of McCune’s computer-generated solution of Robbins problem. Journal of Algebra 208, 526–532 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Dosen, K.: Identity of proofs based on normalization and generality. Bulletin of Symbolic Logic 9, 477–503 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Fitelson, B.: Using Mathematica to understand the computer proof of the Robbins Conjecture. Mathematica In Education and Research 7(1) (1998)Google Scholar
  6. 6.
    Hales, T.: Mathematics in the age of the Turing Machine. Lecture Notes in Logic in Commemoration of the Centennial of the Birth of Alan Turing (to appear, 2012)Google Scholar
  7. 7.
    Hoder, K., Voronkov, A.: Sine Qua Non for Large Theory Reasoning. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 299–314. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    McCune, W.W.: Solution of the Robbins Problem. Journal of Automated Reasoning 19(3), 263–276 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Meng, J., Paulson, L.C.: Lightweight relevance filtering for machine-generated resolution problems. J. Applied Logic 7(1), 41–57 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Paulson, L.C., Blanchette, J.: Three years of experience with Sledgehammer, a practical link between automated and interactive theorem provers. In: 8th IWIL (2010)Google Scholar
  11. 11.
    Phillips, J.D., Stanovský, D.: Automated Theorem Proving in Loop Theory. In: Sutcliffe, G., Colton, S., Schulz, S. (eds.) ESARM. CEUR Workshop Proceedings, vol. 378, pp. 42–53. (2008)Google Scholar
  12. 12.
    Pudlák, P.: Semantic selection of premisses for automated theorem proving. In: Sutcliffe, G., Urban, J., Schulz, S. (eds.) ESARLT. CEUR Workshop Proceedings, vol. 257. (2007)Google Scholar
  13. 13.
    Schulz, S.: E – a brainiac theorem prover. J. of AI Communications 15(2-3), 111–126 (2002)zbMATHGoogle Scholar
  14. 14.
    Urban, J.: MPTP 0.2: Design, implementation, and initial experiments. J. Autom. Reasoning 37(1-2), 21–43 (2006)zbMATHCrossRefGoogle Scholar
  15. 15.
    Urban, J., Hoder, K., Voronkov, A.: Evaluation of Automated Theorem Proving on the Mizar Mathematical Library. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 155–166. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Urban, J., Sutcliffe, G., Pudlák, P., Vyskočil, J.: MaLARea SG1–Machine learner for automated reasoning with semantic guidance. In: IJCAR, pp. 441–456 (2008)Google Scholar
  17. 17.
    Urban, J., Vyskočil, J., Štěpánek, P.: MaLeCoP Machine Learning Connection Prover. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS, vol. 6793, pp. 263–277. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Vyskočil, J., Stanovský, D., Urban, J.: Automated Proof Compression by Invention of New Definitions. In: Clarke, E.M., Voronkov, A. (eds.) LPAR-16 2010. LNCS, vol. 6355, pp. 447–462. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jesse Alama
    • 1
  • Daniel Kühlwein
    • 2
  • Josef Urban
    • 2
  1. 1.New University of LisbonPortugal
  2. 2.Radboud University NijmegenNetherlands

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