International Symposium on Frontiers of Combining Systems

Frontiers of Combining Systems pp 341-356 | Cite as

Lemmatization for Stronger Reasoning in Large Theories

  • Cezary Kaliszyk
  • Josef Urban
  • Jiří Vyskočil
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9322)

Abstract

In this work we improve ATP performance in large theories by the reuse of lemmas derived in previous related problems. Given a large set of related problems to solve, we run automated theorem provers on them, extract a large number of lemmas from the proofs found and post-process the lemmas to make them usable in the remaining problems. Then we filter the lemmas by several tools and extract their proof dependencies, and use machine learning on such proof dependencies to add the most promising generated lemmas to the remaining problems. On such enriched problems we run the automated provers again, solving more problems. We describe this method and the techniques we used, and measure the improvement obtained. On the MPTP2078 large-theory benchmark the method yields 6.6% and 6.2% more problems proved in two different evaluation modes.

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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. Reasoning 52(2), 191–213 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Blanchette, J.C.: Redirecting proofs by contradiction. In: Blanchette, J.C., Urban, J. (eds.) PxTP@CADE. EPiC Series, vol. 14, pp. 11–26. EasyChair (2013)Google Scholar
  3. 3.
    Blanchette, J.C., Kaliszyk, C., Paulson, L.C., Urban, J.: Hammering towards QED. J. Formalized Reasoning (in press, 2015)Google Scholar
  4. 4.
    Fiedler, A.: P.rex: An interactive proof explainer. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 416–420. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Kaliszyk, C., Urban, J.: Learning-assisted automated reasoning with Flyspeck. J. Autom. Reasoning 53(2), 173–213 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kaliszyk, C., Urban, J.: Learning-assisted theorem proving with millions of lemmas. Journal of Symbolic Computation 69, 109–128 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kaliszyk, C., Urban, J.: MizAR 40 for Mizar 40. J. Automated Reasoning (in press, 2015)Google Scholar
  9. 9.
    Kinyon, M., Veroff, R., Vojtěchovský, P.: Loops with abelian inner mapping groups: An application of automated deduction. In: Bonacina, M.P., Stickel, M.E. (eds.) Automated Reasoning and Mathematics. LNCS, vol. 7788, pp. 151–164. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  10. 10.
    Kovács, L., Voronkov, A.: First-order theorem proving and Vampire. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 1–35. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Kuehlwein, D., Urban, J.: Learning from multiple proofs: First experiments. In: Fontaine, P., Schmidt, R.A., Schulz, S. (eds.) PAAR 2012. EPiC Series, vol. 21, pp. 82–94. EasyChair (2013)Google Scholar
  12. 12.
    Kühlwein, D., Blanchette, J.C., Kaliszyk, C., Urban, J.: MaSh: Machine learning for sledgehammer. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 35–50. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Meier, A.: System description: Tramp: transformation of machine-found proofs into nd-proofs at the assertion level. In: McAllester, D. (ed.) CADE 2000. LNCS(LNAI), vol. 1831, pp. 460–464. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Phillips, J.D., Stanovský, D.: Automated theorem proving in quasigroup and loop theory. AI Commun. 23(2–3), 267–283 (2010)MathSciNetMATHGoogle Scholar
  16. 16.
    Puzis, Y., Gao, Y., Sutcliffe, G.: Automated generation of interesting theorems. In: Sutcliffe, G., Goebel, R. (eds.) FLAIRS, pp. 49–54. AAAI Press (2006)Google Scholar
  17. 17.
    Schulz, S.: Learning search control knowledge for equational deduction. DISKI, vol. 230. Infix Akademische Verlagsgesellschaft (2000)Google Scholar
  18. 18.
    Schulz, S.: System description: E 1.8. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19 2013. LNCS, vol. 8312, pp. 735–743. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  19. 19.
    Smolka, S.J., Blanchette, J.C.: Robust, semi-intelligible Isabelle proofs from ATP proofs. In: Blanchette, J.C., Urban, J. (eds.) PxTP 2013. EPiC Series, vol. 14, pp. 117–132. EasyChair (2013)Google Scholar
  20. 20.
    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)CrossRefGoogle Scholar
  21. 21.
    Urban, J.: MoMM - fast interreduction and retrieval in large libraries of formalized mathematics. Int. J. on Artificial Intelligence Tools 15(1), 109–130 (2006)CrossRefGoogle Scholar
  22. 22.
    Urban, J.: BliStr: The Blind Strategymaker. CoRR, abs/1301.2683 (2014) (accepted to PAAR 2014)Google Scholar
  23. 23.
    Urban, J., Sutcliffe, G.: ATP-based cross-verification of Mizar proofs: Method, systems, and first experiments. MCS 2(2), 231–251 (2008)MathSciNetMATHGoogle Scholar
  24. 24.
    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
  25. 25.
    Urban, J., Sutcliffe, G., Trac, S., Puzis, Y.: Combining mizar and TPTP semantic presentation and verification Tools. Studies in Logic, Grammar and Rhetoric 18(31), 121–136 (2009)Google Scholar
  26. 26.
    Veroff, R.: Using hints to increase the effectiveness of an automated reasoning program: Case studies. J. Autom. Reasoning 16(3), 223–239 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Cezary Kaliszyk
    • 1
  • Josef Urban
    • 2
  • Jiří Vyskočil
    • 3
  1. 1.University of InnsbruckInnsbruckAustria
  2. 2.Radboud University NijmegenNijmegenNetherlands
  3. 3.Czech Technical University in PraguePragueCzech Republic

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