Internal Guidance for Satallax

  • Michael Färber
  • Chad Brown
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9706)


We propose a new internal guidance method for automated theorem provers based on the given-clause algorithm. Our method influences the choice of unprocessed clauses using positive and negative examples from previous proofs. To this end, we present an efficient scheme for Naive Bayesian classification by generalising label occurrences to types with monoid structure. This makes it possible to extend existing fast classifiers, which consider only positive examples, with negative ones. We implement the method in the higher-order logic prover Satallax, where we modify the delay with which propositions are processed. We evaluated our method on a simply-typed higher-order logic version of the Flyspeck project, where it solves 26 % more problems than Satallax without internal guidance.



We would like to thank Sebastian Joosten and Cezary Kaliszyk for reading initial drafts of the paper, and especially Josef Urban for inspiring discussions and inviting the authors to Prague. Furthermore, we would like to thank the anonymous IJCAR referees for their valuable comments.

This work has been supported by the Austrian Science Fund (FWF) grant P26201 as well as by the European Research Council (ERC) grant AI4REASON.


  1. [Bie08]
    Biere, A.: PicoSAT essentials. JSAT 4(2–4), 75–97 (2008)zbMATHGoogle Scholar
  2. [Bro12]
    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)CrossRefGoogle Scholar
  3. [HAB+15]
    Hales, T.C., Adams, M., Bauer, G., Dang, D.T., Harrison, J., Le Hoang, T., 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, abs/1501.02155 (2015)Google Scholar
  4. [HV11]
    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
  5. [KE95]
    Kennedy, J., Eberhart, R.: Particle swarm optimization. In: IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948, November 1995Google Scholar
  6. [KSUV15]
    Kaliszyk, C., Schulz, S., Urban, J., Vyskocil, J.: System description: E.T. 0.1. In: Felty, A.P., Middeldorp, A. (eds.) CADE-25. LNCS (LNAI), vol. 9195, pp. 389–398. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  7. [KU14]
    Kaliszyk, C., Urban, J.: Learning-assisted automated reasoning with Flyspeck. J. Autom. Reasoning 53(2), 173–213 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [KU15]
    Kaliszyk, C., Urban, J.: FEMaLeCoP: fairly efficient machine learning connection prover. In: Davis, M., et al. (eds.) LPAR-20 2015. LNCS, vol. 9450, pp. 88–96. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48899-7_7 CrossRefGoogle Scholar
  9. [KV13]
    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
  10. [Kü14]
    Daniel, A.K.: Machine learning for automated reasoning. Ph.D. thesis, Radboud Universiteit Nijmegen, April 2014Google Scholar
  11. [Ott08]
    Otten, J.: \(\sf leanCoP 2.0\) and \(\sf ileanCoP 1.2\): high performance lean theorem proving in classical and intuitionistic logic (system descriptions). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 283–291. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. [SB10]
    Sutcliffe, G., Benzmüller, C.: Automated reasoning in higher-order logic using the TPTP THF infrastructure. J. Formalized Reasoning 3(1), 1–27 (2010)MathSciNetzbMATHGoogle Scholar
  13. [SBP13]
    Sultana, N., Blanchette, J.C., Paulson, L.C.: LEO-II, Satallax on the Sledgehammer test bench. J. Appl. Logic 11(1), 91–102 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Sch00]
    Schulz, S.: Learning Search Control Knowledge for Equational Deduction. DISKI, vol. 230. Akademische Verlagsgesellschaft Aka GmbH Berlin, Berlin (2000)Google Scholar
  15. [Sch13]
    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
  16. [Urb15]
    Urban, J.: BliStr: the blind Strategy maker. In: Gottlob, G., Sutcliffe, G., Voronkov, A. (eds.) GCAI 32015, Global Conference on Artificial Intelligence. EPiC Series in Computing, vol. 36, pp. 312–319. EasyChair (2015)Google Scholar
  17. [UVŠ11]
    Urban, J., Vyskočil, J., Štěpánek, P.: \(\sf MaLeCoP\) machine learning connection prover. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS, vol. 6793, pp. 263–277. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. [Ver96]
    Veroff, R.: Using hints to increase the effectiveness of an automated reasoning program: case studies. J. Autom. Reasoning 16(3), 223–239 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Universität InnsbruckInnsbruckAustria
  2. 2.Czech Technical University in PraguePragueCzech Republic

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