Journal of Automated Reasoning

, Volume 55, Issue 3, pp 285–294 | Cite as

Automating Boolean Set Operations in Mizar Proof Checking with the Aid of an External SAT Solver

  • Adam NaumowiczEmail author
Open Access


In this paper we present the results of an experiment with employing an external SAT solver to strengthen the notion of obviousness of the Mizar proof checker. The presented extension of the Mizar system is based on a version of MiniSAT, called Logic2CNF. The SAT-enhanced Mizar checker is programmed to automatically spawn a new Logic2CNF process whenever it needs to justify any goal that can be solved by reducing it into a corresponding propositional satisfiability problem (equalities based on Boolean operations or set inclusion). The external tool is interfaced within the implementation of Mizar’s requirements directives.


Mizar SAT solvers Proof assistants Boolean operations 


  1. 1.
    Alama, J., Kohlhase, M., Mamane, L., Naumowicz, A., Rudnicki, P., Urban, J.: Licensing the Mizar mathematical library. In: Lecture Notes in Computer Science of MKM’11, vol. 6824, pp. 149–163. Springer, Berlin (2011)Google Scholar
  2. 2.
    Armand, M., Faure, G., Grégoire, B., Keller, C., Théry, L., Werner, B.: A modular integration of SAT/SMT solvers to Coq through proof witnesses. In: Lecture Notes in Computer Science, vol. 7086, pp. 135–150. Springer, Berlin (2011)Google Scholar
  3. 3.
    Caminati, M.B., Rosolini, G.: Custom automations in Mizar. J. Autom. Reason. 50(2), 147–160 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Adam Grabowski: Automated discovery of properties of rough sets. Fundamenta Informaticae 128(1–2), 65–79 (2013)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formalized Reason. 3(2), 153–245 (2010)zbMATHGoogle Scholar
  6. 6.
    Grabowski, A., Schwarzweller, C.: Translating mathematical vernacular into knowledge repositories. In: Proceedings of the 4th International Conference on Mathematical Knowledge Management MKM’05, pp. 49–64. Springer, Berlin (2006)Google Scholar
  7. 7.
    Grabowski, A., Schwarzweller, C.: Revisions as an essential tool to maintain mathematical repositories. In: Calculemus ’07 /MKM ’07, pp. 235–249. Springer, Berlin (2007)Google Scholar
  8. 8.
    Grabowski, A., Schwarzweller, C.: Towards automatically categorizing mathematical knowledge. In: Proceedings of Federated Conference on Computer Science and Information Systems – FedCSIS 2012, 9–12 September, pp. 63–68, Wroclaw (2012)Google Scholar
  9. 9.
    Kaliszyk, C., Urban, J.: Mizar 40 for Mizar 40. CoRR (2013). arXiv:1310.2805
  10. 10.
    Kornilowicz, A.: Tentative experiments with ellipsis in Mizar, vol. 7362, pp. 453–457. Springer (2012)Google Scholar
  11. 11.
    Kornilowicz, A.: On rewriting rules in Mizar. J. Autom. Reason. 50(2), 203–210 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Naumowicz, A.: An example of formalizing recent mathematical results in Mizar. J. Appl. Log. 4(4), 396–413 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Naumowicz, A.: Evaluating prospective built-in elements of computer algebra in Mizar. Studies in Logic, Grammar and Rhetoric 10(23), 191–200 (2007)Google Scholar
  14. 14.
    Naumowicz, A.: Interfacing external CA systems for Gröbner bases computation in Mizar proof checking. Int. J. Comput. Math. 87(1), 1–11 (2010)zbMATHCrossRefGoogle Scholar
  15. 15.
    Naumowicz, A.: SAT-enhanced Mizar proof checking. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) CICM of Lecture Notes in Computer Science, vol. 8543, pp. 449–452. Springer (2014)Google Scholar
  16. 16.
    Naumowicz, A., Bylinski, C.: Improving Mizar texts with properties and requirements. In: Lecture Notes in Computer Science of MKM’04, vol. 3119, pp. 290–301 (2004)Google Scholar
  17. 17.
    Naumowicz, A., Korniłowicz, A.: A brief overview of Mizar. In: Lecture Notes in Computer Science of TPHOLs’09, vol. 5674, pp. 67–72. Springer, Berlin (2009)Google Scholar
  18. 18.
    Pak, K.: Improving legibility of natural deduction proofs is not trivial. Logical Methods in Computer Science 10(3), 1–30 (2014)CrossRefGoogle Scholar
  19. 19.
    Pak, K.: Methods of lemma extraction in natural deduction proofs. J. Autom. Reason. 50(2), 217–228 (2013)zbMATHCrossRefGoogle Scholar
  20. 20.
    Trybulec, A., Kornilowicz, A., Naumowicz, A., Kuperberg, K.: Formal mathematics for mathematicians. J. Autom. Reason. 50(2), 119–121 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Urban, J., Rudnicki, P., Sutcliffe, G.: ATP and presentation service for Mizar formalizations. J. Autom. Reason. 50(2), 229–241 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Weber, T.: Integrating a SAT solver with an LCF-style theorem prover. In: Proceedings of the 3rd International Workshop on Pragmatical Aspects of Decision Procedures in Automated Reasoning PDPAR 2005 (2005)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 (, 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.Institute of InformaticsUniversity of BialystokBialystokPoland

Personalised recommendations