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


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.


  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)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Adam Grabowski: Automated discovery of properties of rough sets. Fundamenta Informaticae 128(1–2), 65–79 (2013)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formalized Reason. 3(2), 153–245 (2010)

    MATH  Google 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)

  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)

  11. 11.

    Kornilowicz, A.: On rewriting rules in Mizar. J. Autom. Reason. 50(2), 203–210 (2013)

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Naumowicz, A.: An example of formalizing recent mathematical results in Mizar. J. Appl. Log. 4(4), 396–413 (2006)

    MATH  MathSciNet  Article  Google 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)

    MATH  Article  Google 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)

  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)

  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)

    Article  Google Scholar 

  19. 19.

    Pak, K.: Methods of lemma extraction in natural deduction proofs. J. Autom. Reason. 50(2), 217–228 (2013)

    MATH  Article  Google Scholar 

  20. 20.

    Trybulec, A., Kornilowicz, A., Naumowicz, A., Kuperberg, K.: Formal mathematics for mathematicians. J. Autom. Reason. 50(2), 119–121 (2013)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Urban, J., Rudnicki, P., Sutcliffe, G.: ATP and presentation service for Mizar formalizations. J. Autom. Reason. 50(2), 229–241 (2013)

    MATH  MathSciNet  Article  Google 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)

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Correspondence to Adam Naumowicz.

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To the memory of Andrzej Trybulec

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Naumowicz, A. Automating Boolean Set Operations in Mizar Proof Checking with the Aid of an External SAT Solver. J Autom Reasoning 55, 285–294 (2015).

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  • Mizar
  • SAT solvers
  • Proof assistants
  • Boolean operations