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Journal of Automated Reasoning

, Volume 55, Issue 3, pp 257–268 | Cite as

Definitional Expansions in Mizar

In memoriam of Andrzej Trybulec, a pioneer of computerized formalization
  • Artur Korniłowicz
Open Access
Article

Abstract

The Mizar Verifier uses definitional expansions for controlling proof structures. In this paper we propose another use of definitional expansions—enriching verified inferences by expansions of definitions of formulae included in the inferences and increasing the number of premises accessible by Checker. This introduces more knowledge to the reasoning, which helps to draw more conclusions. Some statistics about influence of such expansions on the Mizar Mathematical Library are presented.

Keywords

Proof assistant Formal verification Definitional expansion Mizar 

References

  1. 1.
    Alama, J., Kohlhase, M., Mamane, L., Naumowicz, A., Rudnicki, P., Urban, J.: Licensing the Mizar Mathematical Library. In: Davenport, J.H. et al. (eds.) Proceedings of Calculemus/MKM 2011, LNCS, vol. 6824, pp. 149–163. Springer-Verlag, Berlin, Heidelberg. doi:  10.1007/978-3-642-22673-1_11 (2011)
  2. 2.
    Bancerek, G.: Information retrieval and rendering with MML query. In: Borwein, J. et al. (eds.) Mathematical Knowledge Management, LNCS, vol. 4108, pp. 266–279. Springer Berlin Heidelberg. doi:  10.1007/11812289_21 (2006)
  3. 3.
    Bancerek, G., Urban, J.: Integrated semantic browsing of the Mizar Mathematical Library for authoring Mizar articles. In: Asperti, A. et al. (eds.) MKM 2004, Bialowieza, Poland, September 2004, Proceedings, LNCS, vol. 3119, pp. 44–57. Springer. doi:  10.1007/978-3-540-27818-4_4 (2004)
  4. 4.
    Barker-Plummer, D.: Gazing: An approach to the problem of definition and lemma use. J. Autom. Reasoning 8(3), 311–344 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bishop, M., Andrews, P.B.: Selectively instantiating definitions. In: CADE-15, Lindau, Germany, July, 1998, Proceedings, LNCS, vol. 1421, pp. 365–380. Springer (1998)Google Scholar
  6. 6.
    Bledsoe, W.W.: The UT interactive prover. Memo ATP-17B, Mathematics Department. University of Texas (1983)Google Scholar
  7. 7.
    Bylinski, C., Alama, J.: New developments in parsing Mizar. In: Jeuring, J. et al. (eds.) Intelligent Computer Mathematics 11th International Conference LNAI vol. 7362, pp. 427–431 Springer-Verlag Berlin Heidelberg (2012), doi:  10.1007/978-3-642-31374-5_30
  8. 8.
    Cairns, P., Gow, J.: Using and parsing the Mizar language. Electronic Notes in Theoretical Computer Science 93, 60–69 (2004). doi:  10.1016/j.entcs.2003.12.028. http://www.sciencedirect.com/science/article/pii/S1571066104000131
  9. 9.
    Davis, M.: Obvious logical inferences. In: Proceedings of the Seventh International Joint Conference on Artificial Intelligence, pp. 530–531 (1981)Google Scholar
  10. 10.
    Giunchiglia, F., Walsh, T.: Theorem proving with definitions. In: Proceedings of AISB 89, pp. 433–435 (1989)Google Scholar
  11. 11.
    Grabowski, A., Naumowicz, A.: Mizar in a nutshell. J. Formal. Reasoning, Special Issue: User Tutorials I 3(2), 153–245 (2010)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Grabowski, A., Schwarzweller, C.: Translating mathematical vernacular into knowledge repositories. In: Proceedings of MKM’05, pp. 49–64. Springer-Verlag, Berlin, Heidelberg. doi:  10.1007/11618027_4 (2006)
  13. 13.
    Grabowski, A., Schwarzweller, C.: Revisions as an essential tool to maintain mathematical repositories. In: Proceedings of Calculemus ’07 / MKM ’07, pp. 235–249. Springer-Verlag, Berlin, Heidelberg. doi:  10.1007/978-3-540-73086-6_20(2007)
  14. 14.
    Grabowski, A., Schwarzweller, C.: Towards automatically categorizing mathematical knowledge. In: Ganzha, M. et al. (eds.) FedCSIS 2012, Wroclaw, Poland, September 2012, Proceedings, pp. 63–68 (2012)Google Scholar
  15. 15.
    Grätzer, G.: General Lattice Theory. Academic Press, New York (1978)CrossRefGoogle Scholar
  16. 16.
    Iancu, M., Kohlhase, M., Rabe, F., Urban, J.: The Mizar Mathematical Library in OMDoc Translation and applications. J. Autom. Reasoning 50(2), 191–202 (2013). doi:  10.1007/s10817-012-9271-4 zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Jaskowski, S.: On the Rules of Suppositions in Formal Logic. Studia Logica. Nakladem Seminarjum Filozoficznego Wydzialu Matematyczno-Przyrodniczego Uniwersytetu Warszawskiego (1934). http://books.google.pl/books?id=6w0vRAAACAAJ
  18. 18.
    Kieffer, S., Avigad, J., Friedman, H.: A language for mathematical knowledge management. In: Grabowski, A. et al. (eds.) Computer Reconstruction of the Body of Mathematics, Studies in Logic, Grammar and Rhetoric, vol. 18(31), pp. 51–66. Bialystok (2009)Google Scholar
  19. 19.
    Kornilowicz, A.: On rewriting rules in Mizar. J. Autom. Reasoning 50(2), 203–210 (2013). doi:  10.1007/s10817-012-9261-6 zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Naumowicz, A.: Interfacing external CA systems for Grobner bases computation in Mizar proof checking. Int. J. Comput. Math. 87(1), 1–11 (2010). doi:  10.1080/00207160701864459 zbMATHCrossRefGoogle Scholar
  21. 21.
    Naumowicz, A. In: Watt, S.M. et al. (eds.) : SAT-enhanced Mizar proof checking (2014). doi:  10.1007/978-3-319-08434-3_37
  22. 22.
    Naumowicz, A., Byliński, C.: Improving Mizar texts with properties and requirements. In: Asperti, A. et al. (eds.) MKM 2004 Proceedings, LNCS, vol. 3119, pp. 290–301. doi:  10.1007/978-3-540-27818-4_21 (2004)
  23. 23.
    Naumowicz, A., Kornilowicz, A., et al.: A brief overview of Mizar. In: Berghofer, S. (ed.) Proceedings of TPHOLs’09, LNCS, vol. 5674, pp. 67–72. Springer-Verlag, Berlin, Heidelberg. doi:  10.1007/978-3-642-03359-9_5 (2009)
  24. 24.
    Pak, K.: Methods of lemma extraction in natural deduction proofs. J. Autom. Reasoning 50(2), 217–228 (2013). doi:  10.1007/s10817-012-9267-0 zbMATHCrossRefGoogle Scholar
  25. 25.
    Pak, K.: Improving legibility of natural deduction proofs is not trivial. Logical Methods in Comput. Sc. 10(3), 1–30 (2014). doi:  10.2168/LMCS-10(3:23)2014 Google Scholar
  26. 26.
    Trybulec, A., Kornilowicz, A., Naumowicz, A., Kuperberg, K.: Formal mathematics for mathematicians. J. Autom. Reasoning 50(2), 119–121 (2013). doi:  10.1007/s10817-012-9268-z MathSciNetCrossRefGoogle Scholar
  27. 27.
    Urban, J.: XML-izing Mizar: Making semantic processing and presentation of MML easy. In: Kohlhase, M. (ed.) MKM 2005, Bremen, Germany July 2005, LNCS, vol. 3863, pp. 346–360 Springer . doi:  10.1007/11618027_23 (2005)
  28. 28.
    Urban, J., Hoder, K., Voronkov, A.: Evaluation of automated theorem proving on the Mizar Mathematical Library. In: Fukuda, K. et al. (eds.) ICMS 2010, Kobe, Japan. LNCS, vol. 6327, pp. 155–166. Springer. doi:  10.1007/978-3-642-15582-6_30 (2010)
  29. 29.
    In: Wiedijk, F. (ed.) : The Seventeen Provers of the World, Foreword by Dana S. Scott, LNCS, vol. 3600. Springer (2006)Google Scholar
  30. 30.
    Woronowicz, E.: Relations and their basic properties. Formalized Mathematics 1(1), 73–83 (1990). http://fm.mizar.org/1990-1/pdf1-1/relat_1.pdf Google Scholar
  31. 31.
    Wos, L.: Automated Reasoning: 33 Basic Research Problems. Prentice-Hall, Englewood Cliffs. N.J (1987)Google Scholar
  32. 32.
    Wos, L.: The problem of definition expansion and contraction. J. Autom. Reasoning 3(4), 433–435 (1987)CrossRefGoogle Scholar

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© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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 BiałystokBiałystokPoland

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