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Journal of Automated Reasoning

, Volume 50, Issue 2, pp 191–202 | Cite as

The Mizar Mathematical Library in OMDoc: Translation and Applications

  • Mihnea Iancu
  • Michael Kohlhase
  • Florian Rabe
  • Josef Urban
Article

Abstract

The Mizar Mathematical Library is one of the largest libraries of formalized and mechanically verified mathematics. Its language is highly optimized for authoring by humans. As in natural languages, the meaning of an expression is influenced by its (mathematical) context in a way that is natural to humans, but harder to specify for machine manipulation. Thus its custom file format can make the access to the library difficult. Indeed, the Mizar system itself is currently the only system that can fully operate on the Mizar library. This paper presents a translation of the Mizar library into the OMDoc format (Open Mathematical Documents), an XML-based representation format for mathematical knowledge. OMDoc is geared towards machine support and interoperability by making formula structure and context dependencies explicit. Thus, the Mizar library becomes accessible for a wide range of OMDoc-based tools for formal mathematics and knowledge management. We exemplify interoperability by indexing the translated library in the MathWebSearch engine, which provides an “applicable theorem search” service (almost) out of the box.

Keywords

Mizar OMDoc Representation Translation Math search Logical frameworks 

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References

  1. 1.
    Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.): Intelligent Computer Mathematics. In: 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, 5–10 July 2010. Proceedings, Lecture Notes in Computer Science, vol. 6167. Springer (2010)Google Scholar
  2. 2.
    Bancerek, G.: On the structure of Mizar types. Electron. Notes Theor. Comp. Sci. 85(7), 69–85 (2003)CrossRefGoogle Scholar
  3. 3.
    Bancerek, G.: Automatic translation in formalized mathematics. Mech. Math. Its Appl. 5(2), 19–31 (2006)Google Scholar
  4. 4.
    Bancerek, G.: Information retrieval and rendering with MML query. In: Borwein, J., Farmer, W.M. (eds.) Mathematical Knowledge Management (MKM). LNAI, vol. 4108, pp. 266–279. Springer (2006)Google Scholar
  5. 5.
    Bancerek, G., Kohlhase, M.: Towards a Mizar mathematical library in OMDoc format. In: Matuszewski, R., Zalewska, A. (eds.) From Insight to Proof: Festschrift in Honour of Andrzej Trybulec, Studies in Logic, Grammar and Rhetoric, vol. 10:23, pp. 265–275. University of Białystok (2007)Google Scholar
  6. 6.
    Codescu, M., Horozal, F., Kohlhase, M., Mossakowski, T., Rabe, F.: Project abstract: logic atlas and integrator (Latin). In: Davenport, J., Farmer, W., Rabe, F., Urban, J. (eds.) Intelligent Computer Mathematics. LNAI, vol. 6824, pp. 289–291. Springer (2011)Google Scholar
  7. 7.
    Dahn, I., Wernhard, C.: First order proof problems extracted from an article in the Mizar Mathematical Library. In: Furbach, U., Bonacina, M.P. (eds.) Proceedings of the International Workshop on First Order Theorem Proving. RISC-Linz Report Series, no. 97-50, pp. 58–62. Johannes Kepler Universität Linz (1997)Google Scholar
  8. 8.
    Grabowski, A., Kornilowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formaliz. Reason. 3(2), 153–245 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. J. Assoc. Comput. Mach. 40(1), 143–184 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Horozal, F., Kohlhase, M., Rabe, F.: Extending MKM formats at the statement level. In: Jeuring, J., Campbell, J.A., Carette, J., Dos Reis, G., Sojka, P., Wenzel, M., Sorge, V. (eds.) Intelligent Computer Mathematics. LNAI, vol. 7362, pp. 65–80. Springer (2012).Google Scholar
  11. 11.
    Iancu, M., Kohlhase, M., Rabe, F.: Translating the Mizar Mathematical Library into OMDoc format. KWARC report, Jacobs University Bremen. https://svn.omdoc.org/repos/latin/public/Mizar2OMDoc-Report.pdf (2011). Accessed 29 Sept 2012
  12. 12.
    Iancu, M., Rabe, F.: Formalizing foundations of mathematics. Math. Struct. Comput. Sci. 21(4), 883–911 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kohlhase, M., Matican, B.A., Prodescu, C.C.: MathWebSearch 0.5 – Scaling an Open Formula Search Engine. Jeuring, J., Campbell, J.A., Carette, J., Dos Reis, G., Sojka, P., Wenzel, M., Sorge, V. (eds.) Intelligent Computer Mathematics. LNAI, vol. 7362, pp. 342–357. Springer (2012)Google Scholar
  14. 14.
    Kohlhase, M.: OMDoc—an open markup format for mathematical documents [version 1.2]. In: LNAI, vol. 4180. Springer (2006)Google Scholar
  15. 15.
    Kohlhase, M., Rabe, F., Zholudev, V.: Towards MKM in the large: modular representation and scalable software architecture. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.) Intelligent Computer Mathematics. LNAI, vol. 6167. Springer (2010)Google Scholar
  16. 16.
    Kohlhase, M., Şucan, I.: A search engine for mathematical formulae. In: Ida, T., Calmet, J., Wang, D. (eds.) Proceedings of Artificial Intelligence and Symbolic Computation, AISC’2006. LNAI, vol. 4120, pp. 241–253. Springer (2006)Google Scholar
  17. 17.
    Pfenning, F., Schürmann, C.: System description: Twelf—a meta-logical framework for deductive systems. Lect. Notes Comput. Sci. 1632, 202–206 (1999)CrossRefGoogle Scholar
  18. 18.
    Rabe, F.: The MMT System. See https://trac.kwarc.info/MMT/ (2008). Accessed 29 Sept 2012
  19. 19.
    Rabe, F., Kohlhase, M.: A scalable module system. See http://arxiv.org/abs/1105.0548 (2011). Accessed 29 Sept 2012
  20. 20.
    Trybulec, A., Blair, H.: Computer assisted reasoning with Mizar. In: Proceedings of the 9th International Joint Conference on Artificial Intelligence, pp. 26–28 (1985)Google Scholar
  21. 21.
    Trybulec, A., Rudnicki, P.: On equivalents of well-foundedness. J. Autom. Reasoning 23(3–4), 197–234 (1999)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Trybulec, A.: Tarski Grothendieck set theory. Formaliz. Math. 1(1), 9–11 (1990)Google Scholar
  23. 23.
    Urban, J., Alama, J., Rudnicki, P., Geuvers, H.: A wiki for Mizar: Motivation, considerations, and initial prototype. In: Autexier et al. (eds.) 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, 5–10 July 2010. Proceedings, Lecture Notes in Computer Science, vol. 6167, pp. 455–469. Springer (2010)Google Scholar
  24. 24.
    Urban, J.: MoMM—fast interreduction and retrieval in large libraries of formalized mathematics. Int. J. Artif. Intell. Tools 15(1), 109–130 (2006)CrossRefGoogle Scholar
  25. 25.
    Urban, J.: MPTP 0.2: Design, implementation, and initial experiments. J. Autom. Reasoning 37(1–2), 21–43 (2006)zbMATHGoogle Scholar
  26. 26.
    Urban, J.: XML-izing Mizar: making semantic processing and presentation of MML easy. In: Kohlhase, M. (ed.) Mathematical Knowledge Management, MKM’05. LNAI, vol. 3863, pp. 346–360. Springer (2006)Google Scholar
  27. 27.
    Urban, J., Sutcliffe, G.: ATP-based cross-verification of Mizar proofs: method, systems, and first experiments. Math. Comput. Sci. 2(2), 231–251 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Urban, J., Sutcliffe, G.: Automated reasoning and presentation support for formalizing mathematics in Mizar. In: Autexier et al. (eds.) 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, 5–10 July 2010. Proceedings, Lecture Notes in Computer Science, vol. 6167, pp. 132–146. Springer (2010)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Mihnea Iancu
    • 1
  • Michael Kohlhase
    • 1
  • Florian Rabe
    • 1
  • Josef Urban
    • 2
  1. 1.Computer ScienceJacobs University BremenBremenGermany
  2. 2.Computing and Information SciencesRadboud UniversityNijmegenThe Netherlands

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