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Lobachevskii Journal of Mathematics

, Volume 35, Issue 4, pp 348–354 | Cite as

Mathematical knowledge representation: semantic models and formalisms

  • A. M. Elizarov
  • A. V. Kirillovich
  • E. K. Lipachev
  • O. A. Nevzorova
  • V. D. Solovyev
  • N. G. Zhiltsov
Article

Abstract

The paper provides a survey of semantic methods for solution of fundamental tasks in mathematical knowledge management. Ontological models and formalisms are discussed. We propose an ontology of mathematical knowledge, covering wide range of fields of mathematics. We demonstrate applications of this representation in mathematical formula search, and learning.

Keywords and phrases

Ontology engineering mathematical knowledge metadata extraction information retrieval math formula search 

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References

  1. 1.
    D. E. Knuth, The TEX book (Addison-Wesley Publishing Company, 1986).zbMATHGoogle Scholar
  2. 2.
    CTAN. Comprehensive TEX Archive Network. URL: http://www.ctan.org/
  3. 3.
    Wolfram Mathematica. URL: http://www.wolfram.com/mathematica/
  4. 4.
    WolframAlpha computational knowledge engine. URL: http://www.wolframalpha.com/
  5. 5.
    S. Wolfram, A New Kind of Science (WolframMedia, Inc., 2002).zbMATHGoogle Scholar
  6. 6.
    Math Jax. Beautiful math in all browsers. URL: http://www.mathjax.org/
  7. 7.
    D. Cervone, Notices of the American Mathematical Society 59(2), 312 (2012).CrossRefzbMATHGoogle Scholar
  8. 8.
    ASCIIMathML.js (ver 2.0): Translating ASCII math notation to MathML and graphics. URL: http://www1.chapman.edu/~jipsen/mathml/asciimath.html
  9. 9.
    A. B. Zhizhchenko and A. D. Izaak, Russian Math. Surveys 62(5), 943 (2007).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    S. Kuhn, T. Helmus, R. J. Lancashire, P. Murray-Rust, H. S. Rzepa, C. Steinbeck, and E. L. Willighagen, J. Chem. Inf. Mod. 47(6), 2015 (2007).CrossRefGoogle Scholar
  11. 11.
    P. Murray-Rust, Journal of Cheminformatics, 3, 48 (2011).CrossRefGoogle Scholar
  12. 12.
    MKM-IG. Mathematical Knowledge Management. URL: http://www.mkm-ig.org/
  13. 13.
    W. Sperber, “Search engines and bibliographic databases,” in A Focus on Mathematics, Ed. by B. Wegner and Staff Unit Communications (FIZ Karlsruhe, 26–30, 2008).Google Scholar
  14. 14.
    J. Carette, W. M. Farmer, “A Review of Mathematical Knowledge Management,” in Intelligent Computer Mathematics. Lecture Notes in Computer Science 5625, 233 (2009).CrossRefGoogle Scholar
  15. 15.
    P.D. F. Ion, “Mathematics and the World Wide Web,” in Intelligent Computer Mathematics. Lecture Notes in Computer Science 7961, 230 (2013).CrossRefGoogle Scholar
  16. 16.
    C. Lange, Semantic Web. 4(2), 119 (2013).Google Scholar
  17. 17.
    H. Barendregt and F. Wiedijk, Transactions A of the Royal Society, 363(1835), 2351 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    A.M. Elizarov, E. K. Lipachev, O. A. Nevzorova, and V. D. Solov’ev, Doklady Math. 90(1), 521 (2014).CrossRefGoogle Scholar
  19. 19.
    E. V. Biryal’tsev, A. M. Elizarov, N. G. Zhil’tsov, E. K. Lipachev, O. A. Nevzorova, and V. D. Solov’ev, Automatic Documentation and Mathematical Linguistics 48(2), 81 (2014).CrossRefGoogle Scholar
  20. 20.
    S. Parinov and M. Kogalovsky, Applied Informatics 6 (2009); URL: http://www.ipr-ras.ru/articles/koga-pari09-2.pdf.
  21. 21.
    Liber Mathematicae, URL: http://math.colorado.edu/libermath/.
  22. 22.
    M. J. Pflaum, J. Tuley, arXiv:1102.5720.Google Scholar
  23. 23.
    Computable Document Format (CDF) for Interactive Content, URL: http://wolfram.com/cdf
  24. 24.
    Mizar Project, URL: http://mizar.org/project/.
  25. 25.
    The Coq Proof Assistant, URL: http://coq.inria.fr/.
  26. 26.
    T. Berners-Lee, J. Hendler, and O. Lassila, Scientific American 284(5), 28 (2001).CrossRefGoogle Scholar
  27. 27.
    R. Ausbrooks et al., Mathematical Markup Language (MathML) Version 3.0. W3C Candidate Recommendation of 15 December 2009. World Wide Web Consortium 13 (2009).Google Scholar
  28. 28.
    R. Miner, Notices of the AMS 52, 532 (2005).zbMATHMathSciNetGoogle Scholar
  29. 29.
    Word to LaTeX, LaTeX toWord Converters, URL: http://www.tex2word.com/.
  30. 30.
    A LaTeX to XML/HTML/MathML Converter, URL: http://dlmf.nist.gov/LaTeXML/.
  31. 31.
    M. Kohlhase, OMDoc-An open markup format for mathematical documents [Version 1.2] (Berlin: Springer, 2006).CrossRefGoogle Scholar
  32. 32.
    M. Iancu, M. Kohlhase, F. Rabe, and J. Urban, Journal of Automated Reasoning 50(2), 191 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    C. David, M. Kohlhase, C. Lange, F. Rabe, N. Zhiltsov, and V. Zholudev, Proc. 7th Extended SemanticWeb Conference (ESWC) 370 (2010); URL: http://arxiv.org/pdf/1004.3390.pdf.Google Scholar
  34. 34.
    M. Kohlhase, STEX: Semantic Markup in TEX/LATEX (2005); URL: https://svn.kwarc.info/repos/stex/trunk/sty/stex.pdf.Google Scholar
  35. 35.
    M. Kohlhase, Math. Comput. Sci. 2, 279 (2008).CrossRefzbMATHGoogle Scholar
  36. 36.
    F. Kamareddine and J. B. Wells, Electr. Notes Theor. Comput. Sci. 205(C), 5 (2008); URL: http://www.sciencedirect.com/science/article/pii/S1571066108001680.CrossRefMathSciNetGoogle Scholar
  37. 37.
    V. Solovyev and N. Zhiltsov, Proceedings of the International Conference on Web Intelligence, Mining and Semantics (WIMS’11). ACM, 21:1–21:9 (2011).Google Scholar
  38. 38.
    S. Auer, C. Bizer, G. Kobilarov, J. Lehmann, R. Cyganiak, and Z. Ives, in The Semantic Web (Springer Berlin Heidelberg, 2007), p. 722.CrossRefGoogle Scholar
  39. 39.
    K. Aberer, A. Boyarsky, P. Cudré-Mauroux, G. Demartini, and O. Ruchayskiy, in 10th International Semantic Web Conference (ISWC 2011 — Demo, 2011).Google Scholar
  40. 40.
    N. Sloane, Notices of the AMS 50(8), 912 (2003).zbMATHMathSciNetGoogle Scholar
  41. 41.
    R. Thomas, MSOR Connections 4(3) (2004).Google Scholar
  42. 42.
    M. R. Genesereth and N. J. Nilsson, Logical Foundations of Artificial Intelligence (Morgan Kaufmann, Los Altos, CA, 1987).zbMATHGoogle Scholar
  43. 43.
    T. R. Gruber, Knowledge Acquisition 5(2), 199 (1993).CrossRefGoogle Scholar
  44. 44.
    R. Studer, R. Benjamins, and D. Fensel, Data & Knowledge Engineering. 25(1–2), 161 (1998).CrossRefzbMATHGoogle Scholar
  45. 45.
    N. Guarino, D. Oberle, S. Staab, in Handbook on Ontologies, 2nd ed. (Springer, New York, 2009), pp. 1–17.CrossRefGoogle Scholar
  46. 46.
    J. Angele, M. Kifer, and G. Lausen, in Handbook on Ontologies, 2nd ed. (Springer, New York, 2009), pp. 45–68.CrossRefGoogle Scholar
  47. 47.
    F. Baader, I. Horrocks, and U. Sattler, in Handbook on Ontologies, 2nd ed. (Springer, New York, 2009), pp. 21–43.CrossRefGoogle Scholar
  48. 48.
    P. Hitzler, M. Krötzsch, B. Parsia, P. F. Patel-Schneider, and S. Rudolph (eds,.) OWL 2 Web Ontology Language Primer, http://www.w3.org/TR/owl2-primer/.
  49. 49.
    O. Nevzorova, N. Zhiltsov, A. Kirillovich, and E. Lipachev, KESW 2014, CCIS 468. 105–119 (2014); URL: http://arxiv.org/abs/1407.4833.Google Scholar
  50. 50.
    A. M. Elizarov, E. K. Lipachev, and M. A. Malakhaltsev, Web Technologies for Mathematicians: The Basics of MathML. A Practical Guide (Fizmatlit, Moscow, 2010) [In Russian].Google Scholar
  51. 51.
    O. Nevzorova, N. Zhiltsov, D. Zaikin, O. Zhibrik, A. Kirillovich, V. Nevzorov, and E. Birialtsev, in 12th Int. Semantic Web Conference, Sydney, NSW, Australia, October 21–25, 2013, Proceedings (Springer, Berlin, 2013), Part 1, Vol. 8218, pp. 379–394.Google Scholar
  52. 52.
    D. Velleman, How to Prove It: A Structured Approach, 2nd ed. (Cambridge University Press, 2006).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • A. M. Elizarov
    • 1
  • A. V. Kirillovich
    • 1
  • E. K. Lipachev
    • 1
  • O. A. Nevzorova
    • 1
  • V. D. Solovyev
    • 1
  • N. G. Zhiltsov
    • 1
  1. 1.Higher Institute of Information Technologies and Information SystemsKazanTatarstan, Russia

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