Advertisement

Searching for Geometric Theorems Using Features Retrieved from Diagrams

  • Wenya An
  • Xiaoyu Chen
  • Dongming Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)

Abstract

Searching for knowledge objects from knowledge bases is a basic problem that need be investigated in the context of knowledge management. For geometric knowledge objects such as theorems, natural language representations may not exactly reveal the features and structures of geometric entities, and that is why keyword-based searching is often unsatisfactory. To obtain high-quality results of searching for theorems in plane Euclidean geometry with images of diagrams as input, we propose a method using geometric features retrieved from the images. The method consists of four main steps: (1) retrieve geometric features, with formal representations, from an input image of a diagram D using pattern recognition and numerical verification; (2) construct a graph G corresponding to D from the retrieved features and weaken G to match graphs produced from formal representations of theorems in OpenGeo, an open geometric knowledge base; (3) calculate the degree of relevance between G and the graph for each theorem found from OpenGeo; (4) rank the resulting theorems according to their degrees of relevance. This method, based on graph matching, takes into account the structures of diagrams and works effectively. It is capable of finding out theorems of higher degree of relevance and may have potential applications in geometric knowledge management and education.

Keywords

Theorem searching Graph matching Degree of relevance Knowledge management 

Notes

Acknowledgements

This work has been supported by the project SKLSDE-2015ZX-18 and the NSFC project 11371047.

References

  1. 1.
    Chen, X.: Representation and automated transformation of geometric statements. J. Syst. Sci. Complexity 27(2), 382–412 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chen, X., Song, D., Wang, D.: Automated generation of geometric theorems from images of diagrams. Ann. Math. Artif. Intell. 74(3–4), 333–358 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, X., Wang, D.: Formalization and specification of geometric knowledge objects. Math. Comput. Sci. 7(4), 439–454 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chou, S.-C.: Mechanical Geometry Theorem Proving. D. Reidel, Dordrecht (1988)MATHGoogle Scholar
  5. 5.
    Dehmer, M., Emmert-Streib, F., Kilian, J.: A similarity measure for graphs with low computational complexity. Appl. Math. Comput. 182, 447–459 (2006)MathSciNetMATHGoogle Scholar
  6. 6.
    Einwohner, T.H., Fateman, R.J.: Searching techniques for integral tables. In: 1995 International Symposium on Symbolic and Algebraic Computation, pp. 133–139. ACM (1995)Google Scholar
  7. 7.
    Giugno, R., Shasha, D.: GraphGrep: a fast and universal method for querying graphs. In: 16th International Conference in Pattern Recognition, pp. 112–115. IEEE (2002)Google Scholar
  8. 8.
    Grottke, S., Jeschke, S., Natho, N., Seiler, R.: mArachna: a classification scheme for semantic retrieval in elearning environments in mathematics. Recent Research Developments in Learning Technologies (2005)Google Scholar
  9. 9.
    Haralambous, Y., Quaresma, P.: Querying geometric figures using a controlled language, ontological graphs and dependency lattices. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) Intelligent Computer Mathematics (CICM 2014). LNAI, vol. 8543, pp. 298–311. Springer, Heidelberg (2014)Google Scholar
  10. 10.
    Hashimoto, H., Hijikata, Y., Nishida, S.: Incorporating breadth first search for indexing MathML objects. In: The International Conference on Systems, Man and Cybernetics, pp. 3519–3523. IEEE (2008)Google Scholar
  11. 11.
    Kamali, S., Tompa, F.W.: Improving mathematics retrieval. In: Towards a Digital Mathematics Library, pp. 37–48 (2009)Google Scholar
  12. 12.
    Kamali, S., Tompa, F.W.: A new mathematics retrieval system. In: 19th ACM International Conference on Information and Knowledge Management, pp. 1413–1416. ACM (2010)Google Scholar
  13. 13.
    Kohlhase, M., Sucan, I.: A search engine for mathematical formulae. In: Calmet, J., Ida, T., Wang, D. (eds.) Artificial Intelligence and Symbolic Computation (AISC 2006). LNAI, vol. 4120, pp. 241–253. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Kumar, S., Manjeet, S., Avik, D.: OWL-based ontology indexing and retrieving algorithms for Semantic Search Engine. In: The 7th International Conference of Computing and Convergence Technology, pp. 1135–1140. IEEE (2012)Google Scholar
  15. 15.
    Raymond, J.W., Gardiner, E.J., Willett, P.: RASCAL: calculation of graph similarity using maximum common edge subgraphs. Comput. J. 45, 631–644 (2002)CrossRefMATHGoogle Scholar
  16. 16.
    Singhal, A.: Modern information retrieval: a brief overview. IEEE Data Eng. Bull. 24, 35–43 (2001)Google Scholar
  17. 17.
    Song, D., Wang, D., Chen, X.: Discovering geometric theorems from scanned and photographed images of diagrams. In: Botana, F., Quaresma, P. (eds.) Automated Deduction in Geometry (ADG 2014). LNAI, vol. 9201, pp. 149–165. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  18. 18.
    Wang, D., Chen, X., An, W., et al.: OpenGeo: an open geometric knowledge base. In: Hong, H., Yap, C. (eds.) Mathematical Software (ICMS 2014). LNCS, vol. 8592, pp. 240–245. Springer, Heidelberg (2014)Google Scholar
  19. 19.
    Wical, K.: Concept knowledge base search and retrieval system. Patent 6,038,560. U.S (2000)Google Scholar
  20. 20.
    Woods, W.A.: Knowledge base retrieval. In: Brodie, M.L., Mylopoulos, J. (eds.) On Knowledge Base Management Systems, pp. 179–195. Springer, New York (1986)CrossRefGoogle Scholar
  21. 21.
    Zanibbi, R., Blostein, D.: Recognition and retrieval of mathematical expressions. Int. J. Doc. Anal. Recogn. 15, 331–357 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.LMIB – SKLSDE – School of Mathematics and Systems ScienceBeihang UniversityBeijingChina

Personalised recommendations