Advertisement

Rough Concept Analysis – Theory Development in the Mizar System

  • Adam Grabowski
  • Christoph Schwarzweller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3119)

Abstract

Theories play an important role in building mathematical knowledge repositories. Organizing knowledge in theories is an obvious approach to cope with the growing number of definitions, theorems, and proofs. However, they are also a matter of subject on their own: developing a new piece of mathematics often relies on extending or combining already developed theories in this way reusing definitions as well as theorems. We believe that this aspect of theory development is crucial for mathematical knowledge management.

In this paper we investigate the facilities of the Mizar system concerning extending and combining theories based on structure and attribute definitions. As an example we consider the formation of rough concept analysis out of formal concept analysis and rough sets.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bancerek, G.: Development of the theory of continuous lattices in Mizar. In: Kerber, M., Kohlhase, M. (eds.) Proceedings of the Symposium on Calculemus 2000, pp. 65–80 (2001)Google Scholar
  2. 2.
    Bancerek, G.: On the structure of Mizar types. In: Geuvers, H., Kamareddine, F. (eds.) Proc. of MLC 2003. ENTCS, vol. 85(7) (2003)Google Scholar
  3. 3.
    Buchberger, B.: Mathematical Knowledge Management in Theorema. In: Buchberger, B., Caprotti, O. (eds.) Proc. of MKM 2001, Linz, Austria (2001)Google Scholar
  4. 4.
    Cruz-Filipe, L., Geuvers, H., Wiedijk, F.: C-CoRN, the Constructive Coq Repository at Nijmegen, http://www.cs.kun.nl/~freek/notes/
  5. 5.
    Farmer, W., Guttman, J., Thayer, F.: Little theories. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 567–581. Springer, Heidelberg (1992)Google Scholar
  6. 6.
    Farmer, W., Guttman, J., Thayer, F.: IMPS – an Interactive Mathematical Proof System. Journal of Automated Reasoning 11, 213–248 (1993)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ganter, B., Wille, R.: Formal concept analysis – mathematical foundations. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  8. 8.
    Grabowski, A.: Basic properties of rough sets and rough membership function. Formalized Mathematics (to appear, 2004), Available from: [13]Google Scholar
  9. 9.
    Hurd, J.: Verification of the Miller-Rabin probabilistic primality test. Journal of Logic and Algebraic Programming 56, 3–21 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Järvinen, J.: Approximations and rough sets based on tolerances. In: Ziarko, W., Yao, Y. (eds.) RSCTC 2000. LNCS (LNAI), vol. 2005, pp. 182–189. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Kent, R.E.: Rough Concept Analysis: a synthesis of rough sets and formal concept analysis. Fundamenta Informaticae 27(2–3), 169–181 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Loos, R., Musser, D., Schupp, S., Schwarzweller, C.: The Tecton concept library, Technical Report WSI 99-2, Wilhelm-Schickard-Institute for Computer Science, University of Tübingen (1999)Google Scholar
  13. 13.
    The Mizar Home Page, http://mizar.org
  14. 14.
    Musser, D., Shao, Z.: The Tecton concept description language (revised version), Technical Report 02-2, Rensselaer Polytechnic Institute (2002)Google Scholar
  15. 15.
    Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  16. 16.
    Owre, S., Shankar, N.: Theory interpretations in PVS, Technical Report, NASA/CR-2001-211024 (2001)Google Scholar
  17. 17.
    Pawlak, Z.: Rough sets. International Journal of Information and Computer Science 11(5), 341–356 (1982)CrossRefzbMATHGoogle Scholar
  18. 18.
    Raczkowski, K., Sadowski, P.: Equivalence relations and classes of abstraction. Formalized Mathematics 1(3), 441–444 (1990), Available in JFM from: [13]Google Scholar
  19. 19.
    Rudnicki, P., Trybulec, A.: Mathematical Knowledge Management in Mizar. In: Buchberger, B., Caprotti, O. (eds.) Proc. of MKM 2001, Linz, Austria (2001)Google Scholar
  20. 20.
    Rudnicki, P., Trybulec, A.: On the integrity of a repository of formalized mathematics. In: Asperti, A., Buchberger, B., Davenport, J.H. (eds.) MKM 2003. LNCS, vol. 2594, pp. 162–174. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  21. 21.
    Saquer, J., Deogun, J.S.: Concept approximations based on rough sets and similarity measures. International Journal on Applications of Mathematics in Computer Science 11(3), 655–674 (2001)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Schwarzweller, C.: Introduction to concept lattices. Formalized Mathematics 7(2), 233–242 (1998), Available in JFM from: [13]Google Scholar
  23. 23.
    Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, Reidel. Dordrecht-Boston (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Adam Grabowski
    • 1
  • Christoph Schwarzweller
    • 2
  1. 1.Institute of MathematicsUniversity of BiałystokBiałystokPoland
  2. 2.Department of Computer ScienceUniversity of GdańskGdańskPoland

Personalised recommendations