The Reverse Doubling Construction

Jean-François Viaud, Karell Bertet, Christophe Demko, Rokia Missaoui

Abstract

It is well known inside the Formal Concept Analysis (FCA) community that a concept lattice could have an exponential size in the data. Hence, the size of concept lattices is a critical issue in the presence of large real-life data sets. In this paper, we propose to investigate factor lattices as a tool to get meaningful parts of the whole lattice. These factor lattices have been widely studied from the early theory of lattices to more recent work in the FCA field. This paper contains two parts. The first one gives background about lattice theory and formal concept analysis, and mainly compatible sub-contexts, arrow-closed sub-contexts and congruence relations. The second part presents a new decomposition called “reverse doubling construction” that exploits the above three notions used for the doubling convex construction investigated by Day. Theoretical results and their proofs are given as well as an illustrative example.

References

  1. Barbut, M. and Monjardet, B., editors (1970). L'ordre et la classification. Algèbre et combinatoire, tome II. Hachette.
  2. Belohlavek, R. and Vychodil, V. (2010). Discovery of optimal factors in binary data via a novel method of matrix decomposition. Journal of Computer and System Sciences, 76(1):3-20.
  3. Bertet, K. and Caspard, N. (2002). Doubling convec sets in lattices: characterizations and recognition algorithms. Technical Report TR-LACL-2002-08, LACL (Laboratory of Algorithms, Complexity and Logic), University of Paris-Est (Paris 12).
  4. Day, A. (1977). Splitting lattices generate all lattices. algebra universalis, 7(1):163-169.
  5. Day, A. (1994). Congruence normality: The characterization of the doubling class of convex sets. algebra universalis, 31(3):397-406.
  6. Day, A., Nation, J., and Tschantz, S. (1989). Doubling convex sets in lattices and a generalized semidistributivity condition. Order, 6(2):175-180.
  7. Demel, J. (1982). Fast algorithms for finding a subdirect decomposition and interesting congruences of finite algebras. Kybernetika (Prague), 18(2):121-130.
  8. Duquenne, V. (2010). Lattice drawings and morphisms. In Formal Concept Analysis, 8th International Conference, ICFCA 2010, Agadir, Morocco, March 15-18, 2010. Proceedings, pages 88-103.
  9. Ferré, S. (2014). Reconciling expressivity and usability in information access from file systems to the semantic web. Rapport hdr, University Rennes 1.
  10. Freese, R. (1997). Computing congruence lattices of finite lattices. Proceedings of the American Mathematical Society, 125(12):3457-3463.
  11. Freese, R. (1999). Algorithms in finite, finitely presented and free lattices. Preprint, July, 22:159-178.
  12. Freese, R. (2008). Computing congruences efficiently. Algebra universalis, 59(3-4):337-343.
  13. Funk, P., Lewien, A., and Snelting, G. (1995). Algorithms for concept lattice decomposition and their applications. Technical report, TU Braunschweig.
  14. Ganter, B. and Wille, R. (1999). Formal concept analysis - mathematical foundations. Springer.
  15. Geyer, W. (1994). The generalized doubling construction and formal concept analysis. algebra universalis, 32(3):341-367.
  16. Mihók, P. and Semanis˜in, G. (2008). Unique factorization theorem and formal concept analysis. In Yahia, S., Nguifo, E., and Belohlavek, R., editors, Concept Lattices and Their Applications, volume 4923 of Lecture Notes in Computer Science, pages 232-239. Springer Berlin Heidelberg.
  17. Nation, J. (1995). Alan day's doubling construction. algebra universalis, 34(1):24-34.
  18. Snelting, G. (2005). Concept lattices in software analysis. In Formal Concept Analysis, Foundations and Applications, pages 272-287.
  19. Viaud, J.-F., Bertet, K., Demko, C., and Missaoui, R. (2015). Subdirect decomposition of contexts into subdirectly irreducible factors. Formal Concept Analysis and Applications FCA&A 2015, page 49.
  20. Visani, M., Bertet, K., and Ogier, J.-M. (2011). Navigala: an Original Symbol Classifier Based on Navigation through a Galois Lattice. International Journal on Pattern Recognition and Artificial Intelligence (IJPRAI).
  21. Wille, R. (1969). Subdirekte produkte und konjunkte summen. Journal für die reine und angewandte Mathematik, 0239 0240:333-338.
  22. Wille, R. (1976). Subdirekte Produkte vollständiger Verbände. J. reine angew. Math., 283/284:53-70.
  23. Wille, R. (1983). Subdirect decomposition of concept lattices. Algebra Universalis, 17:275-287.
  24. Wille, R. (1987). Subdirect product construction of concept lattices. Discrete Mathematics, 63(2-3):305-313.
Download


Paper Citation


in Harvard Style

Viaud J., Bertet K., Demko C. and Missaoui R. (2015). The Reverse Doubling Construction . In Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management - Volume 1: KDIR, (IC3K 2015) ISBN 978-989-758-158-8, pages 350-357. DOI: 10.5220/0005613203500357


in Bibtex Style

@conference{kdir15,
author={Jean-François Viaud and Karell Bertet and Christophe Demko and Rokia Missaoui},
title={The Reverse Doubling Construction},
booktitle={Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management - Volume 1: KDIR, (IC3K 2015)},
year={2015},
pages={350-357},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005613203500357},
isbn={978-989-758-158-8},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management - Volume 1: KDIR, (IC3K 2015)
TI - The Reverse Doubling Construction
SN - 978-989-758-158-8
AU - Viaud J.
AU - Bertet K.
AU - Demko C.
AU - Missaoui R.
PY - 2015
SP - 350
EP - 357
DO - 10.5220/0005613203500357