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Minimal generators, an affordable approach by means of massive computation

  • F. Benito-Picazo
  • P. Cordero
  • M. Enciso
  • A. Mora
Article
  • 37 Downloads

Abstract

Closed sets and minimal generators are fundamental elements to build a complete knowledge representation in formal concept analysis. The enumeration of all the closed sets and their minimal generators from a set of rules or implications constitutes a complex problem, drawing an exponential cost. Even for small datasets, such representation can demand an exhaustive management of the information stored as attribute implications. In this work, we tackle this problem by merging two strategies. On the one hand, we design a pruning, strongly based on logic properties, to drastically reduce the search space of the method. On the other hand, we consider a parallelization of the problem leading to a massive computation by means of a map-reduce like paradigm. In this study we have characterized the type of search space reductions suitable for parallelization. Also, we have analyzed different situations to provide an orientation of the resources (number of cores) needed for both the parallel architecture and the size of the problem in the splitting stage to take advantage in the map stage.

Keywords

Minimal generators Formal concept analysis Parallel methods Logic 

Notes

Acknowledgements

The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the Supercomputing and Bioinnovation Center of the University of Málaga - Andalucía Tech (SCBI), particularly to Dr. Rafael Larrosa and Dr. Darío Guerrero. We also want to mention the orientation provided by Dr. José Antonio Onieva to identify the properties of our algorithm for a better classification of its design.

References

  1. 1.
    Armstrong WW (1974) Dependency structures of data base relationships. In: IFIP Congress, pp 580–583Google Scholar
  2. 2.
    Benito-Picazo F, Cordero P, Enciso M, Mora A (2017) Reducing the search space by closure and simplification paradigms. J Supercomput 73(1):75–87CrossRefGoogle Scholar
  3. 3.
    Buchi JR, Siefkes D (1990) Finite automata, their algebras and grammars. Springer, New York Inc, SecaucuszbMATHGoogle Scholar
  4. 4.
    Codd EF (1970) A relational model of data for large shared data banks. Commun ACM 13(6):377–387CrossRefzbMATHGoogle Scholar
  5. 5.
    Cohen E, Datar M, Fujiwara S, Gionis A, Indyk P, Motwani R, Ullman JD, Yang C (2001) Finding interesting associations without support pruning. IEEE Trans Knowl Data Eng 13(1):64–78CrossRefGoogle Scholar
  6. 6.
    Cordero P, Enciso M, Mora A, Ojeda-Aciego M (2012) Computing minimal generators from implications: a logic-guided approach. In Szathmary L, Priss U (eds) Proceedings of the Ninth International Conference on Concept Lattices and Their Applications, Fuengirola (Málaga), Spain, October 11–14, 2012, volume 972 of CEUR Workshop Proceedings, pp 187–198. CEUR-WS.orgGoogle Scholar
  7. 7.
    Cordero P, Mora A, Enciso M, de Guzmán IP (2002) SLFD logic: elimination of data redundancy in knowledge representation. Lect Notes Comput Sci 2527:141–150CrossRefzbMATHGoogle Scholar
  8. 8.
    de Moraes NRM, Dias SM, Freitas HC, Zárate LE (2016) Parallelization of the next closure algorithm for generating the minimum set of implication rules. Artif Intell Res 5(2):40–54CrossRefGoogle Scholar
  9. 9.
    Doignon J, Falmagne J (1998) Knowledge spaces. Springer, BerlinzbMATHGoogle Scholar
  10. 10.
    Dong GZ, Jiang CY, Pei J, Li JY, Wong L (2005) Mining succinct systems of minimal generators of formal concepts. Proc Database Syst Adv Appl 3453:175–187CrossRefGoogle Scholar
  11. 11.
    Ganter B, Wille R (1999) Formal concept analysis: mathematical foundations. Springer, BerlinCrossRefzbMATHGoogle Scholar
  12. 12.
    Guigues JL, Duquenne V (1986) Famille minimale d’implications informatives résultant d’un tableau de données binaires. Math Sci Hum 24(95):5–18Google Scholar
  13. 13.
    Harper FM, Konstan JA (2015) The movielens datasets: history and context. ACM Trans Interact Intell Syst 5(4):19:1–19:19CrossRefGoogle Scholar
  14. 14.
    Hu X, Wei X, Wang D, Li P (2007) A parallel algorithm to construct concept lattice. In Lei J (ed) Proceedings of the Fourth International Conference on Fuzzy Systems and Knowledge Discovery, FSKD 2007, 24–27 August 2007, Haikou, Hainan, China, vol 2, pp 119–123. IEEE Computer SocietyGoogle Scholar
  15. 15.
    Kuznetsov SO, Obiedkov SA (2002) Comparing performance of algorithms for generating concept lattices. J Exp Theor Artif Intell 14(2–3):189–216CrossRefzbMATHGoogle Scholar
  16. 16.
    Maier D (1983) The theory of relational databases. Computer Science Press, RockvillezbMATHGoogle Scholar
  17. 17.
    Missaoui R, Nourine L, Renaud Y (2010) An inference system for exhaustive generation of mixed and purely negative implications from purely positive ones. In: CEUR Workshop Proceedings, vol 672, pp 271–282Google Scholar
  18. 18.
    Missaoui R, Nourine L, Renaud Y (2012) Computing implications with negation from a formal context. Fundam Inf 115(4):357–375MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mora A, Enciso M, Cordero P, Fortes I (2012) Closure via functional dependence simplification. Int J Comput Math 89(4):510–526MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nishio N, Mutoh A, Inuzuka N (2012) On computing minimal generators in multi-relational data mining with respect to 0-subsumption. In: CEUR Workshop Proceedings, vol 975, pp 50–55Google Scholar
  21. 21.
    Poelmans J, Ignatov DI, Kuznetsov SO, Dedene G (2013) Formal concept analysis in knowledge processing: a survey on applications. Expert Syst Appl 40(16):6538–6560CrossRefGoogle Scholar
  22. 22.
    Qu K, Zhai Y, Liang J, Chen M (2007) Study of decision implications based on formal concept analysis. Int J Gen Syst 36(2):147–156MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rodríguez-Lorenzo E, Cordero P, Enciso M, Mora Á (2017) Canonical dichotomous direct bases. Inf Sci 376:39–53CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Universidad de MálagaMálagaSpain

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