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Representing attribute reduction and concepts in concept lattice using graphs

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Abstract

Concept lattice is an area of research which is based on a set-theoretical model for concepts and conceptual hierarchies. It is better for the studying of concept lattice to minimize the input data before revealing the construction of a concept lattice. Actually, this duty can be done by attribute reduction for a context. Graph is useful in data analysis since it gives us a visual trend on the behavior of our data points and allows us to test some laws in data analysis. This paper is a preliminary attempt to study how directed graph can be used on attribute reduction and conceptual construction in concept lattices. We investigate all the reducible attributes and concepts in a context with the aid of graph theory. For a context, we define a relevant graph on the set of attributes and, further, define a pre-weighted relevant graph. Afterward, using relevant graphs and pre-weighted relevant graphs with the method of deleting vertices in a directed graph, we find all the reducible attributes in a context. After that, we seek out all of concepts and the concept lattice for a given context. All these results may be not only used to improve the visibility and readability of attribute reduction in a context and the construction of concept lattice, but also broaden the applied range of directed graph such as in the field of attribute reduction.

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References

  • Abello J, Pogel AJ, Miller L (2004) Breadth first search graph partitions and concept lattices. J. Univers Comput Sci 10(8):934–954

    MATH  MathSciNet  Google Scholar 

  • Amilhastre J, Vilarem MC, Janssen P (1998) Complexity of minimum biclique cover and minimum biclique decomposition for bipartite deminofree graphs. Discret Appl Math 86:125–144

    Article  MATH  Google Scholar 

  • Apollonio N, Caramia M, Franciosa PG (2015) On the Galois lattice of bipartite distance hereditary graphs. Discret Appl Math 190–191:13–23

    Article  MATH  MathSciNet  Google Scholar 

  • Bělohlǎvek R, Baets BD, Outrata J, Vychodil V (2008) Characterizing trees in concept lattices. Int J Uncertain Fuzz Knowl-Based Syst 16(1):1–15

    Article  MATH  MathSciNet  Google Scholar 

  • Berry A, Sigayret A (2004) Representing a concept lattice by a graph. Discret Appl Math 144:27–42

    Article  MATH  MathSciNet  Google Scholar 

  • Bondy JA, Murty USR (2008) Graph theory. Springer, San Francisco

    Book  MATH  Google Scholar 

  • Davey BA, Priestley HA (2002) Introduction to lattice and order, 2nd edn. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Dias SM, Vieira NJ (2015) Concept lattices reduction: definition, analysis and classification. Expert Syst Appl 42(20):7084–7097

    Article  Google Scholar 

  • Ganter B, Wille R (1999) Formal concept analysis: mathematical foundations. Springer, Berlin (German version in 1996)

    Book  MATH  Google Scholar 

  • Ghosh P, Kundu K, Sarkar D (2010) Fuzzy graph representation of a fuzzy concept lattice. Fuzzy Sets Syst 161:1669–1675

    Article  MATH  MathSciNet  Google Scholar 

  • Mao H (2012) Representing a concept lattice by neighborly ideas. Analele Universitatii de Vest, Timisoara, serial Matematica-Informatica 50(1):77–82

    MATH  MathSciNet  Google Scholar 

  • Mao H (2014) A graph-theoretic method to representing a concept lattice. Appl Math Inf Sci 8(2):553–561

    Article  MathSciNet  Google Scholar 

  • Mao H (2014) Characterization and reduction of concept lattices through matroid theory. Inf Sci 28:338–354

    Article  MATH  MathSciNet  Google Scholar 

  • Poelmans J, Kuznetsov SO, Ignatov DI, Dedene G (2013) Formal concept analysis in knowledge processing: a survey on applications. Expert Syst Appl 40(16):6538–6560

    Article  Google Scholar 

  • Singh PK, Kumar CA (2014) Bipolar fuzzy graph representation of concept lattice. Inform Sci 288:437–448

    Article  MATH  MathSciNet  Google Scholar 

  • Singh PK, Kumar CA (2016) A comprehensive survey on formal concept analysis, its research trends and applications. Int J Appl Math Comput Sci 26(2):495–516

    Article  MATH  MathSciNet  Google Scholar 

  • Škopljanac-Mačina F, Blašković B (2014) Formal concept analysis—overview and applications. Proced Eng 69:1258–1267

    Article  Google Scholar 

  • Wei L, Wan Q (2016) Granular transformation and irreducible element judgment theory based on pictorial diagrams. IEEE Trans Cybern 46(2):380–387

    Article  Google Scholar 

  • Wille R (1982) Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival I (ed) Ordered sets. Reidel, Dordrech, pp 445–470

    Chapter  Google Scholar 

  • Wille R (1989) Lattices in data analysis; how to draw them with a computer. In: Rival I (ed) Algorithms and order. Kluwer Academic Publishers, London, pp 33–58

    Chapter  Google Scholar 

  • Xu WH, Zhang XY, Zhong JM, Zhang WX (2010) Attribute reduction in ordered information systems based on evidence theory. Knowl Inf Syst 25:169–184

    Article  Google Scholar 

  • Zhang, T., Li, H., Hong, W.X., Yuan, X., Wei, X.Y.: Deep first formal concept search. The Scientific World Journal, vol.2014, Article ID 275679, 13 pages (2014)

  • Zhang WX, Wei L, Qi JJ (2005) Attribute reduction theory and approach to concept lattice. Science in China Ser. F. Inf Sci 48(6):712–726

    Google Scholar 

Download references

Acknowledgements

The author sincerely acknowledges the financial support from the Natural Science Foundation of China (61572011) and Natural Science Foundation of Hebei Province (A2013201119).

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Authors and Affiliations

  1. Department of Mathematics, Hebei University, Baoding, 071002, China

    Hua Mao

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  1. Hua Mao
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Correspondence to Hua Mao.

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The authors declare that there is no conflict of interest regarding the publication of this paper.

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Communicated by A. Di Nola.

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Mao, H. Representing attribute reduction and concepts in concept lattice using graphs. Soft Comput 21, 7293–7311 (2017). https://doi.org/10.1007/s00500-016-2441-2

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