Abstract
We describe two algorithms for closure systems. The purpose of the first is to produce all closed sets of a given closure operator. The second constructs a minimal family of implications for the ”logic” of a closure system. These algorithms then are applied to problems in concept analysis: Determining all concepts of a given context and describing the dependencies between attributes. The problem of finding all concepts is equivalent, e.g., to finding all maximal complete bipartite subgraphs of a bipartite graph.
A version of this preprint was published in German as part of the book ”Beiträge zur Begriffsanalyse” (Ganter, Wille and Wolff, eds., BI-Wissenschaftsverlag 1987). Computer programs for FCA were already in use in 1984. The new algorithm was not only more efficient, it could also be used to compute the ”canonical base” of implications in a finite formal context that had recently been discovered by J.L. Guigues and V. Duquenne.
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References
Fay, G.: An algorithm for finite Galois connections. Technical report, Institute for Industrial Economy, Budapest (1973)
Guigues, J.-L., Duquenne, V.: Informative implications derived from a table of binary data. Preprint, Groupe Mathématiques et Psychologie, Université René descartes, Paris (1984)
Norris, E.M.: An algorithm for computing the maximal rectangles in a binary relation. Rev. Roum. Math. Pures et Appl., Tome XXIII(2), 243–250 (1978); Bucarest
Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered sets, pp. 445–470. Reidel Publ. Comp., Dordrecht (1982)
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Ganter, B. (2010). Two Basic Algorithms in Concept Analysis. In: Kwuida, L., Sertkaya, B. (eds) Formal Concept Analysis. ICFCA 2010. Lecture Notes in Computer Science(), vol 5986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11928-6_22
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DOI: https://doi.org/10.1007/978-3-642-11928-6_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11927-9
Online ISBN: 978-3-642-11928-6
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