Using congruence relations to extract knowledge from concept lattices

Abstract It is well-known inside the Formal Concept Analysis (FCA) community that a concept lattice could have an exponential size with respect to the input data. Hence, the size of concept lattices is a critical issue in large real-life data sets. In this paper, we propose to investigate congruence relations as a tool to get meaningful parts of the whole lattice or its implication basis. This paper presents two main theoretical contributions, namely two context (or lattice) decompositions based on congruence relations and new results about implication computation after decomposition.

[1]  Karell Bertet,et al.  Doubling Convex Sets in Lattices: Characterizations and Recognition Algorithms , 2002, Order.

[2]  Ralph Freese.  Computing congruence lattices of finite lattices , 1997 .

[3]  Vincent Duquenne.  Lattice Drawings and Morphisms , 2010, ICFCA.

[4]  Alan Day,et al.  Characterizations of Finite Lattices that are Bounded-Homomqrphic Images or Sublattices of Free Lattices , 1979, Canadian Journal of Mathematics.

[5]  Ralph Freese,et al.  Computing congruences efficiently , 2008 .

[6]  Rudolf Wille.  Subdirekte Produkte und konjunkte Summen. , 1969 .

[7]  G. Grätzer.  General Lattice Theory , 1978 .

[8]  Karell Bertet,et al.  Navigala: an Original Symbol Classifier Based on Navigation through a Galois Lattice , 2011, Int. J. Pattern Recognit. Artif. Intell..

[9]  G. Grätzer.  Lattice Theory: Foundation , 1971 .

[10]  Vincent Duquenne,et al.  Attribute-incremental construction of the canonical implication basis , 2007, Annals of Mathematics and Artificial Intelligence.

[11]  James B. Nation,et al.  On implicational bases of closure systems with unique critical sets , 2012, Discret. Appl. Math..

[12]  Rudolf Wille,et al.  Subdirect product construction of concept lattices , 1987, Discret. Math..

[13]  Angel Mora,et al.  From an Implicational System to its Corresponding D-basis , 2015, CLA.

[14]  James B. Nation,et al.  Alan Day's doubling construction , 1995 .

[15]  Alan Day,et al.  Doubling convex sets in lattices and a generalized semidistributivity condition , 1989 .

[16]  Gregor Snelting.  Concept Lattices in Software Analysis , 2005, Formal Concept Analysis.

[17]  Rudolf Wille,et al.  Subdirect decomposition of concept lattices , 1983, ICFCA.

[18]  Bernhard Ganter,et al.  Two Basic Algorithms in Concept Analysis , 2010, ICFCA.

[19]  Vincent Duquenne,et al.  Familles minimales d'implications informatives résultant d'un tableau de données binaires , 1986 .

[20]  Bernhard Ganter,et al.  Formal Concept Analysis: Mathematical Foundations , 1998 .

[21]  Peter Mihók,et al.  Unique Factorization Theorem and Formal Concept Analysis , 2006, CLA.

[22]  George Grätzer,et al.  Joint review of general lattice theory (second edition) and the congruences of a finite lattice: a proof-by picture approach , 2007, SIGA.

[23]  Rokia Missaoui,et al.  A partition-based approach towards constructing Galois (concept) lattices , 2002, Discret. Math..

[24]  Jirí Demel,et al.  Fast algorithms for finding a subdirect decomposition and interesting congruences of finite algebras , 1982, Kybernetika.

[25]  Karell Bertet,et al.  The reverse doubling construction , 2015, 2015 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K).

[26]  R. P. Dilworth,et al.  Algebraic theory of lattices , 1973 .

[27]  W. Geyer,et al.  The generalized doubling construction and formal concept analysis , 1994 .

[28]  Karell Bertet,et al.  Subdirect Decomposition of Contexts into Subdirectly Irreducible Factors , 2015, FCA&A@ICFCA.

[29]  James B. Nation,et al.  Discovery of the D-basis in binary tables based on hypergraph dualization , 2015, Theor. Comput. Sci..

[30]  Vilém Vychodil,et al.  Discovery of optimal factors in binary data via a novel method of matrix decomposition , 2010, J. Comput. Syst. Sci..

[31]  Alan Day.  Splitting lattices generate all lattices , 1977 .

[32]  Alan Day,et al.  Congruence normality: The characterization of the doubling class of convex sets , 1994 .

[33]  Alan Day,et al.  A Simple Solution to the Word Problem for Lattices , 1970, Canadian Mathematical Bulletin.

[34]  Petko Valtchev,et al.  On the Merge of Factor Canonical Bases , 2008, ICFCA.

[35]  Robert Rand,et al.  Ordered direct implicational basis of a finite closure system , 2011, ISAIM.