Fast algorithms for implication bases and attribute exploration using proper premises

A central task in formal concept analysis is the enumeration of a small base for the implications that hold in a formal context. The usual stem base algorithms have been proven to be costly in terms of runtime. Proper premises are an alternative to the stem base. We present a new algorithm for the fast computation of proper premises. It is based on a known link between proper premises and minimal hypergraph transversals. Two further improvements are made, which reduce the number of proper premises that are obtained multiple times and redundancies within the set of proper premises. We have evaluated our algorithms within an application related to refactoring of model variants. In this application an implicational base needs to be computed, and runtime is more crucial than minimal cardinality. In addition to the empirical tests, we provide heuristic evidence that an approach based on proper premises will also be beneficial for other applications. Finally, we show how our algorithms can be extended to an exploration algorithm that is based on proper premises.

[1]  Sebastian Rudolph.  Some Notes on Pseudo-closed Sets , 2007, ICFCA.

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

[3]  Vladimir Gurvich,et al.  An Efficient Incremental Algorithm for Generating All Maximal Independent Sets in Hypergraphs of Bounded Dimension , 2000, Parallel Process. Lett..

[4]  Klaus Kabitzsch,et al.  Extraction of feature models from formal contexts , 2011, SPLC '11.

[5]  Matthias Hagen,et al.  Algorithmic and Computational Complexity Issues of MONET , 2008 .

[6]  Adolfo Jonathan Salinas-López,et al.  CEUR Workshop Proceedings , 2015 .

[7]  Daniel Borchmann.  A General Form of Attribute Exploration , 2012, ArXiv.

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

[9]  Felix Distel,et al.  On the complexity of enumerating pseudo-intents , 2011, Discret. Appl. Math..

[10]  Dimitrios Gunopulos,et al.  Data mining, hypergraph transversals, and machine learning (extended abstract) , 1997, PODS '97.

[11]  Georg Gottlob,et al.  New results on monotone dualization and generating hypergraph transversals , 2002, STOC '02.

[12]  Claude Berge,et al.  Hypergraphs - combinatorics of finite sets , 1989, North-Holland mathematical library.

[13]  Heiko Reppe.  Attribute Exploration Using Implications with Proper Premises , 2008, ICCS.

[14]  James Bailey,et al.  A fast algorithm for computing hypergraph transversals and its application in mining emerging patterns , 2003, Third IEEE International Conference on Data Mining.

[15]  Peter Buneman,et al.  Semistructured data , 1997, PODS.

[16]  Patrick Brézillon,et al.  Lecture Notes in Artificial Intelligence , 1999 .

[17]  Charu C. Aggarwal,et al.  Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery, DMKD 2003, San Diego, California, USA, June 13, 2003 , 2003, DMKD.

[18]  Elias C. Stavropoulos,et al.  Journal of Graph Algorithms and Applications an Efficient Algorithm for the Transversal Hypergraph Generation , 2022 .

[19]  Sergei O. Kuznetsov,et al.  On the Intractability of Computing the Duquenne?Guigues Bas , 2004, J. Univers. Comput. Sci..

[20]  Leonid Khachiyan,et al.  On the Complexity of Dualization of Monotone Disjunctive Normal Forms , 1996, J. Algorithms.

[21]  Mohammed J. Zaki,et al.  Theoretical Foundations of Association Rules , 2007 .

[22]  Georg Gottlob,et al.  New Results on Monotone Dualization and Generating Hypergraph Transversals , 2003, SIAM J. Comput..

[23]  Klaus Kabitzsch,et al.  Automatic variation-point identification in function-block-based models , 2010, GPCE '10.

[24]  David Maier,et al.  The Theory of Relational Databases , 1983 .

[25]  Khaled M. Elbassioni,et al.  On Berge Multiplication for Monotone Boolean Dualization , 2008, ICALP.

[26]  Heikki Mannila,et al.  Algorithms for Inferring Functional Dependencies from Relations , 1994, Data Knowl. Eng..

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

[28]  Karell Bertet,et al.  The multiple facets of the canonical direct unit implicational basis , 2010, Theor. Comput. Sci..

[29]  Sergei O. Kuznetsov,et al.  Recognizing Pseudo-intents is coNP-complete , 2010, CLA.

[30]  KEN TAKATA,et al.  A Worst-Case Analysis of the Sequential Method to List the Minimal Hitting Sets of a Hypergraph , 2007, SIAM J. Discret. Math..

[31]  Felix Distel.  Hardness of Enumerating Pseudo-intents in the Lectic Order , 2010, ICFCA.

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

[33]  Jinyan Li,et al.  Mining border descriptions of emerging patterns from dataset pairs , 2005, Knowledge and Information Systems.