Article

Crisply generated fuzzy concepts

Authors:

Radim Bělohlávek,

Vladimír Sklenář,

Jiří ZacpalAuthors Info & Claims

ICFCA'05: Proceedings of the Third international conference on Formal Concept Analysis

Pages 269 - 284

https://doi.org/10.1007/978-3-540-32262-7_19

Published: 14 February 2005 Publication History

Abstract

In formal concept analysis of data with fuzzy attributes, both the extent and the intent of a formal (fuzzy) concept may be fuzzy sets. In this paper we focus on so-called crisply generated formal concepts. A concept $\langle{A,B}\rangle \in \mathcal{B}(X, Y, I)$ is crisply generated if A = D^↓ (and so B = D^↓↑) for some crisp (i.e., ordinary) set D ⊆ Y of attributes (generator). Considering only crisply generated concepts has two practical consequences. First, the number of crisply generated formal concepts is considerably less than the number of all formal fuzzy concepts. Second, since crisply generated concepts may be identified with a (ordinary, not fuzzy) set of attributes (the largest generator), they might be considered “the important ones” among all formal fuzzy concepts. We present basic properties of the set of all crisply generated concepts, an algorithm for listing all crisply generated concepts, a version of the main theorem of concept lattices for crisply generated concepts, and show that crisply generated concepts are just the fixed points of pairs of mappings resembling Galois connections. Furthermore, we show connections to other papers on formal concept analysis of data with fuzzy attributes. Also, we present examples demonstrating the reduction of the number of formal concepts and the speed-up of our algorithm (compared to listing of all formal concepts and testing whether a concept is crisply generated).

References

[1]

Belohlávek R.: Fuzzy concepts and conceptual structures: induced similarities. In Proc. Joint Conf. Inf. Sci.'98, Vol. I, pages 179-182, Durham, NC, 1998.

[2]

Belohlávek R.: Fuzzy Galois connections. Math. Log. Quart. 45(4)(1999), 497-504.

[3]

Belohlávek R.: Similarity relations in concept lattices. J. Logic Comput. 10(6):823- 845, 2000.

[4]

Belohlávek R.: Reduction and a simple proof of characterization of fuzzy concept lattices. Fundamenta Informaticae 46(4)(2001), 277-285.

Digital Library

[5]

Belohlávek R.: Fuzzy closure operators. J. Math. Anal. Appl. 262(2001), 473-489.

[6]

Belohlávek R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer, Academic/Plenum Publishers, New York, 2002.

Digital Library

[7]

Belohlávek R.: Concept lattices and order in fuzzy logic. Ann. Pure Appl. Logic 128(2004), 277-298.

[8]

Belohlávek R.: Proc. Fourth Int. Conf. on Recent Advances in Soft Computing. Nottingham, United Kingdom, 12-13 December, 2002, pp. 200-205.

[9]

Belohlávek R.: What is a fuzzy concept lattice (in preparation).

[10]

Belohlávek R., Funioková T., Vychodil V.: Galois connections with hedges (submitted). Preliminary version to appear in Proc. 8-th Fuzzy Days, Dortmund, September 2004.

[11]

Burusco A., Fuentes-Gonzáles R.: The study of the L-fuzzy concept lattice. Mathware & Soft Computing, 3:209-218, 1994.

[12]

Ganter B., Wille R.: Formal Concept Analysis. Mathematical Foundations. Springer, Berlin, 1999.

Digital Library

[13]

Hájek P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998.

[14]

Höhle U.: On the fundamentals of fuzzy set theory. J. Math. Anal. Appl. 201(1996), 786-826.

[15]

Johnson D. S., Yannakakis M., Papadimitrou C. H.: On generating all maximal independent sets. Inf. Processing Letters 15(1988), 129-133.

Digital Library

[16]

Klir G. J., Yuan B.: Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice Hall, Upper Saddle River, NJ, 1995.

Digital Library

[17]

Krajči S.: Cluster based efficient generation of fuzzy concepts. Neural Network World 5(2003), 521-530.

[18]

Pollandt S.: Fuzzy Begriffe. Springer, Berlin, 1997.

[19]

Ore O.: Galois connections. Trans. Amer. Math. Soc. 55 (1944), 493-513.

[20]

Wille R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival I.: Ordered Sets. Reidel, Dordrecht, Boston, 1982, 445-470.

[21]

Wolff K. E.: Concepts in fuzzy scaling theory: order and granularity. Fuzzy Sets and Systems 132(2002), 63-75.

Digital Library

[22]

Yahia S., Jaoua A.: Discovering knowledge from fuzzy concept lattice. In: Kandel A., Last M., Bunke H.: Data Mining and Computational Intelligence, pp. 167-190, Physica-Verlag, 2001.

Digital Library

Cited By

Šostak AKrastiņš MUļjane I(2023)Conceptuality Degree of Oriented Crisply Generated Fuzzy PreconceptsFuzzy Logic and Technology, and Aggregation Operators10.1007/978-3-031-39965-7_8(86-98)Online publication date: 4-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-39965-7_8
Xie JYang MLi JZheng Z(2018)Rule acquisition and optimal scale selection in multi-scale formal decision contexts and their applications to smart cityFuture Generation Computer Systems10.1016/j.future.2017.03.01183:C(564-581)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1016/j.future.2017.03.011
Martin THe H(2012)Bisociative discovery in business process modelsBisociative Knowledge Discovery10.5555/2363300.2363337(452-471)Online publication date: 1-Jan-2012
https://dl.acm.org/doi/10.5555/2363300.2363337
Show More Cited By

Index Terms

Crisply generated fuzzy concepts
1. Computing methodologies
  1. Artificial intelligence
    1. Knowledge representation and reasoning
      1. Probabilistic reasoning
      2. Vagueness and fuzzy logic

Index terms have been assigned to the content through auto-classification.

Recommendations

Relations between Proto-fuzzy Concepts, Crisply Generated Fuzzy Concepts, and Interval Pattern Structures
Concept Lattices and Their Applications

Relationships between proto-fuzzy concepts, crisply generated fuzzy concepts, and pattern structures are considered. It is shown that proto-fuzzy concepts are closely related to crisply generated fuzzy concepts in the sense that the mappings involved in ...
Conceptuality Degree of Oriented Crisply Generated Fuzzy Preconcepts
Fuzzy Logic and Technology, and Aggregation Operators
Abstract
A serious obstacle for the use of fuzzy concept analysis in practical issues is the problem of matching between sets of objects and properties in fuzzy environment. To overcome this problem several modified versions of fuzzy concept analysis were ...
Crisply Generated Complex Fuzzy Concepts Analysis Using Shannon Entropy
Abstract
Recently, the properties of complex fuzzy concept lattice given a way to represent the uncertainty and its fluctuation. It given a well established mathematical model for understanding the dark data sets with respect to phase of time. The complex ...

Comments

Information & Contributors

Information

Published In

cover image Guide Proceedings

ICFCA'05: Proceedings of the Third international conference on Formal Concept Analysis

February 2005

416 pages

ISBN:3540245251

Editors:
Bernhard Ganter
Institute of Algebra, University of Technology, Dresden, Germany
,
Robert Godin
Département d'informatique, UQAM, Montréal, (Qc), Canada

Sponsors

Le Centre de Recherche en Informatique de Lens: Le Centre de Recherche en Informatique de Lens
L'Institut Universitaire de Technologie de Lens: L'Institut Universitaire de Technologie de Lens
La Ville de Lens: La Ville de Lens
L'Université d'Artois: L'Université d'Artois
La Commun Aupole de Lens-Liévin: La Commun Aupole de Lens-Liévin

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 14 February 2005

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

7
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 11 Aug 2024

Other Metrics

View Author Metrics

Citations

Cited By

Šostak AKrastiņš MUļjane I(2023)Conceptuality Degree of Oriented Crisply Generated Fuzzy PreconceptsFuzzy Logic and Technology, and Aggregation Operators10.1007/978-3-031-39965-7_8(86-98)Online publication date: 4-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-39965-7_8
Xie JYang MLi JZheng Z(2018)Rule acquisition and optimal scale selection in multi-scale formal decision contexts and their applications to smart cityFuture Generation Computer Systems10.1016/j.future.2017.03.01183:C(564-581)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1016/j.future.2017.03.011
Martin THe H(2012)Bisociative discovery in business process modelsBisociative Knowledge Discovery10.5555/2363300.2363337(452-471)Online publication date: 1-Jan-2012
https://dl.acm.org/doi/10.5555/2363300.2363337
Macko J(2012)Formal concept analysis as a framework for business intelligence technologiesProceedings of the 10th international conference on Formal Concept Analysis10.1007/978-3-642-29892-9_20(195-210)Online publication date: 7-May-2012
https://dl.acm.org/doi/10.1007/978-3-642-29892-9_20
Krupka MLastovicka J(2012)Concept lattices of incomplete dataProceedings of the 10th international conference on Formal Concept Analysis10.1007/978-3-642-29892-9_19(180-194)Online publication date: 7-May-2012
https://dl.acm.org/doi/10.1007/978-3-642-29892-9_19
Belohlavek R(2011)What is a fuzzy concept lattice? IIProceedings of the 13th international conference on Rough sets, fuzzy sets, data mining and granular computing10.5555/2026782.2026787(19-26)Online publication date: 25-Jun-2011
https://dl.acm.org/doi/10.5555/2026782.2026787
Bělohlávek ROutrata JVychodil V(2006)Thresholds and shifted attributes in formal concept analysis of data with fuzzy attributesProceedings of the 14th international conference on Conceptual Structures: inspiration and Application10.1007/11787181_9(117-130)Online publication date: 16-Jul-2006
https://dl.acm.org/doi/10.1007/11787181_9

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Table of Contents