Abstract
Formal Concept Analysis uses a simple representation framework called ‘formal context’. In the classical setting, a formal context specifies existing Boolean relationships between a set of objects and their corresponding properties. Formal concepts are then defined as pairs consisting of a set of objects and a set of properties that mutually characterize each other through a Galois connection. Another Galois connection is also introduced in this setting on the basis of operators induced by a recent possibility theory reading of Formal Concept Analysis. It is shown that this second Galois connection enables us to characterize independent sub-contexts inside the formal context. The second part of the paper discusses an extension of Formal Concept Analysis that has not been much studied, namely the situation where one may be uncertain on the fact that an object possesses or not a Boolean property. Uncertainty is here represented in the possibilistic representation framework.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Belohlavek, R.: Fuzzy Galois connections. Math. Logic Quart. 45, 497–504 (1999)
Belohlavek, R.: Fuzzy Relational Systems. Kluwer Acad. Pub., Dordrecht (2002)
Belohlavek, R., Vychodil, V.: What is a fuzzy concept lattice. In: Proc. CLAV ’05, Olomuc, pp. 34–45. Czech Republic (2005)
Bosc, P., Pivert, O.: About projection-selection-join queries addressed to possibilistic relational databases. IEEE Trans. Fuzzy Systems 13(1), 124–139 (2005)
Ferré, S., Ridoux, O.: Introduction to logical information systems. Information Processing and Mgmt. 40, 383–419 (2004)
Burmeister, P., Holzer, R.: Treating incomplete knowledge in formal concepts analysis. In: Ganter, B., Stumme, G., Wille, R. (eds.) Formal Concept Analysis. LNCS (LNAI), vol. 3626, pp. 11–126. Springer, Heidelberg (2005)
Burusco, A., Fuentes-Gonzalez, R.: The study of the L-fuzzy concept lattice. Mathware & Soft Comput. 3, 209–218 (1994)
Djouadi, Y., Prade, H.: Interval-valued fuzzy formal concept analysis. In: Rauch, J., Raś, Z.W., Berka, P., Elomaa, T. (eds.) ISMIS 2009. LNCS, vol. 5722, pp. 592–601. Springer, Heidelberg (2009)
Djouadi, Y., Prade, H.: Interval-valued fuzzy Galois connections: algebraic requirements and concept lattice construction. Fund. Inform. (to appear)
Djouadi, Y., Dubois, D., Prade, H.: On the possible meanings of degrees when making formal concept analysis fuzzy. In: EUROFUSE’09, Preference Modelling and Decision Analysis Workshop, Pamplona, Spain, pp. 253–258 (2009)
Djouadi, Y., Dubois, D., Prade, H.: Différentes extensions floues de l’analyse formelle de concepts. Actes Renc. Franc. sur la Logique Floue et ses Applications (LFA 2009) Cépadues edn., Toulouse, pp. 141–148 (2009)
Dubois, D.: On ignorance and contradiction considered as truth-values. Logic Journal of the IGPL 16(2), 195–216 (2008)
Dubois, D., Prade, H.: Possibility Theory. Plenum Press, New York (1988)
Dubois, D., Dupin de Saint Cyr, F., Prade, H.: A possibilty-theoretic view of formal concept analysis. Fundamenta Informaticae 75(1-4), 195–213 (2007)
Dubois, D., Prade, H.: Possibility theory: qualitative and quantitative aspects. In: Gabbay, D., Smets, P. (eds.) Quantified Representation of Uncertainty and Imprecision. Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 1, pp. 169–226. Kluwer Acad. Publ., Dordrecht (1998)
Dubois, D., Prade, H.: Possibility theory and formal concept analysis in information systems. In: Proc. IFSA’09, International Fuzzy Systems Association World Congress, Lisbon, Portugal, pp. 1021–1026 (2009)
Düntsch, I., Orlowska, E.: Mixing modal and sufficiency operators. Bulletin of the Section of Logic, Polish Academy of Sciences 28(2), 99–106 (1999)
Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Heidelberg (1999)
Medina, J., Ojeda-Aciego, M., Ruiz-Calvino, J.: Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets and Systems 160(2), 130–144 (2009)
Messai, N., Devignes, M., Napoli, A., Tabbone, M.: Many-valued concept lattices for conceptual clustering and information retrieval. In: Proc. 18th Europ. Conf. on Artif. Intellig., Patras, pp. 722–727 (2008)
Pollandt, S.: Fuzzy Begriffe. Springer, Heidelberg (1997)
Yao, Y.Y., Chen, Y.: Rough Set Approximations in Formal Concept Analysis. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets V. LNCS, vol. 4100, pp. 285–305. Springer, Heidelberg (2006)
Wille, R.: Restructuring Lattice Theory: an Approach Based on Hierarchies of Concepts. In: Rival, I. (ed.) Ordered Sets, pp. 445–470. Reidel, Dordrecht (1982)
Wolff, K.E.: A first course in Formal Concept Analysis - How to understand line diagrams. In: Faulbaum, F. (ed.) SoftStat’93, Advances in Statistical Software 4, pp. 429–438. Gustav Fischer Verlag, Stuttgart (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Djouadi, Y., Dubois, D., Prade, H. (2010). Possibility Theory and Formal Concept Analysis: Context Decomposition and Uncertainty Handling. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds) Computational Intelligence for Knowledge-Based Systems Design. IPMU 2010. Lecture Notes in Computer Science(), vol 6178. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14049-5_27
Download citation
DOI: https://doi.org/10.1007/978-3-642-14049-5_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14048-8
Online ISBN: 978-3-642-14049-5
eBook Packages: Computer ScienceComputer Science (R0)