Advertisement

First Notes on Maximum Entropy Entailment for Quantified Implications

  • Francesco Kriegel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10308)

Abstract

Entropy is a measure for the uninformativeness or randomness of a data set, i.e., the higher the entropy is, the lower is the amount of information. In the field of propositional logic it has proven to constitute a suitable measure to be maximized when dealing with models of probabilistic propositional theories. More specifically, it was shown that the model of a probabilistic propositional theory with maximal entropy allows for the deduction of other formulae which are somehow expected by humans, i.e., allows for some kind of common sense reasoning.

In order to pull the technique of maximum entropy entailment to the field of Formal Concept Analysis, we define the notion of entropy of a formal context with respect to the frequency of its object intents, and then define maximum entropy entailment for quantified implication sets, i.e., for sets of partial implications where each implication has an assigned degree of confidence. Furthermore, then this entailment technique is utilized to define so-called maximum entropy implicational bases (ME-bases), and a first general example of such a ME-base is provided.

Keywords

Maximum entropy Formal context Partial implication Formal Concept Analysis Implicational base Uncertain knowledge 

Notes

Acknowledgements

The author gratefully thanks the anonymous reviewers for their constructive hints and helpful remarks.

References

  1. 1.
    Adaricheva, K.V., Nation, J.B., Rand, R.: Ordered direct implicational basis of a finite closure system. Discrete Appl. Math. 161(6), 707–723 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balcázar, J.L.: Minimum-size bases of association rules. In: Daelemans, W., Goethals, B., Morik, K. (eds.) ECML PKDD 2008. LNCS, vol. 5211, pp. 86–101. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-87479-9_24 CrossRefGoogle Scholar
  3. 3.
    Beierle, C., et al.: Extending and completing probabilistic knowledge and beliefs without bias. KI 29(3), 255–262 (2015)Google Scholar
  4. 4.
    Borchmann, D.: Deciding entailment of implications with support and confidence in polynomial space. CoRR abs/1201.5719 (2012)Google Scholar
  5. 5.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Guigues, J.-L., Duquenne, V.: Famille minimale d’implications informatives résultant d’un tableau de données binaires. Mathématiques et Sci. Hum. 95, 5–18 (1986)Google Scholar
  7. 7.
    Kriegel, F.: Probabilistic implicational bases in FCA and probabilistic bases of GCIs in \({\cal{EL}}^\bot \). In: Yahia, S.B., Konecny, J. (eds.) Proceedings of the Twelfth International Conference on Concept Lattices and Their Applications, Clermont-Ferrand, France, 13–16 October 2015, vol. 1466. CEUR Workshop Proceedings. CEUR-WS.org, pp. 193–204 (2015)Google Scholar
  8. 8.
    Luxenburger, M.: Implikationen, Abhängigkeiten und Galois Abbildungen - Beiträge zur formalen Begriffsanalyse. Ph.D. thesis. Technische Hochschule Darmstadt (1993)Google Scholar
  9. 9.
    Paris, J.B.: The Uncertain Reasoner’s Companion - A Mathematical Perspective. Cambridge Tracts in Theoretical Computer Science, vol. 39. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  10. 10.
    Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1949)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Theoretical Computer ScienceTechnische Universität DresdenDresdenGermany

Personalised recommendations