Advertisement

Joining Implications in Formal Contexts and Inductive Learning in a Horn Description Logic

  • Francesco KriegelEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11511)

Abstract

A joining implication is a restricted form of an implication where it is explicitly specified which attributes may occur in the premise and in the conclusion, respectively. A technique for sound and complete axiomatization of joining implications valid in a given formal context is provided. In particular, a canonical base for the joining implications valid in a given formal context is proposed, which enjoys the property of being of minimal cardinality among all such bases. Background knowledge in form of a set of valid joining implications can be incorporated. Furthermore, an application to inductive learning in a Horn description logic is proposed, that is, a procedure for sound and complete axiomatization of \({\mathsf {Horn\text {-}}}\mathcal {M} \) concept inclusions from a given interpretation is developed. A complexity analysis shows that this procedure runs in deterministic exponential time.

Keywords

Inductive learning Data mining Axiomatization Formal Concept Analysis Joining implication Horn description logic Concept inclusion 

Notes

Acknowledgments

The author gratefully thanks Sebastian Rudolph for the very idea of learning in Horn description logics as well as for a helpful discussion on basics of Horn description logics. The author further thanks the reviewers for their constructive remarks.

References

  1. 1.
    Baader, F., Brandt, S., Lutz, C.: Pushing the \(\cal{EL}\) envelope. In: Kaelbling, L.P., Saffiotti, A. (eds.) IJCAI-05, Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence, Edinburgh, Scotland, UK, July 30–August 5 2005, pp. 364–369. Professional Book Center (2005)Google Scholar
  2. 2.
    Baader, F., Horrocks, I., Lutz, C., Sattler, U.: An Introduction to Description Logic. Cambridge University Press, New York (2017)CrossRefGoogle Scholar
  3. 3.
    Belohlávek, R., Vychodil, V.: Closure-based constraints in formal concept analysis. Discrete Appl. Math. 161(13–14), 1894–1911 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borchmann, D.: Learning terminological knowledge with high confidence from erroneous data. Doctoral thesis, Technische Universität Dresden, Dresden, Germany (2014)Google Scholar
  5. 5.
    Borchmann, D., Distel, F., Kriegel, F.: Axiomatisation of general concept inclusions from finite interpretations. J. Appl. Non-Class. Logics 26(1), 1–46 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Calvanese, D., De Giacomo, G., Lembo, D., Lenzerini, M., Rosati, R.: Data complexity of query answering in description logics. In: Doherty, P., Mylopoulos, J., Welty, C.A. (eds.) Proceedings, Tenth International Conference on Principles of Knowledge Representation and Reasoning, Lake District of the United Kingdom, 2–5 June 2006, pp. 260–270. AAAI Press (2006)Google Scholar
  7. 7.
    Dantsin, E., Eiter, T., Gottlob, G., Voronkov, A.: Complexity and expressive power of logic programming. ACM Comput. Surv. 33(3), 374–425 (2001)CrossRefGoogle Scholar
  8. 8.
    De Giacomo, G., Lenzerini, M.: A uniform framework for concept definitions in description logics. J. Artif. Intell. Res. 6, 87–110 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Distel, F.: Learning description logic knowledge bases from data using methods from formal concept analysis. Doctoral thesis, Technische Universität Dresden, Dresden, Germany (2011)Google Scholar
  10. 10.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1999).  https://doi.org/10.1007/978-3-642-59830-2CrossRefzbMATHGoogle Scholar
  11. 11.
    Guigues, J.L., Duquenne, V.: Famille minimale d’implications informatives résultant d’un tableau de données binaires. Mathématiques et Sciences Humaines 95, 5–18 (1986)Google Scholar
  12. 12.
    Hernich, A., Lutz, C., Papacchini, F., Wolter, F.: Horn-Rewritability vs. PTime query evaluation in ontology-mediated querying. In: Lang, J. (ed.) Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, IJCAI 2018, Stockholm, Sweden, 13–19 July 2018, pp. 1861–1867. ijcai.org (2018)Google Scholar
  13. 13.
    Hitzler, P., Krötzsch, M., Rudolph, S.: Foundations of Semantic Web Technologies. Chapman and Hall/CRC Press, Boca Raton (2010)Google Scholar
  14. 14.
    Hustadt, U., Motik, B., Sattler, U.: Data complexity of reasoning in very expressive description logics. In: Kaelbling, L.P., Saffiotti, A. (eds.) IJCAI-05, Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence, Edinburgh, Scotland, UK, July 30 - August 5 2005, pp. 466–471. Professional Book Center (2005)Google Scholar
  15. 15.
    Hustadt, U., Motik, B., Sattler, U.: Reasoning in description logics by a reduction to disjunctive datalog. J. Autom. Reason. 39(3), 351–384 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kriegel, F.: Concept Explorer FX (2010–2019), Software for Formal Concept Analysis with Description Logic Extensions. https://github.com/francesco-kriegel/conexp-fx
  17. 17.
    Kriegel, F.: NextClosures with constraints. In: Huchard, M., Kuznetsov, S. (eds.) Proceedings of the Thirteenth International Conference on Concept Lattices and Their Applications, Moscow, Russia, 18–22 July 2016. CEUR Workshop Proceedings, vol. 1624, pp. 231–243. CEUR-WS.org (2016)Google Scholar
  18. 18.
    Kriegel, F.: Acquisition of terminological knowledge from social networks in description logic. In: Missaoui, R., Kuznetsov, S.O., Obiedkov, S. (eds.) Formal Concept Analysis of Social Networks. LNSN, pp. 97–142. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-64167-6_5CrossRefGoogle Scholar
  19. 19.
    Kriegel, F.: Most specific consequences in the description logic \(\cal{EL}\). LTCS-Report 18–11, Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universität Dresden, Dresden, Germany (2018, accepted for publication in Discrete Applied Mathematics). https://tu-dresden.de/inf/lat/reports#Kr-LTCS-18-11
  20. 20.
    Kriegel, F.: Joining implications in formal contexts and inductive learning in a horn description logic (Extended Version). LTCS-Report 19–02, Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universität Dresden, Dresden, Germany (2019). https://tu-dresden.de/inf/lat/reports#Kr-LTCS-19-02
  21. 21.
    Kriegel, F.: Most specific consequences in the description logic \(\cal{EL}\). Discrete Applied Mathematics (2019).  https://doi.org/10.1016/j.dam.2019.01.029
  22. 22.
    Kriegel, F., Borchmann, D.: NextClosures: parallel computation of the canonical base with background knowledge. Int. J. Gen. Syst. 46(5), 490–510 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Krisnadhi, A., Lutz, C.: Data complexity in the \(\cal{EL}\) family of description logics. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 333–347. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-75560-9_25CrossRefGoogle Scholar
  24. 24.
    Krötzsch, M., Rudolph, S., Hitzler, P.: Complexities of horn description logics. ACM Trans. Comput. Logic 14(1), 2:1–2:36 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kupferman, O., Sattler, U., Vardi, M.Y.: The complexity of the graded \({\mu }\)-Calculus. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 423–437. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45620-1_34CrossRefGoogle Scholar
  26. 26.
    Kuznetsov, S.O., Obiedkov, S.A.: Some decision and counting problems of the Duquenne-Guigues basis of implications. Discrete Appl. Math. 156(11), 1994–2003 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Rudolph, S.: Relational exploration: combining description logics and formal concept analysis for knowledge specification. Doctoral thesis, Technische Universität Dresden, Dresden, Germany (2006)Google Scholar
  28. 28.
    Schild, K.: A correspondence theory for terminological logics: preliminary report. In: Mylopoulos, J., Reiter, R. (eds.) Proceedings of the 12th International Joint Conference on Artificial Intelligence, Sydney, Australia, 24–30 August 1991, pp. 466–471. Morgan Kaufmann (1991)Google Scholar
  29. 29.
    Stumme, G.: Attribute exploration with background implications and exceptions. In: Bock, H.H., Polasek, W. (eds.) Studies in Classification, Data Analysis, and Knowledge Organization, pp. 457–469. Springer, Heidelberg (1996).  https://doi.org/10.1007/978-3-642-80098-6_39CrossRefzbMATHGoogle Scholar
  30. 30.
    Tobies, S.: Complexity results and practical algorithms for logics in knowledge representation. Doctoral thesis, Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, Germany (2001)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Theoretical Computer ScienceTechnische Universität DresdenDresdenGermany

Personalised recommendations