Minimal Distance-Based Generalisation Operators for First-Order Objects

  • Vicent Estruch
  • César Ferri
  • Jose Hernández-Orallo
  • María José Ramírez-Quintana
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4455)


Distance-based methods have been a successful family of machine learning techniques since the inception of the discipline. Basically, the classification or clustering of a new individual is determined by the distance to one or more prototypes. From a comprehensibility point of view, this is not especially problematic in propositional learning where prototypes can be regarded as a good generalisation (pattern) of a group of elements. However, for scenarios with structured data, this is no longer the case. In recent work, we developed a framework to determine whether a pattern computed by a generalisation operator is consistent w.r.t. a distance. In this way, we can determine which patterns can provide a good representation of a group of individuals belonging to a metric space. In this work, we apply this framework to analyse and define minimal distance-based generalisation operators (mg operators) for first-order data. We show that Plotkin’s lgg is a mg operator for atoms under the distance introduced by J. Ramon, M. Bruynooghe and W. Van Laer. We also show that this is not the case for clauses with the distance introduced by J. Ramon and M. Bruynooghe. Consequently, we introduce a new mg operator for clauses, which could be used as a base to adapt existing bottom-up methods in ILP.


Cost Function Logic Programming Nerve Function Function Symbol Generalisation Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Estruch, V.: A distance-based generalisation framework for model-based learning from structured data. PhD thesis, Technical University of Valencia (2007),
  2. 2.
    Estruch, V., Ferri, C., Hernández-Orallo, J., Ramírez-Quintana, M.J.: Distance-based generalisation. In: Kramer, S., Pfahringer, B. (eds.) ILP 2005. LNCS (LNAI), vol. 3625, pp. 87–102. Springer, Heidelberg (2005)Google Scholar
  3. 3.
    Estruch, V., Ferri, C., Hernández-Orallo, J., Ramírez-Quintana, M.J.: Distance-based generalisation for graphs. In: Proc. of the WS of Mining and Learning with Graphs, MLG06 (2006)Google Scholar
  4. 4.
    Gaertner, T., Lloyd, J.W., Flach, P.A.: Kernels and distances for structured data. Machine Learning 57(3), 205–232 (2004)zbMATHCrossRefGoogle Scholar
  5. 5.
    Lavrac, N., Dzeroski, S.: Inductive Logic Programming: Techniques and Applications. Ellis Horwood, New York (1994)zbMATHGoogle Scholar
  6. 6.
    Lloyd, J.W.: Foundations of logic programming (2nd extended edn.). Springer, New York (1987)Google Scholar
  7. 7.
    Muggleton, S.: Inductive Logic Programming. New Generation Computing 8(4), 295–318 (1991)zbMATHCrossRefGoogle Scholar
  8. 8.
    Muggleton, S.H.: Inductive logic programming: Issues, results, and the challenge of learning language in logic. Artificial Intelligence 114(1–2), 283–296 (1999)zbMATHCrossRefGoogle Scholar
  9. 9.
    Nienhuys-Cheng, S-H.: Distance between Herbrand interpretations: A measure for approximations to a target concept. In: Džeroski, S., Lavrač, N. (eds.) Inductive Logic Programming. LNCS, vol. 1297, pp. 213–226. Springer, Heidelberg (1997)Google Scholar
  10. 10.
    Nienhuys-Cheng, S-H., de Wolf, R.: Foundations of Inductive Logic Programming. In: Džeroski, S., Lavrač, N. (eds.) Inductive Logic Programming. LNCS, vol. 1297, Springer, Heidelberg (1997)Google Scholar
  11. 11.
    Plotkin, G.: A note on inductive generalization. Machine Intelligence 5, 153–163 (1970)MathSciNetGoogle Scholar
  12. 12.
    Ramon, J., Bruynooghe, M.: A framework for defining distances between first-order logic objects. In: Page, D.L. (ed.) Inductive Logic Programming. LNCS, vol. 1446, pp. 271–280. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. 13.
    Ramon, J., Bruynooghe, M., Van Laer, W.: Distance measures between atoms. In: CompulogNet Area Meeting on Computational Logic and Machine Learning, pp. 35–41. University of Manchester, UK (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Vicent Estruch
    • 1
  • César Ferri
    • 1
  • Jose Hernández-Orallo
    • 1
  • María José Ramírez-Quintana
    • 1
  1. 1.DSIC, Univ. Politècnica de València , Camí de Vera s/n, 46020 ValènciaSpain

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