Analogical Proportions in a Lattice of Sets of Alignments Built on the Common Subwords in a Finite Language

  • Laurent Miclet
  • Nelly Barbot
  • Baptiste Jeudy
Part of the Studies in Computational Intelligence book series (SCI, volume 548)


We define the locally maximal subwords and locally minimal superwords common to a finite set of words. We also define the corresponding sets of alignments. We give a partial order relation between such sets of alignments, as well as two operations between them. We show that the constructed family of sets of alignments has the lattice structure. The study of analogical proportion in lattices gives hints to use this structure as a machine learning basis, aiming at inducing a generalization of the set of words.


Locally maximal subwords Alignments Algebraic structure of sets of alignments on a set of words (lattice) Analogical proportion 


  1. 1.
    Bayoudh, S., Mouchère, H., Miclet, L., Anquetil, E.: Learning a classifier with very few examples: analogy based and knowledge based generation of new examples for character recognition. In: European Conference on Machine Learning, Springer LNAI 4701 (2007)Google Scholar
  2. 2.
    de la Higuera, C.: Grammatical Inference. Cambridge University Press, Cambridge (2010)Google Scholar
  3. 3.
    Fraser, C., Irving, R., Middendorf, M.: Maximal common subsequences and minimal common supersequences. Inf. Comput. 124, 145–153 (1996)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Gusfield, D.: Algorithms on Strings, Trees, and Sequences. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar
  5. 5.
    Ben Hassena, A.: Apprentissage par analogie de structures d’arbres. PhD thesis, UniversitT de Rennes 1 (2011)Google Scholar
  6. 6.
    Irving, R., Fraser, C.: Two algorithms for the longest common subsequence of three (and more) strings. In: Proceedings of 3rd Symposium on Combinatorial Pattern Matching. Springer LCNS 644, pp. 214–229 (1992)Google Scholar
  7. 7.
    Kuznetsov, O.: Machine learning on the basis of formal concept analysis. Autom. Remote Control 62(10), 1543–1564 (2001)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Lepage, Y.: De l’analogie rendant compte de la commutation en linguistique. UniversitT de Grenoble, Grenoble, Habilitation a diriger les recherches (2003)Google Scholar
  9. 9.
    Maier, D.: The complexity of some problems on subsequences and supersequences. JACM 25, 332–336 (1978)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Miclet, L., Bayoudh, S., Delhay, A.: Analogical dissimilarity: definition, algorithms and two experiments in machine learning. J. Artif. Intell. Res. 32, 793–824 (2008)MATHMathSciNetGoogle Scholar
  11. 11.
    Stroppa, v., Yvon, F.: Analogical learning and formal proportions: definitions and methodological issues. Technical Report ENST-2005-D004, +cole Nationale SupTrieure des TTlTcommunications, June (2005)Google Scholar
  12. 12.
    Wagner, R., Fisher, M.: The string-to-string correction problem. J. ACM 21(1), 168–173 (1974)CrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.IRISA-DylissRennesFrance
  2. 2.IRISA-CordialLannionFrance
  3. 3.Laboratoire Hubert CurienUniversité de Saint-ÉtienneSaint-ÉtienneFrance

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