Abstract
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.
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Notes
- 1.
Other terms for subword are subsequence and partial word. A factor, or substring is a subword of \(u\) built by contiguous letters of \(u\).
- 2.
A superword of \(u\), also called a supersequence must not be confused with a superstring of \(u\), in which the letters of \(u\) are contiguous. In other words, \(u\) is a factor (a substring) of any superstring of \(u\). See [4], pages 4, 309 and 426.
- 3.
- 4.
A consequence of this assertion is : let \(LCS(u,v)\) be a longest common subword to \(u\) and \(v\) and \(SCS(u,v)\) be a shortest common superword to \(u\) and \(v\). Then we have: \(|LCS(u,v)|+|SCS(u,v)|=|u|+|v|\).
- 5.
Remember that this support, that we have denoted \(word(A)\), is a subset of \(U\).
- 6.
The elementary operation is the comparison.
References
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)
de la Higuera, C.: Grammatical Inference. Cambridge University Press, Cambridge (2010)
Fraser, C., Irving, R., Middendorf, M.: Maximal common subsequences and minimal common supersequences. Inf. Comput. 124, 145–153 (1996)
Gusfield, D.: Algorithms on Strings, Trees, and Sequences. Cambridge University Press, Cambridge (1997)
Ben Hassena, A.: Apprentissage par analogie de structures d’arbres. PhD thesis, UniversitT de Rennes 1 (2011)
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)
Kuznetsov, O.: Machine learning on the basis of formal concept analysis. Autom. Remote Control 62(10), 1543–1564 (2001)
Lepage, Y.: De l’analogie rendant compte de la commutation en linguistique. UniversitT de Grenoble, Grenoble, Habilitation a diriger les recherches (2003)
Maier, D.: The complexity of some problems on subsequences and supersequences. JACM 25, 332–336 (1978)
Miclet, L., Bayoudh, S., Delhay, A.: Analogical dissimilarity: definition, algorithms and two experiments in machine learning. J. Artif. Intell. Res. 32, 793–824 (2008)
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)
Wagner, R., Fisher, M.: The string-to-string correction problem. J. ACM 21(1), 168–173 (1974)
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Miclet, L., Barbot, N., Jeudy, B. (2014). Analogical Proportions in a Lattice of Sets of Alignments Built on the Common Subwords in a Finite Language. In: Prade, H., Richard, G. (eds) Computational Approaches to Analogical Reasoning: Current Trends. Studies in Computational Intelligence, vol 548. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54516-0_10
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