On the Parameterised Complexity of Learning Patterns

Conference paper

Abstract

Angluin (1980) showed that there is a consistent and conservative learner for the class of non-erasing pattern languages; however, most of these learners are NP-hard. In the current work, the complexity of consistent polynomial time learners for the class of non-erasing pattern languages is revisited, with the goal to close one gap left by Angluin, namely the question on what happens if the learner is not required to output each time a consistent pattern of maximum possible length. It is shown that consistent learners are non-uniformly W[1]-hard inside the fixed-parameter hierarchy of Downey and Fellows (1999), and that there is also a W[1]-complete such learner. Only when one requires that the learner is in addition both, conservative and class-preserving, then one can show that the learning task is NP-hard for certain alphabet-sizes.

References

  1. 1.
    Angluin, D.: Finding patterns common to a set of strings. J. of Comput. Syst. Sci. 21, 46–62 (1980)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Angluin, D.: Inductive inference of formal languages from positive data. Inform. Control 45, 117–135 (1980)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Case, J., Kötzing, T.: Dynamically delayed postdictive completeness and consistency in learning. In: Freund, Y., Györfi, L., Turán, G., Zeugmann, T. (eds) ALT 2008. LNAI, vol. 5254, pp. 389–403. Springer, Berlin (2008)Google Scholar
  4. 4.
    Case, J., Kötzing, T.: Difficulties in forcing fairness of polynomial time inductive inference. In: Gavaldà, R., Lugosi, G., Zeugmann, T., Zilles, S. (eds) ALT 2009. LNAI, vol. 5809, pp. 263–277. Springer, Berlin (2009)Google Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Monographs in computer science. Springer, Berlin (1999)CrossRefGoogle Scholar
  6. 6.
    Erlebach, T., Rossmanith, P., Stadtherr, H., Steger, A., Zeugmann, T.: Learning one-variable pattern languages very efficiently on average, in parallel, and by asking queries. Theoret. Comput. Sci. 261, 119–156 (2001)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Gold, M.: Language identification in the limit. Inform. Control 10, 447–474 (1967)CrossRefMATHGoogle Scholar
  8. 8.
    Lange, S., Wiehagen, R.: Polynomial-time inference of arbitrary pattern languages. New Genera. Comput. 8, 361–370 (1991)CrossRefMATHGoogle Scholar
  9. 9.
    Lange, S., Zeugmann, T., Zilles, S.: Learning indexed families of recursive languages from positive data: a survey. Theoret. Comput. Sci. 397, 194–232 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Reidenbach, D.: A non-learnable class of E-pattern languages. Theoret. Comput. Sci. 350, 91–102 (2006)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Rossmanith, P., Zeugmann, T.: Stochastic finite learning of the pattern languages. Mach. Learn. 44, 67–91 (2001)CrossRefMATHGoogle Scholar
  12. 12.
    Shinohara, T.: Polynomial time inference of extended regular pattern languages. In: RIMS Symposium on Software Science and Engineering, Proc. LNCS, 147. pp. 115–127, Springer, Berlin (1983)Google Scholar
  13. 13.
    Shinohara, T., Arikawa, S.: Pattern inference. In: Algorithmic Learning for Knowledge-Based Systems, LNAI, 961. pp. 259–291, Springer, Berlin (1995)Google Scholar
  14. 14.
    Zeugmann, T.: Lange and Wiehagen’s pattern language learning algorithm: an average-case analysis with respect to its total learning time. Ann. Math. Artif. Intell. 23, 117–145 (1998)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Zeugmann, T., Lange, S.: A guided tour across the boundaries of learning recursive languages. In: Algorithmic Learning for Knowledge-Based Systems, LNAI, 961. pp. 190–258, Springer, Berlin (1995)Google Scholar

Copyright information

© Springer-Verlag London Limited  2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of Computer ScienceNational University of SingaporeSingaporeRepublic of Singapore
  2. 2.ERATO Minato ProjectJapan Science and Technology Agency Hokkaido UniversitySapporoJapan
  3. 3.Division of Computer ScienceHokkaido UniversitySapporoJapan

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