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The VC-Dimension of Subclasses of Pattern Languages

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Algorithmic Learning Theory (ALT 1999)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1720))

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Abstract

This paper derives the Vapnik Chervonenkis dimension of several natural subclasses of pattern languages. For classes with unbounded VC-dimension, an attempt is made to quantify the “rate of growth” of VC-dimension for these classes. This is achieved by computing, for each n, size of the “smallest” witness set of n elements that is shattered by the class. The paper considers both erasing (empty substitutions allowed) and nonerasing (empty substitutions not allowed) pattern languages. For erasing pattern languages, optimal bounds for this size — within polynomial order — are derived for the case of 1 variable occurrence and unary alphabet, for the case where the number of variable occurrences is bounded by a constant, and the general case of all pattern languages. The extent to which these results hold for nonerasing pattern languages is also investigated. Some results that shed light on efficient learning of subclasses of pattern languages are also given.

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Author information

Authors and Affiliations

  1. School of Computer Science and Engineering, University of New South Wales, Sydney, 2052, NSW, Australia

    Andrew Mitchell & Arun Sharma

  2. FIN/IWS, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Tobias Scheffer

  3. Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany

    Frank Stephan

Authors
  1. Andrew Mitchell
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  2. Tobias Scheffer
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  3. Arun Sharma
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  4. Frank Stephan
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Editor information

Editors and Affiliations

  1. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, 152-8552, Japan

    Osamu Watanabe

  2. Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku, Tokyo, 169-8050, Japan

    Takashi Yokomori

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© 1999 Springer-Verlag Berlin Heidelberg

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Mitchell, A., Scheffer, T., Sharma, A., Stephan, F. (1999). The VC-Dimension of Subclasses of Pattern Languages. In: Watanabe, O., Yokomori, T. (eds) Algorithmic Learning Theory. ALT 1999. Lecture Notes in Computer Science(), vol 1720. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46769-6_8

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  • DOI: https://doi.org/10.1007/3-540-46769-6_8

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66748-3

  • Online ISBN: 978-3-540-46769-4

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