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On the Learnability of Vector Spaces

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Algorithmic Learning Theory (ALT 2002)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2533))

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Abstract

The central topic of the paper is the learnability of the recursively enumerable subspaces of V ∞/V , where V ∞ is the standard recursive vector space over the rationals with countably infinite dimension, and V is a given recursively enumerable subspace of V ∞. It is shown that certain types of vector spaces can be characterized in terms of learnability properties: V ∞/V is behaviourally correct learnable from text iff V is finitely dimensional, V ∞/V is behaviourally correct learnable from switching type of information iff V is finite-dimensional, 0-thin, or 1-thin. On the other hand, learnability from an informant does not correspond to similar algebraic properties of a given space. There are 0-thin spaces W 1 and W 2 such that W 1 is not explanatorily learnable from informant and the infinite product (W 1)∞ is not behaviourally correct learnable, while W 2 and the infinite product (W 2)∞ are both explanatorily learnable from informant.

Valentina Harizanov was partially supported by the UFF grant of the George Washington University.

Frank Stephan was supported by the Deutsche Forschungsgemeinschaft (DFG) Heisenberg grant Ste 967/1-1.

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Author information

Authors and Affiliations

  1. The Department of Mathematics, The George Washington University, Funger Hall, 2201 G Street, DC, 20052, Washington, USA

    Valentina S. Harizanov

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

    Frank Stephan

Authors
  1. Valentina S. Harizanov
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  2. Frank Stephan
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Editor information

Editors and Affiliations

  1. Dipartimento di Tecnologie dell’Informazione, Università degli Studi di Milano, via Bramante 65, 26013, Crema (CR), Italy

    Nicolò Cesa-Bianchi

  2. Department of Computer Science, Tokyo Institute of Technology, 2-12-1, Ohokayama Meguro Ward, 152-8552, Tokyo, Japan

    Masayuki Numao

  3. Institut für Theoretische Informatik, Universität zu Lübeck, Wallstr. 40, 23560, Lübeck, Germany

    Rüdiger Reischuk

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© 2002 Springer-Verlag Berlin Heidelberg

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Harizanov, V.S., Stephan, F. (2002). On the Learnability of Vector Spaces. In: Cesa-Bianchi, N., Numao, M., Reischuk, R. (eds) Algorithmic Learning Theory. ALT 2002. Lecture Notes in Computer Science(), vol 2533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36169-3_20

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  • DOI: https://doi.org/10.1007/3-540-36169-3_20

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00170-6

  • Online ISBN: 978-3-540-36169-5

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