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An Algebraic Perspective on Boolean Function Learning

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Algorithmic Learning Theory (ALT 2009)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5809))

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Abstract

In order to systematize existing results, we propose to analyze the learnability of boolean functions computed by an algebraically defined model, programs over monoids. The expressiveness of the model, hence its learning complexity, depends on the algebraic structure of the chosen monoid. We identify three classes of monoids that can be identified, respectively, from Membership queries alone, Equivalence queries alone, and both types of queries. The algorithms for the first class are new to our knowledge, while those for the other two are combinations or particular cases of known algorithms. Learnability of these three classes captures many previous learning results. Moreover, by using nontrivial taxonomies of monoids, we can argue that using the same techniques to learn larger classes of boolean functions seems to require proving new circuit lower bounds or proving learnability of DNF formulas.

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Author information

Authors and Affiliations

  1. Department of Software (LSI), U. Politècnica de Catalunya, Barcelona, Spain

    Ricard Gavaldà

  2. School of Computer Science, McGill University, Montréal, Québec, Canada

    Denis Thérien

Authors
  1. Ricard Gavaldà
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  2. Denis Thérien
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Editor information

Editors and Affiliations

  1. Research Group, Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya,, LARCA Jordi Girona Salgado 1-3, 08034, Barcelona, Spain

    Ricard Gavaldà

  2. ICREA and Department of Economics, Pompeu Fabra Universitat, Ramon Trias Fargas 25-27, 08005, Barcelona, Spain

    Gábor Lugosi

  3. Division of Computer Science, Hokkaido University, N-14, W-9, 060-0814, Sapporo, Japan

    Thomas Zeugmann

  4. Department of Computer Science, University of Regina, S4S 0A2, Regina, Saskatchewan, Canada

    Sandra Zilles

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© 2009 Springer-Verlag Berlin Heidelberg

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Gavaldà, R., Thérien, D. (2009). An Algebraic Perspective on Boolean Function Learning. In: Gavaldà, R., Lugosi, G., Zeugmann, T., Zilles, S. (eds) Algorithmic Learning Theory. ALT 2009. Lecture Notes in Computer Science(), vol 5809. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04414-4_19

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  • DOI: https://doi.org/10.1007/978-3-642-04414-4_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04413-7

  • Online ISBN: 978-3-642-04414-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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