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Learnability of Quantified Formulas

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Computational Learning Theory (EuroCOLT 1999)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1572))

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Abstract

We consider the following classes of quantified formulas. Fix a set of basic relations called a basis. Take conjunctions of these basic relations applied to variables and constants in arbitrary ways. Finally, quantify existentially or universally some of the variables. We introduce some conditions on the basis that guarantee efficient learnability. Furthermore, we show that with certain restrictions on the basis the classification is complete. We introduce, as an intermediate tool, a link between this class of quantified formulas and some well-studied structures in Universal Algebra called clones. More precisely, we prove that the computational complexity of the learnability of these formulas is completely determined by a simple algebraic property of the basis of relations, their clone of polymorphisms. Finally, we use this technique to give a simpler proof of the already known dichotomy theorem over boolean domains and we present an extension of this theorem to bases with infinite size.

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Author information

Authors and Affiliations

  1. Departament LSI, Universitat Politécnica de Catalunya, Barcelona, Spain

    Víctor Dalmau

  2. Department of Computer Science, Royal Holloway, University of London, London, UK

    Peter Jeavons

Authors
  1. Víctor Dalmau
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  2. Peter Jeavons
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Editor information

Editors and Affiliations

  1. Lehrstuhl für Informatik II, Universität Dortmund, D-44221, Dortmund, Germany

    Paul Fischer

  2. Fakultät für Mathematik, Ruhr Universität Bochum, D-44780, Bochum, Germany

    Hans Ulrich Simon

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© 1999 Springer-Verlag Berlin Heidelberg

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Dalmau, V., Jeavons, P. (1999). Learnability of Quantified Formulas. In: Fischer, P., Simon, H.U. (eds) Computational Learning Theory. EuroCOLT 1999. Lecture Notes in Computer Science(), vol 1572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49097-3_6

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  • DOI: https://doi.org/10.1007/3-540-49097-3_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65701-9

  • Online ISBN: 978-3-540-49097-5

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