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Learnability and Definability in Trees and Similar Structures

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Abstract

We prove upper bounds for combinatorial parameters of finite relational structures, related to the complexity of learning a definable set. We show that monadic second-order (MSO) formulas with parameters have bounded Vapnik–Chervonenkis dimension over structures of bounded clique-width, and first-order formulas with parameters have bounded Vapnik–Chervonenkis dimension over structures of bounded local clique-width (this includes planar graphs). We also show that MSO formulas of a fixed size have bounded strong consistency dimension over MSO formulas of a fixed larger size, for labeled trees. These bounds imply positive learnability results for the PAC and equivalence query learnability of a definable set over these structures. The proofs are based on bounds for related definability problems for tree automata.

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Authors and Affiliations

  1. Institut f\”ur Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

    Martin Grohe

  2. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA and Research Group on AI, Hungarian Academy of Sciences, Szeged, Hungary

    Gy. Turán

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  1. Martin Grohe
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  2. Gy. Turán
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Correspondence to Martin Grohe or Gy. Turán.

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Grohe, M., Turán, G. Learnability and Definability in Trees and Similar Structures. Theory Comput Systems 37, 193–220 (2004). https://doi.org/10.1007/s00224-003-1112-8

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  • DOI: https://doi.org/10.1007/s00224-003-1112-8

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