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On Minimal Grammar Problems for Finite Languages

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Book cover Developments in Language Theory (DLT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11088))

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Abstract

We investigate the grammatical complexity of finite languages w.r.t. context-free grammars and variants thereof. For fixed alphabets, it is shown that both the minimal number of productions and the minimal size of a context-free grammar generating a finite language cannot be approximated within a factor of \(o(p^{1/6})\) and \(o(s^{1/7})\), respectively, unless . Here, p is the number of productions and s the size of the given grammar. Similar inapproximability results also hold for linear context-free and right-linear (or regular) grammars. As a byproduct, we show that the language of all cubes of a given length requires an exponential number of context-free productions and we also investigate upper and lower bounds on the complexity of the operations union and concatenation for finite languages.

S. Wolfsteiner—This research was completed while the author was on leave at the Institut für Informatik, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany, in 2017 and is supported by the Vienna Science Fund (WWTF) project VRG12-004.

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Authors and Affiliations

  1. knowledgepark GmbH, Leonrodstr. 68, 80636, München, Germany

    Hermann Gruber

  2. Institut für Informatik, Universität Giessen, Arndtstr. 2, 35392, Giessen, Germany

    Markus Holzer

  3. Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstr. 8–10, 1040, Wien, Austria

    Simon Wolfsteiner

Authors
  1. Hermann Gruber
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  2. Markus Holzer
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  3. Simon Wolfsteiner
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Corresponding author

Correspondence to Markus Holzer .

Editor information

Editors and Affiliations

  1. National Archives of Japan, Tokyo, Japan

    Mizuho Hoshi

  2. The University of Electro-Communications, Tokyo, Japan

    Shinnosuke Seki

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Gruber, H., Holzer, M., Wolfsteiner, S. (2018). On Minimal Grammar Problems for Finite Languages. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_28

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  • DOI: https://doi.org/10.1007/978-3-319-98654-8_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-98653-1

  • Online ISBN: 978-3-319-98654-8

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