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On the Density of Regular and Context-Free Languages

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 6196)

Abstract

The density of a language is defined as the function \(d_L(n) = | L \cap \Sigma^n |\) and counts the number of words of a certain length accepted by L. The study of the density of regular and context-free languages has attracted some attention culminating in the fact that such languages are either sparse, when the density can be bounded by a polynomial, or dense otherwise. We show that for all nonambiguous context-free languages the number of accepted words of a given length n can also be computed recursively using a finite combination of the number of accepted words smaller than n, or \(d_L = \sum_{j=1}^k u_j d_L (n-j) \). This extends an old result by Chomsky and provides us with a more expressive description and new insights into possible applications of the density function for such languages as well as possible characterizations of the density of higher languages.

Keywords

  • context-free languages
  • regular languages
  • density

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Hartwig, M. (2010). On the Density of Regular and Context-Free Languages. In: Thai, M.T., Sahni, S. (eds) Computing and Combinatorics. COCOON 2010. Lecture Notes in Computer Science, vol 6196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14031-0_35

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  • DOI: https://doi.org/10.1007/978-3-642-14031-0_35

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14030-3

  • Online ISBN: 978-3-642-14031-0

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