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Widths of Regular and Context-Free Languages

Authors: David Mestel

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)


Abstract
Given a partially-ordered finite alphabet Sigma and a language L subseteq Sigma^*, how large can an antichain in L be (where L is given the lexicographic ordering)? More precisely, since L will in general be infinite, we should ask about the rate of growth of maximum antichains consisting of words of length n. This fundamental property of partial orders is known as the width, and in a companion work [Mestel, 2019] we show that the problem of computing the information leakage permitted by a deterministic interactive system modeled as a finite-state transducer can be reduced to the problem of computing the width of a certain regular language. In this paper, we show that if L is regular then there is a dichotomy between polynomial and exponential antichain growth. We give a polynomial-time algorithm to distinguish the two cases, and to compute the order of polynomial growth, with the language specified as an NFA. For context-free languages we show that there is a similar dichotomy, but now the problem of distinguishing the two cases is undecidable. Finally, we generalise the lexicographic order to tree languages, and show that for regular tree languages there is a trichotomy between polynomial, exponential and doubly exponential antichain growth.

Cite as

David Mestel. Widths of Regular and Context-Free Languages. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{mestel:LIPIcs.FSTTCS.2019.49,
  author =	{Mestel, David},
  title =	{{Widths of Regular and Context-Free Languages}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{49:1--49:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.49},
  URN =		{urn:nbn:de:0030-drops-116111},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.49},
  annote =	{Keywords: Formal languages, combinatorics on words, information flow}
}
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