License: Creative Commons Attribution-NoDerivs 3.0 Unported license (CC BY-ND 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2008.1354
URN: urn:nbn:de:0030-drops-13541
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2008/1354/
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Gelade, Wouter ; Neven, Frank

Succinctness of the Complement and Intersection of Regular Expressions

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Abstract

We study the succinctness of the complement and intersection of
regular expressions. In particular, we show that when constructing
a regular expression defining the complement of a given regular
expression, a double exponential size increase cannot be avoided.
Similarly, when constructing a regular expression defining the
intersection of a fixed and an arbitrary number of regular
expressions, an exponential and double exponential size increase,
respectively, can in worst-case not be avoided. All mentioned
lower bounds improve the existing ones by one exponential and are
tight in the sense that the target expression can be constructed in
the corresponding time class, i.e., exponential or double
exponential time. As a by-product, we generalize a theorem by
Ehrenfeucht and Zeiger stating that there is a class of DFAs which
are exponentially more succinct than regular expressions, to a
fixed four-letter alphabet. When the given regular expressions are
one-unambiguous, as for instance required by the XML Schema
specification, the complement can be computed in polynomial time
whereas the bounds concerning intersection continue to hold. For
the subclass of single-occurrence regular expressions, we prove a
tight exponential lower bound for intersection.

BibTeX - Entry

@InProceedings{gelade_et_al:LIPIcs:2008:1354,
  author =	{Wouter Gelade and Frank Neven},
  title =	{{Succinctness of the Complement and Intersection of Regular Expressions}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{325--336},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Susanne Albers and Pascal Weil},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2008/1354},
  URN =		{urn:nbn:de:0030-drops-13541},
  doi =		{10.4230/LIPIcs.STACS.2008.1354},
  annote =	{Keywords: }
}

Collection: 25th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2008
Date of publication: 06.02.2008

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