License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.128
URN: urn:nbn:de:0030-drops-62636
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6263/
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Chistikov, Dmitry ; Haase, Christoph

The Taming of the Semi-Linear Set

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LIPIcs-ICALP-2016-128.pdf (0.6 MB)

Abstract

Semi-linear sets, which are rational subsets of the monoid (Z^d,+), have numerous applications in theoretical computer science. Although semi-linear sets are usually given implicitly, by formulas in Presburger arithmetic or by other means, the effect of Boolean operations on semi-linear sets in terms of the size of description has primarily been studied for explicit representations. In this paper, we develop a framework suitable for implicitly presented semi-linear sets, in which the size of a semi-linear set is characterized by its norm—the maximal magnitude of a generator.

We put together a toolbox of operations and decompositions for semi-linear sets which gives bounds in terms of the norm (as opposed to just the bit-size of the description), a unified presentation, and simplified proofs. This toolbox, in particular, provides exponentially better bounds for the complement and set-theoretic difference. We also obtain bounds on unambiguous decompositions and, as an application of the toolbox, settle the complexity of the equivalence problem for exponent-sensitive commutative grammars.

BibTeX - Entry

@InProceedings{chistikov_et_al:LIPIcs:2016:6263,
  author =	{Dmitry Chistikov and Christoph Haase},
  title =	{{The Taming of the Semi-Linear Set}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{128:1--128:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6263},
  URN =		{urn:nbn:de:0030-drops-62636},
  doi =		{10.4230/LIPIcs.ICALP.2016.128},
  annote =	{Keywords: semi-linear sets, convex polyhedra, triangulations, integer linear programming, commutative grammars}
}

Keywords: semi-linear sets, convex polyhedra, triangulations, integer linear programming, commutative grammars
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016

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