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.STACS.2019.43
URN: urn:nbn:de:0030-drops-102823
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10282/
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Kiefer, Stefan ; Mascle, Corto

On Finite Monoids over Nonnegative Integer Matrices and Short Killing Words

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Abstract

Let n be a natural number and M a set of n x n-matrices over the nonnegative integers such that M generates a finite multiplicative monoid. We show that if the zero matrix 0 is a product of matrices in M, then there are M_1, ..., M_{n^5} in M with M_1 *s M_{n^5} = 0. This result has applications in automata theory and the theory of codes. Specifically, if X subset Sigma^* is a finite incomplete code, then there exists a word w in Sigma^* of length polynomial in sum_{x in X} |x| such that w is not a factor of any word in X^*. This proves a weak version of Restivo's conjecture.

BibTeX - Entry

@InProceedings{kiefer_et_al:LIPIcs:2019:10282,
  author =	{Stefan Kiefer and Corto Mascle},
  title =	{{On Finite Monoids over Nonnegative Integer Matrices and Short Killing Words}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{43:1--43:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Rolf Niedermeier and Christophe Paul},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10282},
  doi =		{10.4230/LIPIcs.STACS.2019.43},
  annote =	{Keywords: matrix semigroups, unambiguous automata, codes, Restivo's conjecture}
}

Keywords: matrix semigroups, unambiguous automata, codes, Restivo's conjecture
Collection: 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)
Issue Date: 2019
Date of publication: 12.03.2019


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