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DOI: 10.4230/LIPIcs.FSTTCS.2013.475

On the Pseudoperiodic Extension of u^l = v^m w^n

Authors: Florin Manea, Mike Müller, and Dirk Nowotka

Published in: LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

Abstract

We investigate the solution set of the pseudoperiodic extension of the classical Lyndon and Sch\"utzenberger word equations. Consider u_1 ... u_l = v_1 ... v_m w_1 ... w_n, where u_i is in {u, theta(u)} for all 1 <= i <= l, v_j is in {v, theta(v)} for all 1 <= j <= m, w_k is in {w, theta(w)} for all 1 <= k <= n and u, v and w are variables, and theta is an antimorphic involution. A solution is called pseudoperiodic, if u,v,w are in {t, theta(t)}^+ for a word t. [Czeizler et al./I&C/2011] established that for small values of l, m, and n non-periodic solutions exist, and that for large enough values all solutions are pseudoperiodic. However, they leave a gap between those bounds which we close for a number of cases. Namely, we show that for l = 3 and either m,n >= 12 or m,n >= 5 and either m and n are not both even or not all u_i's are equal, all solutions are pseudoperiodic.

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Florin Manea, Mike Müller, and Dirk Nowotka. On the Pseudoperiodic Extension of u^l = v^m w^n. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 475-486, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{manea_et_al:LIPIcs.FSTTCS.2013.475,
  author =	{Manea, Florin and M\"{u}ller, Mike and Nowotka, Dirk},
  title =	{{On the Pseudoperiodic Extension of u^l = v^m w^n}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)},
  pages =	{475--486},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-64-4},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{24},
  editor =	{Seth, Anil and Vishnoi, Nisheeth K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.475},
  URN =		{urn:nbn:de:0030-drops-43948},
  doi =		{10.4230/LIPIcs.FSTTCS.2013.475},
  annote =	{Keywords: Word equations, Pseudoperiodicity, Lyndon-Sch\"{u}tzenberger equation}
}

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On the Pseudoperiodic Extension of u^l = v^m w^n

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