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.MFCS.2019.43
URN: urn:nbn:de:0030-drops-109871
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Lohrey, Markus ; WeiƟ, Armin

The Power Word Problem

LIPIcs-MFCS-2019-43.pdf (0.5 MB)


In this work we introduce a new succinct variant of the word problem in a finitely generated group G, which we call the power word problem: the input word may contain powers p^x, where p is a finite word over generators of G and x is a binary encoded integer. The power word problem is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G). The main result of the paper states that the power word problem for a finitely generated free group F is AC^0-Turing-reducible to the word problem for F. Moreover, the following hardness result is shown: For a wreath product G Wr Z, where G is either free of rank at least two or finite non-solvable, the power word problem is complete for coNP. This contrasts with the situation where G is abelian: then the power word problem is shown to be in TC^0.

BibTeX - Entry

  author =	{Markus Lohrey and Armin Wei{\ss}},
  title =	{{The Power Word Problem}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{43:1--43:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-109871},
  doi =		{10.4230/LIPIcs.MFCS.2019.43},
  annote =	{Keywords: word problem, compressed word problem, free groups}

Keywords: word problem, compressed word problem, free groups
Collection: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 20.08.2019

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