The Power Word Problem

Authors Markus Lohrey, Armin Weiß

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Markus Lohrey
  • Universität Siegen, Germany
Armin Weiß
  • Universität Stuttgart, Germany


We thank Laurent Bartholdi for pointing out the result [Bartholdi et al, 2003, Theorem 6.6] on the bound of the order of elements in the Grigorchuk group, which allowed us to establish Theorem 10.

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Markus Lohrey and Armin Weiß. The Power Word Problem. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • word problem
  • compressed word problem
  • free groups


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