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The Power Word Problem in Graph Products

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Developments in Language Theory (DLT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13257))

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Abstract

The power word problem of a group G asks whether an expression \(p_1^{x_1} \dots p_n^{x_n}\), where the \(p_i\) are words and the \(x_i\) binary encoded integers, is equal to the identity of G. We show that the power word problem in a fixed graph product is \({{{\textsf {{AC}}}}}^0\)-Turing-reducible to the word problem of the free group \(F_2\) and the power word problem of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups \(\mathcal {C}\), the uniform power word problem in a graph product can be solved in \({{{\textsf {{AC}}}}}^0({ {{\mathsf {{C}}}}_=\!\;\!{{\mathsf {{L}}}} }^{\mathrm {PowWP}\mathcal {C}})\). As a consequence of our results, the uniform knapsack problem in graph groups is \({{{\textsf {{NP}}}}}\)-complete.

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Authors and Affiliations

  1. FMI, Universität Stuttgart, Stuttgart, Germany

    Florian Stober & Armin Weiß

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  1. Florian Stober
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  2. Armin Weiß
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Correspondence to Armin Weiß .

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Editors and Affiliations

  1. University of Stuttgart, Stuttgart, Baden-Württemberg, Germany

    Volker Diekert

  2. Ural Federal University, Ekaterinburg, Russia

    Mikhail Volkov

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Stober, F., Weiß, A. (2022). The Power Word Problem in Graph Products. In: Diekert, V., Volkov, M. (eds) Developments in Language Theory. DLT 2022. Lecture Notes in Computer Science, vol 13257. Springer, Cham. https://doi.org/10.1007/978-3-031-05578-2_23

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  • DOI: https://doi.org/10.1007/978-3-031-05578-2_23

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-05577-5

  • Online ISBN: 978-3-031-05578-2

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