License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2021.74
URN: urn:nbn:de:0030-drops-145148
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14514/
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Mattes, Caroline ; Weiß, Armin

Parallel Algorithms for Power Circuits and the Word Problem of the Baumslag Group

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LIPIcs-MFCS-2021-74.pdf (1 MB)

Abstract

Power circuits have been introduced in 2012 by Myasnikov, Ushakov and Won as a data structure for non-elementarily compressed integers supporting the arithmetic operations addition and (x,y) ↦ x⋅2^y. The same authors applied power circuits to give a polynomial-time solution to the word problem of the Baumslag group, which has a non-elementary Dehn function.
In this work, we examine power circuits and the word problem of the Baumslag group under parallel complexity aspects. In particular, we establish that the word problem of the Baumslag group can be solved in NC - even though one of the essential steps is to compare two integers given by power circuits and this, in general, is shown to be ?-complete. The key observation is that the depth of the occurring power circuits is logarithmic and such power circuits can be compared in NC.

BibTeX - Entry

@InProceedings{mattes_et_al:LIPIcs.MFCS.2021.74,
  author =	{Mattes, Caroline and Wei{\ss}, Armin},
  title =	{{Parallel Algorithms for Power Circuits and the Word Problem of the Baumslag Group}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{74:1--74:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14514},
  URN =		{urn:nbn:de:0030-drops-145148},
  doi =		{10.4230/LIPIcs.MFCS.2021.74},
  annote =	{Keywords: Word problem, Baumslag group, power circuit, parallel complexity}
}

Keywords: Word problem, Baumslag group, power circuit, parallel complexity
Collection: 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
Issue Date: 2021
Date of publication: 18.08.2021

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