2 Search Results for "Mattes, Caroline"

Document
DOI: 10.4230/LIPIcs.MFCS.2021.74
Parallel Algorithms for Power Circuits and the Word Problem of the Baumslag Group

Authors: Caroline Mattes and Armin Weiß

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

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 𝖯-complete. The key observation is that the depth of the occurring power circuits is logarithmic and such power circuits can be compared in NC.
Cite as

Caroline Mattes and Armin Weiß. Parallel Algorithms for Power Circuits and the Word Problem of the Baumslag Group. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 74:1-74:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@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/entities/document/10.4230/LIPIcs.MFCS.2021.74},
  URN =		{urn:nbn:de:0030-drops-145148},
  doi =		{10.4230/LIPIcs.MFCS.2021.74},
  annote =	{Keywords: Word problem, Baumslag group, power circuit, parallel complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.102
Hardness of Equations over Finite Solvable Groups Under the Exponential Time Hypothesis

Authors: Armin Weiß

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in 𝖯 for nilpotent groups while it is 𝖭𝖯-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of Fitting length two. In this work, we present the first lower bounds for the equation satisfiability problem in finite solvable groups: under the assumption of the exponential time hypothesis, we show that it cannot be in 𝖯 for any group of Fitting length at least four and for certain groups of Fitting length three. Moreover, the same hardness result applies to the equation identity problem.
Cite as

Armin Weiß. Hardness of Equations over Finite Solvable Groups Under the Exponential Time Hypothesis. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 102:1-102:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{wei:LIPIcs.ICALP.2020.102,
  author =	{Wei{\ss}, Armin},
  title =	{{Hardness of Equations over Finite Solvable Groups Under the Exponential Time Hypothesis}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{102:1--102:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.102},
  URN =		{urn:nbn:de:0030-drops-125093},
  doi =		{10.4230/LIPIcs.ICALP.2020.102},
  annote =	{Keywords: equations in groups, solvable groups, exponential time hypothesis}
}
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