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Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131)

Authors: Volker Diekert, Olga Kharlampovich, Markus Lohrey, and Alexei Myasnikov

Published in: Dagstuhl Reports, Volume 9, Issue 3 (2019)

Abstract

Since its early days, combinatorial group theory was deeply interwoven with computability theory. In the last 20 years we have seen many new successful interactions between group theory and computer science. On one hand, groups played an important rule in many developments in complexity theory and automata theory. On the other hand, concepts from these computer science fields as well as efficient algorithms, cryptography, and data compression led to the formulation of new questions in group theory. The Dagstuhl Seminar Algorithmic Problems in Group Theory was aimed at bringing together researchers from group theory and computer science so that they can share their expertise. This report documents the material presented during the course of the seminar.

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Volker Diekert, Olga Kharlampovich, Markus Lohrey, and Alexei Myasnikov. Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131). In Dagstuhl Reports, Volume 9, Issue 3, pp. 83-110, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{diekert_et_al:DagRep.9.3.83,
  author =	{Diekert, Volker and Kharlampovich, Olga and Lohrey, Markus and Myasnikov, Alexei},
  title =	{{Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131)}},
  pages =	{83--110},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2019},
  volume =	{9},
  number =	{3},
  editor =	{Diekert, Volker and Kharlampovich, Olga and Lohrey, Markus and Myasnikov, Alexei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.3.83},
  URN =		{urn:nbn:de:0030-drops-112939},
  doi =		{10.4230/DagRep.9.3.83},
  annote =	{Keywords: algorithmic group theory; generic-case complexity; circuit complexity; diophantine theories}
}

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DOI: 10.4230/LIPIcs.MFCS.2017.23

TC^0 Circuits for Algorithmic Problems in Nilpotent Groups

Authors: Alexei Myasnikov and Armin Weiß

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract

Recently, Macdonald et. al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in LOGSPACE. Here we follow their approach and show that all these problems are complete for the uniform circuit class TC^0 - uniformly for all r-generated nilpotent groups of class at most c for fixed r and c. Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform TC^0, while all the other problems we examine are shown to be TC^0-Turing reducible to the problem of computing greatest common divisors and expressing them as linear combinations.

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Alexei Myasnikov and Armin Weiß. TC^0 Circuits for Algorithmic Problems in Nilpotent Groups. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{myasnikov_et_al:LIPIcs.MFCS.2017.23,
  author =	{Myasnikov, Alexei and Wei{\ss}, Armin},
  title =	{{TC^0 Circuits for Algorithmic Problems in Nilpotent Groups}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{23:1--23:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.23},
  URN =		{urn:nbn:de:0030-drops-81089},
  doi =		{10.4230/LIPIcs.MFCS.2017.23},
  annote =	{Keywords: nilpotent groups, TC^0, abelian groups, word problem, conjugacy problem, subgroup membership problem, greatest common divisors}
}

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Documents authored by Myasnikov, Alexei

Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131)

Abstract

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TC^0 Circuits for Algorithmic Problems in Nilpotent Groups

Abstract

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