research-article

Groups with ALOGTIME-hard Word Problems and PSPACE-complete Compressed Word Problems

Authors:

Laurent Bartholdi,

Michael Figelius,

Armin WeißAuthors Info & Claims

ACM Transactions on Computation Theory, Volume 14, Issue 3-4

Article No.: 11, Pages 1 - 41

https://doi.org/10.1145/3569708

Published: 01 February 2023 Publication History

Abstract

We give lower bounds on the complexity of the word problem for a large class of non-solvable infinite groups that we call strongly efficiently non-solvable groups. This class includes free groups, Grigorchuk’s group, and Thompson’s groups. We prove that these groups have an NC¹-hard word problem and that for some of them (including Grigorchuk’s group and Thompson’s groups) the compressed word problem (which is equivalent to the circuit evaluation problem) is PSPACE-complete.

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Cited By

Lohrey MStober FWeiß A(2024)The Power Word Problem in Graph ProductsTheory of Computing Systems10.1007/s00224-024-10173-z68:3(403-464)Online publication date: 7-May-2024
https://dl.acm.org/doi/10.1007/s00224-024-10173-z
Lohrey M(2023)Complexity of word problems for HNN-extensionsJournal of Computer and System Sciences10.1016/j.jcss.2023.02.005135:C(145-157)Online publication date: 1-Aug-2023
https://dl.acm.org/doi/10.1016/j.jcss.2023.02.005
Mattes CWeiß A(2023)Parallel algorithms for power circuits and the word problem of the Baumslag groupComputational Complexity10.1007/s00037-023-00244-x32:2Online publication date: 13-Sep-2023
https://dl.acm.org/doi/10.1007/s00037-023-00244-x

Index Terms

Groups with ALOGTIME-hard Word Problems and PSPACE-complete Compressed Word Problems
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
2. Theory of computation
  1. Computational complexity and cryptography
    1. Algebraic complexity theory
    2. Circuit complexity

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cover image ACM Transactions on Computation Theory

ACM Transactions on Computation Theory Volume 14, Issue 3-4

December 2022

122 pages

ISSN:1942-3454

EISSN:1942-3462

DOI:10.1145/3582881

Editor:
Ryan O'Donnell
Carnegie Mellon University, USA

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Publication History

Published: 01 February 2023

Online AM: 01 December 2022

Accepted: 24 October 2022

Revised: 10 October 2022

Received: 29 August 2021

Published in TOCT Volume 14, Issue 3-4

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Cited By

Lohrey MStober FWeiß A(2024)The Power Word Problem in Graph ProductsTheory of Computing Systems10.1007/s00224-024-10173-z68:3(403-464)Online publication date: 7-May-2024
https://dl.acm.org/doi/10.1007/s00224-024-10173-z
Lohrey M(2023)Complexity of word problems for HNN-extensionsJournal of Computer and System Sciences10.1016/j.jcss.2023.02.005135:C(145-157)Online publication date: 1-Aug-2023
https://dl.acm.org/doi/10.1016/j.jcss.2023.02.005
Mattes CWeiß A(2023)Parallel algorithms for power circuits and the word problem of the Baumslag groupComputational Complexity10.1007/s00037-023-00244-x32:2Online publication date: 13-Sep-2023
https://dl.acm.org/doi/10.1007/s00037-023-00244-x

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