Groups with ALOGTIME-Hard Word Problems and PSPACE-Complete Circuit Value Problems

Authors Laurent Bartholdi , Michael Figelius , Markus Lohrey , Armin Weiß

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Author Details

Laurent Bartholdi
  • ENS Lyon, Unité de Mathématiques Pures et Appliquées, France
  • Universität Göttingen, Mathematisches Institut, Germany
Michael Figelius
  • Universität Siegen, Germany
Markus Lohrey
  • Universität Siegen, Germany
Armin Weiß
  • Universität Stuttgart, Institut für Formale Methoden der Informatik (FMI), Germany


The authors are grateful to Schloss Dagstuhl and the organizers of Seminar 19131 for the invitation, where this work began.

Cite AsGet BibTex

Laurent Bartholdi, Michael Figelius, Markus Lohrey, and Armin Weiß. Groups with ALOGTIME-Hard Word Problems and PSPACE-Complete Circuit Value Problems. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 29:1-29:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk’s group and Thompson’s groups, we prove that their word problem is ALOGTIME-hard. For some of these groups (including Grigorchuk’s group and Thompson’s groups) we prove that the circuit value problem (which is equivalent to the circuit evaluation problem) is PSPACE-complete.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Circuit complexity
  • Mathematics of computing → Combinatorics
  • NC^1-hardness
  • word problem
  • G-programs
  • straight-line programs
  • non-solvable groups
  • self-similar groups
  • Thompson’s groups
  • Grigorchuk’s group


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