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The Word Problem for Finitary Automaton Groups

Maximilian Kotowsky, Jan Philipp Wächter

TL;DR

It is shown that the respective compressed word problem, where the state sequence is given as a straight-line program, is PSpace-complete and the problems remain complete for their respective classes if the authors restrict the input alphabet of the automata to a binary one.

Abstract

A finitary automaton group is a group generated by an invertible, deterministic finite-state letter-to-letter transducer whose only cycles are self-loops at an identity state. We show that, for this presentation of finite groups, the uniform word problem is coNP-complete. Here, the input consists of a finitary automaton together with a finite state sequence and the question is whether the sequence acts trivially on all input words. Additionally, we also show that the respective compressed word problem, where the state sequence is given as a straight-line program, is PSpace-complete. In both cases, we give a direct reduction from the satisfiability problem for (quantified) boolean formulae and we further show that the problems remain complete for their respective classes if we restrict the input alphabet of the automata to a binary one.

The Word Problem for Finitary Automaton Groups

TL;DR

It is shown that the respective compressed word problem, where the state sequence is given as a straight-line program, is PSpace-complete and the problems remain complete for their respective classes if the authors restrict the input alphabet of the automata to a binary one.

Abstract

A finitary automaton group is a group generated by an invertible, deterministic finite-state letter-to-letter transducer whose only cycles are self-loops at an identity state. We show that, for this presentation of finite groups, the uniform word problem is coNP-complete. Here, the input consists of a finitary automaton together with a finite state sequence and the question is whether the sequence acts trivially on all input words. Additionally, we also show that the respective compressed word problem, where the state sequence is given as a straight-line program, is PSpace-complete. In both cases, we give a direct reduction from the satisfiability problem for (quantified) boolean formulae and we further show that the problems remain complete for their respective classes if we restrict the input alphabet of the automata to a binary one.
Paper Structure (4 sections, 11 theorems, 37 equations, 9 figures)

This paper contains 4 sections, 11 theorems, 37 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Word Problem
  4. The Compressed Word Problem

Key Result

Proposition 3.1

The uniform word problem for finitary automaton groups is in coNP.

Figures (9)

  • Figure 1: Single, inverted, combined and abbreviated cross diagrams
  • Figure 2: Automaton generating Grigorchuk's group
  • Figure 3: Illustration of the main idea for proving that Grigorchuk's group is contracting
  • Figure 4: Labeling the nodes of a regular binary tree with binary numbers in reverse notation with the corresponding commutators
  • Figure 5: The automaton part for the states $\{ r_n \mid 0 < n \leq N \}$. The dotted states are already defined by the subautomaton $\mathcal{R}$ and the transitions exist for all $a \in \Sigma$ and $b \in \Sigma \setminus \{ \bot, \top \}$.
  • ...and 4 more figures

Theorems & Definitions (38)

  • Example 2.1: Grigorchuk's Group
  • Remark 2.2
  • proof
  • Definition 2.4: compare to waechter2022automaton
  • Example 2.9: The Alternating Group of Degree $5$; compare to waechter2022automaton
  • Example 2.10: SENS Groups
  • Example 2.11: Again: Grigorchuk's Group
  • Proposition 3.1
  • proof
  • Definition 3.2
  • ...and 28 more