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On groups with EDT0L word problem

Alex Bishop, Murray Elder, Alex Evetts, Paul Gallot, Alex Levine

TL;DR

The paper analyzes the word problem in finitely generated groups through the lens of EDT0L languages, proving that the word problem of the infinite cyclic group $( olinebreak \\mathbb{Z},\{1,-1\})$ is not EDT0L and deriving that any group with an EDT0L word problem must be torsion. It establishes that EDT0L properties are invariant under changing generating sets and relate to finite-index submonoids, and it connects finite-index EDT0L languages to restricted non-branching MCFGs (R-MCFGs). A central technical contribution is showing that finite-index EDT0L languages coincide with R-MCFGs and that the EDT0L framework cannot capture the $ olinebreak ext{WP}( olinebreak ext{Z})$ language via a normal-form grammar, using a carefully crafted counterexample word and a decomposition-dynamics analysis. The results significantly progress toward the conjecture that groups with EDT0L word problems are finite and illustrate deep interactions between formal language theory and geometric group theory.

Abstract

We prove that the word problem for the infinite cyclic group is not EDT0L, and obtain as a corollary that a finitely generated group with EDT0L word problem must be torsion. In addition, we show that the property of having an EDT0L word problem is invariant under change of generating set and passing to finitely generated subgroups. This represents significant progress towards the conjecture that all groups with EDT0L word problem are finite (i.e. precisely the groups with regular word problem).

On groups with EDT0L word problem

TL;DR

The paper analyzes the word problem in finitely generated groups through the lens of EDT0L languages, proving that the word problem of the infinite cyclic group is not EDT0L and deriving that any group with an EDT0L word problem must be torsion. It establishes that EDT0L properties are invariant under changing generating sets and relate to finite-index submonoids, and it connects finite-index EDT0L languages to restricted non-branching MCFGs (R-MCFGs). A central technical contribution is showing that finite-index EDT0L languages coincide with R-MCFGs and that the EDT0L framework cannot capture the language via a normal-form grammar, using a carefully crafted counterexample word and a decomposition-dynamics analysis. The results significantly progress toward the conjecture that groups with EDT0L word problems are finite and illustrate deep interactions between formal language theory and geometric group theory.

Abstract

We prove that the word problem for the infinite cyclic group is not EDT0L, and obtain as a corollary that a finitely generated group with EDT0L word problem must be torsion. In addition, we show that the property of having an EDT0L word problem is invariant under change of generating set and passing to finitely generated subgroups. This represents significant progress towards the conjecture that all groups with EDT0L word problem are finite (i.e. precisely the groups with regular word problem).

Paper Structure

This paper contains 18 sections, 34 theorems, 239 equations, 3 figures.

Table of Contents

  1. Introduction
  2. High-level overview of the proof of Theorem \ref{['thm:main']}
  3. Notation
  4. Geodesics in finitely-generated groups
  5. EDT0L and EDT0L of finite index
  6. EDT0L is closed under application of string transducers
  7. EDT0L word problem
  8. Non-branching multiple context-free languages
  9. Proof of the main theorem
  10. Counterexample word
  11. Constants
  12. Functions on words and word weights
  13. Decompositions
  14. Maintaining a decomposition
  15. Mirroring decompositions
  16. ...and 3 more sections

Key Result

Proposition 0

Let $G$ be a group with finite monoid generating sets $X$ and $Y$. If $\mathrm{WP}(G,X)$ (resp. $\mathrm{coWP}(G,X)$) is EDT0L, then $\mathrm{WP}(G,Y)$ (resp. $\mathrm{coWP}(G,Y)$) is EDT0L of finite index. These results also hold if $Y$ instead generates a submonoid of $G$. In particular, this impl

Figures (3)

  • Figure 1: Inclusion diagram of formal languages (see Theorem 15 in Rozenberg1978 and Figure 8 in engelfriet1980stack).
  • Figure 2: The word $\mathcal{W}$ defined in \ref{['sec:main-theorem']} for values $k=2$ and $m=4$.
  • Figure 3: Add one to a number encoded in binary with an end marker.

Theorems & Definitions (91)

  • Proposition 0
  • Conjecture 1: Conjecture 8.2 in CiobanuElderFerov
  • Theorem 2
  • Theorem 3
  • Definition 3.1
  • Definition 3.2: See Rozenberg1978
  • Lemma 3.3
  • Definition 3.4
  • Lemma 3.5: Latteux latteux1977
  • proof
  • ...and 81 more