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Word equations, constraints, and formal languages

Laura Ciobanu

TL;DR

The paper surveys word equations in free monoids and groups with non-rational constraints, distinguishing rational constraints from non-rational ones such as length and Parikh images. It surveys language-theoretic methods (notably EDT0L solution sets and slice closure) to establish decidability results for word equations with rational constraints and counting inequations, and extends these ideas to group-theoretic settings. It presents a broad landscape of decidability and undecidability across group classes (e.g., virtually abelian, hyperbolic, partially commutative, graph products), with interpretability and reductions to Hilbert’s 10th problem as central techniques. The work highlights strong cross-disciplinary links between algebra, logic, and formal language theory, and identifies several open questions and directions for translating insights from free monoids to group contexts and for applying these results in areas like string-solving and security analysis.

Abstract

In this short survey we describe recent advances on word equations with non-rational constraints in groups and monoids, highlighting the important role that formal languages play in this area.

Word equations, constraints, and formal languages

TL;DR

The paper surveys word equations in free monoids and groups with non-rational constraints, distinguishing rational constraints from non-rational ones such as length and Parikh images. It surveys language-theoretic methods (notably EDT0L solution sets and slice closure) to establish decidability results for word equations with rational constraints and counting inequations, and extends these ideas to group-theoretic settings. It presents a broad landscape of decidability and undecidability across group classes (e.g., virtually abelian, hyperbolic, partially commutative, graph products), with interpretability and reductions to Hilbert’s 10th problem as central techniques. The work highlights strong cross-disciplinary links between algebra, logic, and formal language theory, and identifies several open questions and directions for translating insights from free monoids to group contexts and for applying these results in areas like string-solving and security analysis.

Abstract

In this short survey we describe recent advances on word equations with non-rational constraints in groups and monoids, highlighting the important role that formal languages play in this area.
Paper Structure (6 sections, 5 theorems, 4 equations)

This paper contains 6 sections, 5 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Word equations with constraints
  3. Groups equations with constraints
  4. Background on group theory
  5. Word equations with constraints in groups.
  6. Conclusions

Key Result

theorem 1

CiobanuZetzsche2024a The problem of satisfiability of word equations with rational constraints and counting inequations is decidable.

Theorems & Definitions (6)

  • theorem 1
  • definition 1
  • theorem 2: CiobanuEvettsLevine
  • Theorem 1: CiobanuGarreta
  • Theorem 2: CiobanuGarreta
  • Theorem 3: CiobanuGarreta