On groups with EDT0L word problem

2025-05-26

AI Summary

This paper investigates whether the word problem of a group belongs to the class of EDT0L languages—specifically, which groups possess EDT0L word problems. Method: The authors employ techniques from formal language theory, combinatorial group theory, and automatic structure analysis to establish invariance properties and structural constraints. Contribution/Results: First, they prove that the word problem of the infinite cyclic group is not EDT0L, implying that every finitely generated EDT0L group must be a torsion group. Second, they establish, for the first time, the invariance of the EDT0L property under change of finite generating set and passage to finitely generated subgroups. Third, they rigorously confine EDT0L groups to the class of torsion groups, providing critical theoretical support for the long-standing conjecture that all EDT0L groups are finite. Collectively, this work introduces a novel group-theoretic invariance framework for EDT0L properties and significantly advances the classification of word problem complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Natural Language Processing: Lexical Semantics and Morphology

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• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing

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).

Problem

• Prove infinite cyclic group lacks EDT0L word problem
• Show EDT0L word problem implies torsion groups
• Demonstrate EDT0L property invariance under subgroup changes

Innovation

• Proves cyclic group word problem not EDT0L
• Shows EDT0L invariance under generating sets
• Links EDT0L word problem to finite groups