Groups that do and do not have context-sensitive word problem

TL;DR

This paper characterizes groups with context-sensitive word problems using non-deterministic Cannon's algorithms, expanding the understanding of language-theoretic distinctions in group theory.

Contribution

It establishes a precise equivalence between context-sensitive word problems and non-deterministic Cannon's algorithms for groups, and identifies groups that do not admit such algorithms.

Findings

Characterizes groups with context-sensitive word problems

Identifies groups not admitting non-deterministic Cannon's algorithms

Provides new language-theoretic separation results

Abstract

We prove that a group has word problem that is a growing context-sensitive language precisely if its word problem can be solved using a non-deterministic Cannon's algorithm (the deterministic algorithms being defined by Goodman and Shapiro). We generalise their results to find many examples of groups not admitting non-deterministic Cannon's algorithms. This adds to the examples of Kambites and Otto of groups separating context-sensitive and growing context-sensitive word problems, and provides a new language-theoretic separation result.