The word problem for free groups cannot be solved in linear time*

TL;DR

This paper proves that the word problem for free groups cannot be solved in linear time on a standard Turing machine, contrasting with algorithms that run faster on more complex machines.

Contribution

It establishes a lower bound of quadratic time for solving the free group word problem on a standard Turing machine, highlighting a fundamental computational limitation.

Findings

Linear time solution is impossible on a standard Turing machine

Existing algorithms run in linear time on more complex machines

The result clarifies the computational complexity of the free group word problem

Abstract

*by a standard (one-tape) Turing machine. It is well-known that the word problem for hyperbolic groups, whence in particular for free groups, can be solved in linear time. However, these algorithms run on machines more complicated than a standard Turing machine. By contrast, in this note we show that a standard Turing machine cannot solve the word problem for the free group on two generators in less than quadratic time.