On the complexity of the word problem of the R. Thompson group V

TL;DR

This paper investigates the computational complexity of the word problem in Thompson's group V, revealing its classification within formal language theory and establishing its quadratic time complexity.

Contribution

It demonstrates that the word problem of V is the complement of the cyclic closure of certain deterministic context-free languages, linking group theory with formal language classifications.

Findings

The word problem of V is co-context-free.

It belongs to the class logDCFL.

It has quadratic time complexity on a deterministic multitape Turing machine.

Abstract

We analyze the proof by Lehnert and Schweitzer that the word problem of the Thompson group V is co-context-free, and we show that this word problem is the complement of the cyclic closure of a union of reverse deterministic context-free languages. The same is true for any finitely generated subgroup of V. For certain finite generating sets, this word problem is the complement of the cyclic closure of the union of four deterministic context-free languages. Therefore the word problem of V has quadratic time-complexity on a deterministic multitape Turing machine, and belongs to logDCFL.