Logspace Computations in Coxeter Groups and Graph Groups

TL;DR

This paper demonstrates that key normal form computations in Coxeter and graph groups can be performed in logarithmic space, significantly improving the understanding of their computational complexity and applications in group theory and concurrency.

Contribution

It introduces logspace algorithms for computing normal forms in Coxeter groups, right-angled Coxeter groups, and graph groups, advancing the complexity analysis of these problems.

Findings

Normal form length and letter set can be computed in logspace for Coxeter groups.

Short length normal forms in right-angled Coxeter groups are computable in logspace.

Word problem for free partially commutative inverse monoids is in logspace.

Abstract

Computing normal forms in groups (or monoids) is in general harder than solving the word problem (equality testing). However, normal form computation has a much wider range of applications. It is therefore interesting to investigate the complexity of computing normal forms for important classes of groups. For Coxeter groups we show that the following algorithmic tasks can be solved by a deterministic Turing machine using logarithmic work space, only: 1. Compute the length of any geodesic normal form. 2. Compute the set of letters occurring in any geodesic normal form. 3. Compute the Parikh-image of any geodesic normal form in case that all defining relations have even length (i.e., in even Coxeter groups.) 4. For right-angled Coxeter groups we can do actually compute the short length normal form in logspace. (Note that short length normal forms are geodesic.) Next, we apply the results to right-angled Artin groups. They are also known as free partially commutative groups or as graph groups. As a consequence of our result on right-angled Coxeter groups we show that shortlex normal forms in graph groups can be computed in logspace, too. Graph groups play an important role in group theory, and they have a close connection to concurrency theory. As an application of our results we show that the word problem for free partially commutative inverse monoids is in logspace. This result generalizes a result of Ondrusch and the third author on free inverse monoids. Concurrent systems which are deterministic and co-deterministic can be studied via inverse monoids.