Logspace Computations in Graph Groups and Coxeter Groups

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. We show that shortlex normal forms in graph groups and in right-angled Coxeter groups can be computed in logspace. Graph groups are also known as free partially commutative groups or as right-angled Artin groups in the literature. (Artin groups can be realized as subgroups of Coxeter groups.) Graph groups arise in many areas and have a close connection to concurrency theory. The connection is used here. Indeed, for our result we use a representation of group elements by Mazurkiewicz traces. These are directed acyclic node-labelled graphs (i.e. pomsets). They form an algebraic model to describe runs of concurrent systems. Concurrent systems which are deterministic and co-deterministic can be studied via inverse monoids. 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.

All Coxeter groups are linear, so the word problem can be solved in logspace, but it is open (in the non-right-angled case) whether shortlex normal forms can be computed in logspace, or, less demanding, whether they can be computed efficiently in parallel. We show that for all Coxeter groups the set of letters occurring in the shortlex normal form of an element can be computed in logspace.