On deciding whether a monoid is a free monoid or is a group

Summary In general it is undecidable whether or not the monoid described by a given finite presentation is a free monoid or a group. However, these two decision problems are reducible to a very restricted form of the uniform word problem. So whenever a class of presentations is considered for which this restricted form of the uniform word problem is decidable, then the above decision problems become decidable. This holds in particular for the class of all presentations involving finite string-rewriting systems that are noetherian and confluent.