Cancellativity in finitely presented semigroups

The question of whether a monoid presented by a finite Thue system is cancellative is shown to be recursively undecidable, even when the Thue system is Church-Rosser. A decision procedure is described for the case of monadic Church-Rosser Thue systems, and general commutative Thue systems. For canonical commutative systems, the negation of the problem is in NP.