After the discovery of various criteria for discarding critical pairs ahead of time by B. Buchberger and H.M. Möller (see [@Bu79], [@Bu85] and [@Mo85]), this subject found an initial resolution via the *Gebauer-Möller installation* presented in [@GM88] which offers a good compromise between efficiency and the success rate for detecting unnecessary critical pairs.

A very different picture presents itself for Gröbner basis computations for two-sided ideals in non-commutative polynomial rings. The basic Gröbner basis theory in this case was described by G.H. Bergman (see [@Be78]), T. Mora (see [@Mo86] and [@Mo94]) and others, and obstructions, the non-commutative analogue of critical pairs, were studied in [@Mo94]. However, since only a few authors endeavoured to implement efficient versions of Buchberger’s Procedure for the non-commutative polynomial ring (i.e. the free associative algebra), the subject of minimizing the number of obstructions which have to be treated has received comparatively little attention, and merely a few rules were developed. For instance, the package [Plural]{} of the computer algebra system [Singular]{} implements a version of the product and the chain criterion, but not the multiply criterion or the leading word criterion. On the other hand, the system [Magma]{} appears to be based on a variant of 