However, MATPHI works in the setting of commutative polynomial rings using a Gröbner basis as input while TC belongs to the setting of groups using arbitrary relators and subgroup generators as input. In order to compare both methods, we have to use the generalized setting presented in Section 1 - binomial ideals in free group rings - and enable the new procedure to deal with possibly infinite generating sets in a finitary manner.
To encode the input of TC as binomials we associate the relators R and the subgroup generators U with two sets of polynomials and . Essentially we want to check whether the subgroup generated by in is finitely generated and this will be done in an incremental fashion using the fact that for a given finitely generated subgroup of a free group the membership problem can be solved using prefix Gröbner bases and the generating subset of the subgroup is then enlarged by adding polynomials modified by left multiplication with suitable group elements in order to ``approximate'' N(R).
To compute prefix Gröbner bases of subgroups in the free group ring we need the concept of weak prefix saturation: A set is called weakly prefix saturated if for every , we have . This becomes necessary as the ordering on is no longer admissible (see  for the details).
The property of being weakly saturated can be ensured for a set of polynomials by using a procedure to compute a saturating set for a polynomial, i.e. a set such that each right multiple of the polynomial prefix reduces to 0 in one step by a polynomial in the saturating set. For free groups there are saturating sets consisting of at most two polynomials called and . In our setting of binomials u - v, informally is gained from u-v by ``shortening'' the head term u without losing its head position while is derived from by forcing the shortened head term to lose its head position by cutting off its last letter. Then and where , and there exists such that , , and .
Procedure: PREFIX GR¨OBNER BASES OF RIGHT IDEALS IN FREE GROUP RINGS
Given: A finite set . Find: G, the monic reduced prefix Gröbner basis of the right ideal generated by F. [1ex]G := ; while there is such that is prefixreducible by do G := ; f := ; % Compute a normal form (if non-zero with head coefficient 1). if then G := ; endif endwhile
Correctness and termination follow from the results presented in [23,16]. The procedure can be used to solve the subgroup problem for a subgroup in :
However, in general the subgroup of we are interested in is generated by the set where the set of relators is not empty. We have to find a way to treat this possibly infinitely generated subgroup of in a finitary manner in order to verify whether it is in fact finitely generated. The normal closure of a set of relators R can be approached using a result similar to the one presented in  to solve the ideal membership problem for two-sided ideals in solvable polynomial rings using one-sided ideals (compare also Zharkov's idea to compute Janet bases in ). For a set let denote the two-sided and the right ideal generated by F in .