The first framework considered uses a value provided by the user. After new elements have been added to the coset table the Gröbner basis computation is performed. Procedure 2 (see page ) implements this framework.
Unfortunately, the choice of the parameter is not obvious as due to undecidability results it is not known beforehand whether the number of cosets is finite or not. Especially, it is not known how many cosets will be generated. This becomes even more complicated as there are examples where the number of cosets which have to be defined exceeds the number of cosets of the final result. Thus, a reasonably good value can only be chosen if the result (that is total and maximal number of cosets) is already known before the computation is done but in general this is impossible for unknown examples at the moment.
This framework depends on the order in which elements of the same length are considered. Changing the order in which the characters are given to the program or the way in which elements are selected from the borderset changes immediately the results unless .