3.2 Level-Framework

The ''level''-framework which is implemented by Procedure 3 (see page ) is independent of such a value. Instead of a fixed number supplied by the user, an internal condition triggers the Gröbner basis computation.

At the beginning only the letters of the alphabet are added to the border set $B$. This is called level 1. Each element selected from the border set $B$ which is a new coset is added to the coset-set $N$. Further, it is multiplied with all letters of the alphabet except the inverse of its last letter. These Elements have length 2 and are added to the new border set $B_{new}$. Finally, the multiplication table $G$ is extended. If the current border set is empty the new border set becomes current and the Gröbner basis computation is performed. Thus, the border set of level $n$ consist of all elements of length $n$ which were generated by the previous level $n-1$. They are examined before the Gröbner basis of $G$ is computed again. As the recomputation of the Gröbner basis is triggered by the procedure itself this framework is self-adapting with respect to the number of new cosets defined.

As all elements of the current level $n$ are considered before the recomputation of $G$, that is all elements of length $n$, this method is independent of the order in which these elements are considered. Nevertheless, this framework, too, still depends on the method used to select elements from the borderset.