7. Evaluation of the Frameworks and the Strategies

Thirteen examples were tabulated in [1,3]. They were computed using the ''level'' framework introduced in Section 3 and the seven strategies 1-7 introduced in Section 4 for adding elements to the border set. Neither the ''steps''-framework nor the strategies 8 and 9 were examined as they depend on additional parameters and therefore are more difficult to evaluate.

A length-lexicographical ordering (see Section 5) was used with the precedences chosen as depicted in Table 3 (see Appendix B.1, page ). The results together with the findings of [3] are presented in Table 4 - 7 (see Appendix B.2, pages - ) . The first column shows the best results for Felsch-Type enumeration, the second the results for lookahead HLT-Style. The third column shows the results for strategy NONE while columns 4-6 show either the prefix strategies P-ALL, P-G, and P-R or the inverse strategies I-ALL, I-R, and I-R-P.

The NONE-strategy performs better than HLT for 8 out of 13 examples considering the maximal number of cosets defined, and for 10 out of 13 examples considering the total number of cosets defined. Compared to the best results of the Felsch strategy which were presented in [3] it performs better only for two examples considering maximal and total number of cosets.

Adding elements makes things worse for most of the examples considered. But there are notable exceptions. Adding the inverses of the relators (I-R) reduces the number of cosets to be defined for $G(2,4) \vert E$ and $G(2,6) \vert E$ to about $65\%$.

Adding all cyclic permutations of the relators to the set of relators improves the enumeration considerably. Unfortunately, this also slows down the computation in most cases, as the Gröbner bases to be computed are larger and contain more information. The results are shown in Table 8 - 11 (see Appendix B.3, pages - ). Especially, if no elements are added to the border set we get a performance which is at least as good or even much better as without these permutations. In this case, NONE performs better than HLT with the exception of $G^{3,7,16}\vert E$ were the maximal number of cosets defined is higher while there are only half as many cosets defined totally. Compared to the Felsch-strategy, 8 out of 13 examples can be computed defining less cosets. Remarkable are the MacDonald groups $G(n,m) \vert E$ where 3 to 7 times less cosets have to be defined.

Now, adding elements does not improve the behaviour with the following exceptions. Adding the inverses of the relators (I-R) reduces the maximal/total number of cosets to be defined for $E_1 \vert E$ from 157/157 to 97/97 which is about about $38\%$ less. The same strategy reduces the numbers for $Neu \vert <a, c>$ slightly from 1683/1697 to 1637/1671. The same behaviour is found for $G(2,4) \vert E$ where a reduction from 467/467 to 424/424 is achieved and for $G(3,3)\vert E$ where the reduction is from 9753/9753 to 9253/9253 while for $G(2,6) \vert E$ more cosets have to be enumerated. For $G^{3, 7, 17} \vert <ab, c>$ strategy I-R-P leads to 945/998 cosets enumerated compared to 1153/1153 using strategy NONE. For $G^{3,7,16}\vert E$ all other strategies perform better than NONE.