6. Sims' Approach

In Section 2.8. of [15] Sims gives an approach similar to the one in the previous
section but allowing *arbitrary* presentations
for the group .
Instead of defining a specialized completion procedure for
prefix string rewriting he encodes the set *T*_{S} into an ordinary string rewriting system
by adding a new symbol $ and transforming the rules to a new set
.
Then the extended string rewriting system
is completed using KB.

For the cosets of the subgroup of the Dyck group *D*(3,3,2) as presented in Example 2
we get the following situation:
,
,
and
.
Completing the string rewriting system
with respect to the length-lexicographical ordering induced by the precedence
using KB
results in the convergent set
of rules
.
For the sets *T*_{C} and *T* from the previous section we then have
and
.

Let us continue with a comparison of TC and KB as presented in this setting:
When running KB for the free group on
we are in fact
computing a Nielsen reduced set for the subgroup generated by *S*.
The situation is slightly different in the general case, as in contrary to TC, although we
are simulating coset enumeration, it no longer must terminate for finite index.
This is due to the fact that in any case KB will try to complete the defining relators
and there are examples where we have finite index but no finite convergent system for
exists.
However, if we know that the index is finite, it is possible to find a bound on how far we
have to run KB to gain enough information to describe the cosets.
More information on this can be found in Section 3.10. in [15] where the following example is
taken from.
Notice that this example cannot be handled by the approach of Kuhn and Madlener
in the previous section for the chosen string rewriting system presenting the group is not convergent.

Next we provide a procedure which can handle the finite index case *without* applying
additional knowledge.