Let us close this section by sketching how this result closes the gap in comparing TC to KB type procedures in string rewriting. The case of the trivial subgroup has successfully been treated in [2,25] while for the general case a partial solution was presented in  which did not necessarily terminate for subgroups of finite index. Now using Knuth-Bendix techniques for prefix string rewriting systems we can give a procedure analogous to EXTENDED TC SIMULATION and hence to TC. We say the rule with prefix rewrites the word to v if is a prefix of u, say , and . Note that in this setting no free reduction steps are applied due to the fact that pure prefix string rewriting takes place in the free monoid. Therefore, we have to add the inverse relators to the defining relators of the group. Let PREFIXKB be an algorithm which given a finite set of rules , , computes the reduced equivalent convergent system.
llGiven: , . [1ex] ; if then ; else ; ; ; while do ; ; if is not prefix string reducible by G then ; ; ; ; G; ; endif endwhile endif
A thorough comparison of all three methods is provided in .