Knuth-Bendix completion algorithm: word problem

Question

Can someone explain me how to set up an algorithm to find the 12 normal forms of the group $A_4$ by making use of the Knuth-Bendix completion algorithm?

So we have that $RRR=1, SSS=1$ and $RSRS=1$. The goal is to find 12 normal forms (namely the identity, $R$, $RR$, $RS$, etc).

The first step is to look at the given three 'rules' and formulating new rules by trying to combine the old ones. For example, by searching for critical pairs like $RRRSRS$, which can become $SRS$ and $RR$, which gives us the new rule $SRS=RR$. This is straightforward but wouldn't bring us to a solution, because we have to see certain structures. My problem is that I don't know how a computer can see these patterns or can be explained as a step in an algorithm...

Answer 1

I have a Knuth-Bendix program, and I ran it on this example. It completed with the $7$ rewrite rules:

     [R^3 -> 1],
     [S^3 -> 1],
     [S*R*S -> R^2],
     [R*S*R -> S^2],
     [R^2*S^2 -> S*R],
     [S^2*R^2 -> R*S],
     [S*R^2*S -> R*S^2*R]

The $12$ normal form words are:

 1, R, R^2, R^2*S, R*S, R*S^2, R*S^2*R, S, S*R, S*R^2, S^2, S^2*R