Let's say that we have a set of rewrite rules:
$$AB \mapsto AC, A \mapsto B, B \mapsto A$$
Given the two strings $ABC$ and $BCC$
we know we can rewrite
$$ABC \mapsto ACC \mapsto BCC$$
We can define the minimum number of steps it takes for a rewrite from $S_1$ to $S_2$ to be $s(S_1, S_2)$. For example, $s(ABC, BCC) = 2$.
Is there a name for this distance?