A regular expression for the language $L=\{w:(n_a(w)-n_b(w))mod3=1\}$

Question

Assume a language like $L=\{w:(n_a(w)-n_b(w))mod3=1\}$ is given.
How can i find a regular expression for this language using a systematic process?
Note : I can easily draw a DFA accepting this language with 3 states. ( The remainder can be either 0 or 1 or 2. ) But the problem is how to find the regular expression.

Answer 1

The state removal method is probably the simplest to do by hand. In this example, we need only remove one state, $2$. Afterward, the edge $0\to1$ will be labeled $a\mid bb$, the edge $1\to0$ will be labeled $b\mid aa$, and we add loops $0\to 0$ labeled $(ba)^*$ and $1\to1$ labeled $(ab)^*$. The resulting regular expression is $$(ba)^*(a\mid bb)((ab)^*\mid(b\mid aa)(ba)^*(a\mid bb))^*. $$