How do you do convert an ɛ-NFA to a regular expression?
Take for example this ɛ-NFA in which there's an ɛ transition from the final state q2 to the initial state q0:
I first combine eliminate q1 and get the regular expression $ab^*b(ab^*b)^*$:
How do I get from here to a regular expression?
Am I done now and is the regular expression simply $ab^*b(ab^*b)^*$ or should I eliminate the ɛ transition as well to get a correct regular expression?
Let us assume the lower branch is given by $r$ then the whole automaton should be $$ r^+ $$ which is short for $$ r^+ = r(r^*) = r^*r $$ I am not sure about the regexp for the lower branch.
It could be $$ a(b^*|(ba)^*)b $$
so we would end up with $$ (a(b^*|(ba)^*)b)^+ $$