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Find Regular Expression by Arden’s Theorem:

Calculate RE of this machine by Arden’s Theorem:

First we calculate all the state.

Starting state $q_{1}$ accept λ and we can also get $q_{1}$ by $q_{1} . b$ or $q_{2} . a$ .

Now we can get $q_{2}$ by $q_{2} . b$ or $q_{1} . a$ .

Now if apply Arden’s theorem to $q_{2}$ then

$q_{2} = (q_{1} a) b *$ as per Arden’s theorem where r = $q_{2}$ , p = b and q = $q_{1} . a$

Here $q_{1}$ is the final state, then write $q_{1}$ equation and solve as per Arden’s theorem.

Now we can write above equation as r = q + rp = qp* to apply Arden’s theorem.

We can say r = $q_{1}$ , p = (b + ab*a) and q = λ.

Then as per Arden’s theorem we can write,

Regular Expression RE = (b + a*ba)*