Find Regular Expression by Arden’s Theorem:

Calculate RE of this machine by Arden’s Theorem:

First we calculate all the state.

Starting state ${\mathbf{q}}_{\mathbf{1}}$ accept λ and we can also get ${\mathbf{q}}_{\mathbf{1}}$ by ${\mathbf{q}}_{\mathbf{1}}\mathbf{.}\mathbf{b}$ or ${\mathbf{q}}_{\mathbf{2}}\mathbf{.}\mathbf{a}$

Now we can get ${\mathbf{q}}_{\mathbf{2}}$ by ${\mathbf{q}}_{\mathbf{2}}\mathbf{.}\mathbf{b}$ or ${\mathbf{q}}_{\mathbf{1}}\mathbf{.}\mathbf{a}$

Now if apply Arden’s theorem to ${\mathbf{q}}_{\mathbf{2}}$ then

as per Arden’s theorem where r = ${\mathbf{q}}_{\mathbf{2}}$, p = b and q = ${\mathbf{q}}_{\mathbf{1}}\mathbf{.}\mathbf{a}$

Here ${\mathbf{q}}_{\mathbf{1}}$ is the final state, then write ${\mathbf{q}}_{\mathbf{1}}$ 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 = ${\mathbf{q}}_{\mathbf{1}}$, p = (b + ab*a) and q = λ.

Then as per Arden’s theorem we can write,

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