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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 ${\mathbf{q}}_{\mathbf{1}}$ 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

${\mathbf{q}}_{\mathbf{2}}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{\left(}{\mathbf{q}}_{\mathbf{1}}\mathbf{a}\mathbf{\right)}\mathbf{b}\mathbf{*}$ 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)*