Answer

Correct Option: (iii) $R = QP^*$ (Arden’s Theorem)

Explanation

Arden’s Theorem: If $P$ does not contain $\epsilon$, then equation $R = Q + RP$ has unique solution $R = QP^*$.

Proof: $R = Q + (Q + RP)P = Q(\epsilon + P) + RP^2 = \dots = Q(\epsilon + P + P^2 + \dots + P^{k}) + RP^{k+1}$. As $k \to \infty$, $RP^{k+1}$ contributes no finite strings if language, so $R = QP^*$.

• (i) $QP$ missing star → incomplete.
• (ii) $PQ^*$ swaps order (concatenation not commutative).
• (iv) duplicate of (iii).

Condition: $P$ not null ($\epsilon
otin P$) ensures uniqueness.

Final Answer

$R = QP^*$ → Option (iii) (also (iv) if duplicate).