Answer

Correct Option: (i) $\{a^n b^n\}$

Explanation

Deterministic CFL (DCFL): Accepted by DPDA (deterministic pushdown).

• (i) $\{a^n b^n\}$: DPDA can push $a$'s, then pop for $b$'s, deterministic (no choice). DCFL.
• (ii) Arithmetic expressions (e.g., $E \rightarrow E+E \mid E*E$) → inherently ambiguous, CFL but not deterministic (needs guessing).
• (iii) Palindromes $\{ww^R\}$ → not deterministic (needs to guess middle). $\{w\#w^R\}$ is DCFL, but plain palindromes is NPD A.
• (iv) $\{ww^R\}$ similar to (iii) — not DCFL.

Hence only $\{a^n b^n\}$ is deterministic.

Final Answer

$\{a^n b^n\}$ is DCFL → Option (i).