Answer

Correct Option: (i) Complement

Explanation

Languages accepted by Deterministic Pushdown Automata (DPDA) are Deterministic Context-Free Languages (DCFLs).

Closure Properties of DCFLs

Operation	Closed?
Complement	Yes
Union	No
Intersection	No
Intersection with Regular	Yes
Concatenation	No
Kleene Star	No

Reason:

• A DPDA can be complemented by swapping final/non-final states after ensuring it is total and handles end-of-input deterministically, similar to DFA complementation, but adapted for stack handling.
• DCFLs are not closed under union: e.g., $L_1 = \{a^n b^n \mid n\ge0\}$ and $L_2 = \{a^n b^{2n} \mid n\ge0\}$ are DCFLs, but $L_1 \cup L_2$ is not a DCFL.
• Since union fails, (iii) Both is false and (iv) is false.

Contrast with general CFLs (NPDA)

CFLs (non-deterministic) are not closed under complement nor intersection, but are closed under union, concatenation, Kleene star – opposite behaviour.

Final Answer

DCFL is closed under complement only. Hence option (i) Complement is correct.