Answer

Correct Option: (iii) The Halting Problem

Explanation

Decidability:

Problem	Decidable?
DFA acceptance $w \in L(DFA)$?	Decidable — simulate DFA $O(|w|)$
NFA acceptance	Decidable — subset simulation or convert to DFA
Regex matching $w \in L(R)$?	Decidable — convert to DFA/NFA
Emptiness $L(DFA)=\emptyset$?	Decidable
Halting Problem $M$ halts on $w$?	Undecidable (Turing 1936)

Proof sketch: Assume $H(\langle M,w\rangle)$ decides halting, diagonalize with $D(\langle M\rangle)=$ loop if $M$ halts on $\langle M\rangle$ else halt. $D(\langle D\rangle)$ contradicts.

Final Answer

Undecidable is Halting Problem → Option (iii).