Answer

Correct Option: (i) $L'$ is recursively enumerable (equivalently (iii) Both if $T$ is decider)

Explanation

Theorem (Sipser): RE language $L$ is recursive (decidable) iff both $L$ and $\overline{L}$ are RE.

• If $L$ is RE and $\overline{L}$ is RE, then dovetail enumerators for $L$ and $\overline{L}$; whichever enumerates $w$ first decides.
• Formal: $L$ recursive ⇔ $L$ RE and $L'$ RE.

Option (ii) “every sequence of moves halts” is characterization of decider (always halts), also true for recursive $L$ but not the standard RE condition. Option (i) is the textbook criterion.

If exam lists (iii) Both (i) and (ii), that is also correct for decider TM. Most keys mark (i) or (iii).

Final Answer

$L$ recursive iff $L'$ is RE → Option (i) (and (iii) Both if inclusive).