Let L be a language and L’ be its complement. Which one of the following is NOT a viable possibility?
(A) Neither L nor L’ is recursively enumerable (r.e.).
(B) One of L and L’ is r.e. but not recursive; the other is not r.e.
(C) Both L and L’ are r.e. but not recursive.
(D) Both L and L’ are recursive
Answer: (C)
Explanation: A) It is possible if L itself is NOT RE. Then L’ will also not be RE.
B) Suppose there is a language such that turing machine halts on the input. The given language is RE but not recursive and its complement is NOT RE.
C) This is not possible because if we can write enumeration procedure for both languages and it’s complement, then the language becomes recursive.
D) It is possible because L is closed under complement if it is recursive.
Thus, C is the correct choice.
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