GATE | GATE-CS-2005 | Question 56

Let L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?

L1' --> Complement of L1
L2' --> Complement of L2 

(A) L1′ is recursive and L2′ is recursively enumer­able
(B) L1′ is recursive and L2′ is not recursively enumerable
(C) L1′ and L2′ are recursively enumerable
(D) L1′ is recursively enumerable and L2′ is recursive


Answer: (B)

Explanation: Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively enumerable (Source: http://en.wikipedia.org/wiki/Recursively_enumerable_language )

Recursive Languages are closed under complementation, but recursively enumerable are not closed under complementation.  If a languages L is recursively enumerable, then the complement of it is recursively enumerable if and only if  L is also recursive.  Since L2 is recursively enumerable, but not recursive, L2′ is not recursively enumerable.

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