GATE | GATE-CS-2015 (Set 1) | Question 65

For any two languages L1 and L2 such that L1 is context free and L2 is recursively enumerable but not recursive, which of the following is/are necessarily true?

1. L1' (complement of L1) is recursive 
2. L2' (complement of L2) is recursive
3. L1' is context-free 
4. L1' ∪ L2 is recursively enumerable 

(A) 1 only
(B) 3 only
(C) 3 and 4 only
(D) 1 and 4 only


Answer: (D)

Explanation: 1. L1′ (complement of L1) is recursive is true
L1 is context free. Every context free language is also recursive and recursive languages are closed under complement.

4. L1′ ∪ L2 is recursively enumerable is true
Since L1′ is recursive, it is also recursively enumerable and recursively enumerable languages are closed under union.
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: Wiki)

3. L1′ is context-free:
Context-free languages are not closed under complement, intersection, or difference.

2. L2′ (complement of L2) is recursive is false:
Recursively enumerable languages are not closed under set difference or complementation

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