GATE CS 2016 Set 2 — Question 28
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Theory of Computation → Context-Free Languages → Closure Properties (CFL)
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Question
Consider the following types of languages: : Regular, : Context-free, : Recursive, : Recursively enumerable. Which of the following is/are TRUE?I. is recursively enumerable
II. is recursive
III. is context-free
IV. is context-free
II. is recursive
III. is context-free
IV. is context-free
Correct answer
(D) I, II and III only
Solution
Let's analyze each statement:
- I. is recursively enumerable:
- is Recursive. The complement of a Recursive language () is also Recursive.
- Recursive languages are a subset of Recursively Enumerable (RE) languages.
- is RE.
- The union of two RE languages is RE. Since is Recursive (thus RE) and is RE, their union is RE.
- True.
- II. is recursive:
- is Context-Free (CFL). CFLs are a subset of Recursive languages.
- The complement of a CFL () is not necessarily CFL, but it is definitely Recursive (since Recursive languages are closed under complement).
- is Recursive.
- The union of two Recursive languages is Recursive.
- True.
- III. is context-free:
- is Regular. The Kleene star of a Regular language () is Regular.
- is CFL.
- The intersection of a Regular language and a CFL is always a CFL (Closure property).
- True.
- IV. is context-free:
- is Regular.
- is the complement of a CFL. The class of CFLs is not closed under complementation, so is not necessarily Context-Free.
- The union of a Regular language and a non-CFL is not guaranteed to be Context-Free.
- False.
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