GATE CS 2022 Set 1 — Question 23
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Theory of Computation → Context-Free Languages → Closure Properties (CFL)
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Question
Which of the following statements is/are TRUE?
Correct answer
(B) If a language L and its complement L are both recursively enumerable, then L must be recursive.; (C) Complement of a context-free language must be recursive.; (D) If L₁ and L₂ are regular, then L₁ ∩ L₂ must be deterministic context-free.
Solution
Statement (A) is FALSE. Consider , which is a regular language and thus recursively enumerable. Any undecidable language is a subset of . Therefore, a subset of a recursively enumerable language is not necessarily recursive.Statement (B) is TRUE. This is a standard theorem: A language is recursive (decidable) if and only if both and its complement are recursively enumerable.Statement (C) is TRUE. Every context-free language (CFL) is recursive (decidable). The class of recursive languages is closed under complementation. Therefore, the complement of a CFL is recursive.Statement (D) is TRUE. The class of regular languages is closed under intersection. Thus, is regular. Since every regular language is a deterministic context-free language (DCFL), must be deterministic context-free.
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