GATE CS 2019 Set 1 — Question 41
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Theory of Computation → Context-Free Languages → Pumping Lemma (CFL)
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Question
Which one of the following languages over is NOT context-free?
Correct answer
(C) \waⁿ w^R bⁿ w ∈ \a, b\^, n ≥ 0\
Solution
Option (A) is the language of palindromes of even length, which is a standard Context-Free Language (CFL) generated by .Option (B) represents strings where a center is surrounded by and . This can be generated by a context-free grammar like , . Thus, it is a CFL.Option (D) is the union of three languages: , , and . Since each is a CFL and CFLs are closed under union, (D) is a CFL.Option (C) is . This language requires matching with (nested dependency) and simultaneously matching with (another dependency). The structure interleaves these dependencies in a way that a single stack cannot handle. Specifically, to match with , one would need to access across , but must be popped to match . This cross-dependency makes the language not context-free.
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