GATE CS 2021 Set 2 — Question 51
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Theory of Computation → Context-Free Languages → Closure Properties (CFL)
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Question
For a string , we define to be the reverse of . For example, if then .
Which of the following languages is/are context-free?
Which of the following languages is/are context-free?
Correct answer
(B) \w w^R x x^R w, x ∈ \0, 1\^ \; (C) \w x w^R w, x ∈ \0, 1\^ \; (D) \w x x^R w^R w, x ∈ \0, 1\^ \
Solution
Analysis of the options:
- (A) : This language requires matching with and with in a cross-serial manner. A single stack cannot handle this interleaved matching. Thus, it is not context-free.
- (B) : This is the concatenation of two context-free languages and . Since context-free languages are closed under concatenation, this language is context-free.
- (C) : If we take , then the language becomes , which is regular. Even if is not empty, we can match the first part with the last part while ignoring the middle part . This can be done with a PDA. In fact, for any , this language simplifies to because we can always choose to be the first character and to be the last character if they match, or just choose . Thus, it is regular and therefore context-free.
- (D) : This represents nested matching where matches on the outside and matches on the inside. This is a classic context-free language structure (like balanced parentheses).
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