GATE CS 2017 Set 1 — Question 38
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Theory of Computation → Context-Free Languages → Closure Properties (CFL)
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Question
Consider the following languages over the alphabet .
Let and .Which of the following are context-free languages?I.
II.
Let and .Which of the following are context-free languages?I.
II.
Correct answer
(A) I only
Solution
is a Deterministic Context-Free Language (DCFL). A PDA can push 's, pop for 's, and then ignore 's.
is also a DCFL. A PDA can ignore 's, push 's, and pop for 's.I. : The union of two Context-Free Languages (CFLs) is always a CFL. Therefore, is context-free.II. : The intersection of two CFLs is NOT necessarily a CFL.
.
For to be in , number of 's = number of 's.
For to be in , number of 's = number of 's.
Thus, .
This is the standard example of a language that is NOT context-free (can be proved using Pumping Lemma).Therefore, only I is context-free.
is also a DCFL. A PDA can ignore 's, push 's, and pop for 's.I. : The union of two Context-Free Languages (CFLs) is always a CFL. Therefore, is context-free.II. : The intersection of two CFLs is NOT necessarily a CFL.
.
For to be in , number of 's = number of 's.
For to be in , number of 's = number of 's.
Thus, .
This is the standard example of a language that is NOT context-free (can be proved using Pumping Lemma).Therefore, only I is context-free.
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