GATE CS 2025 Set 1 — Question 45
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Theory of Computation → Context-Free Languages → Context-Free Grammars
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Question
Consider the following two languages over the alphabet , where and are natural numbers.Which ONE of the following statements is CORRECT?
Correct answer
(C) L₁ is not a context-free language but L₂ is a context-free language.
Solution
Step 1: Analyze .
We can construct a Context-Free Grammar (CFG) for :
The first rule generates , and the second rule generates in the middle. Together they produce . Since a CFG exists, is a context-free language.Step 2: Analyze .
This language can be rewritten as .
To recognize this language, a machine would need to match the number of 's and 's (using a stack), and then also ensure that the number of 's is at least the number of 's. However, after matching 's and 's, the stack would be empty, and the information about would be lost. Using the pumping lemma for context-free languages, it can be formally proven that is not context-free (similar to the proof for ).Conclusion: is not context-free, but is context-free.
We can construct a Context-Free Grammar (CFG) for :
The first rule generates , and the second rule generates in the middle. Together they produce . Since a CFG exists, is a context-free language.Step 2: Analyze .
This language can be rewritten as .
To recognize this language, a machine would need to match the number of 's and 's (using a stack), and then also ensure that the number of 's is at least the number of 's. However, after matching 's and 's, the stack would be empty, and the information about would be lost. Using the pumping lemma for context-free languages, it can be formally proven that is not context-free (similar to the proof for ).Conclusion: is not context-free, but is context-free.
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