GATE CS 2026 Set 1 — Question 52
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Theory of Computation → Context-Free Languages → Context-Free Grammars
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Question
Consider the following context-free grammar .In the above grammar, is the start symbol, and are terminal symbols, and and are non-terminal symbols.Let
L(G) be the language generated by the grammar . For a string , let be the number of ’s in and be the number of ’s in .Which of the following statements is/are true?Correct answer
(B) For every string s ∈ L(G), n₁(s) ≥ n₂(s); (D) For every string s ∈ L(G), n₁(s) ≤ 2n₂(s)
Solution
1.Analyze : The productions generate the language . For any string , the number of 's () equals the number of 's (). Thus, for some .
2.Analyze :
- Base case: . For , and . Here, .
- Recursive case: . and . The difference .
- By induction, for any string , . Specifically, , where is the number of recursive steps plus 1.
- .
- .
- Since , it follows that for all . Thus, statement (B) is true.
- . Let . We need .
- . Since grows significantly with each recursive step (adding 's), while only increases by 1, the inequality holds for all derivations. Thus, statement (D) is true.
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