GATE CS 2024 Set 2 — Question 52
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Theory of Computation → Context-Free Languages → Context-Free Grammars
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Question
Consider a context-free grammar with the following 3 rules.Let . Let , , denote the number of times occur in , respectively. Which of the following statements is/are TRUE?
Correct answer
(B) nₐ(w) n_c(w) - 2; (C) n_c(w) = n_b(w) + 1
Solution
Let be the number of times the rules , , and are applied in the derivation of a string , respectively.We track the number of non-terminals and the terminals generated:
Substituting the counts: . This matches our derived constraint exactly. Thus, this statement is TRUE.(B)
Substitute :Since and , this becomes:Since is a count of rule applications, . Therefore, is always true. Thus, this statement is TRUE.(A) This is not always true. For example, the derivation has . Thus, this statement is FALSE.(D) This is not always true. For example, has , so . Thus, this statement is FALSE.Correct options are (B) and (C).
1.Start: We begin with 1 .
2.Rule 1 (): Replaces 1 with 1 . Net change in is . Adds 1 .
3.Rule 2 (): Replaces 1 with 2 's. Net change in is . Adds 1 and 1 .
4.Rule 3 (): Replaces 1 with 0 's. Net change in is . Adds 1 .
For a derivation to terminate and produce a string , the final number of 's must be 0. Starting from 1 , the total change must be .The number of terminals in are:- (since rules 1 and 2 produce 'a')
- (only rule 2 produces 'b')
- (only rule 3 produces 'c')
Substituting the counts: . This matches our derived constraint exactly. Thus, this statement is TRUE.(B)
Substitute :Since and , this becomes:Since is a count of rule applications, . Therefore, is always true. Thus, this statement is TRUE.(A) This is not always true. For example, the derivation has . Thus, this statement is FALSE.(D) This is not always true. For example, has , so . Thus, this statement is FALSE.Correct options are (B) and (C).
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