GATE CS 2017 Set 1 — Question 34
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Theory of Computation → Context-Free Languages → Context-Free Grammars
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Question
If is a grammar with productionswhere is the start variable, then which one of the following strings is not generated by ?
Correct answer
(D) babba
Solution
The given grammar is:Let's analyze the number of 's () and 's () generated by each production:
(A) : . (, possible)
(B) : . (, possible)
(C) : . (, possible)
(D) : . (, impossible)Since option (D) has more 's than 's, it cannot be generated by the grammar.
1.: . ()
2.: Adds one . ( increases by 1, unchanged)
3.: Adds one and one . ( increases by 1, increases by 1)
4.: Adds one and one . ( increases by 1, increases by 1)
5.
From the base case (), we have . All recursive steps either add equal numbers of 's and 's (maintaining ) or add an extra (increasing relative to ).Thus, an invariant of this grammar is that for any string generated by , the number of 's must be greater than or equal to the number of 's ().Let's check the options:S → SS: Combines counts.(A) : . (, possible)
(B) : . (, possible)
(C) : . (, possible)
(D) : . (, impossible)Since option (D) has more 's than 's, it cannot be generated by the grammar.
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