Question

Which of the following string is not generated by the grammar :
$S \to SaSbS | \epsilon$ ?

• A. aabb
• B. abab
• C. aababb
• D. aaabb

Hint:

For grammars generating balanced strings, always check the count first. If $count(a) \neq count(b)$, the string is immediately invalid.

Answer 1

The Correct Option is D

Solution: To determine which string cannot be generated, we must analyze the structural properties of the language defined by the given context-free grammar (CFG). 1. Language Identification: The grammar $S \to SaSbS | \epsilon$ is a classic representation of the language of balanced parentheses (Dyck language), where 'a' acts as an opening parenthesis and 'b' acts as a closing parenthesis. Every time an 'a' is generated by the rule $SaSbS$, it is paired with a corresponding 'b'. 2. Property of Balanced Strings: A fundamental property of any string generated by this grammar is that the total count of 'a's must exactly equal the total count of 'b's. 3. Evaluating the Options:
• aabb: 2 'a's and 2 'b's. This follows the balanced structure $( ( ) )$.
• abab: 2 'a's and 2 'b's. This follows the balanced structure $( ) ( )$.
• aababb: 3 'a's and 3 'b's. This follows the structure $( ( ) ) ( )$.
• aaabb: 3 'a's and 2 'b's. Since the number of 'a's ($n_a = 3$) does not equal the number of 'b's ($n_b = 2$), it violates the fundamental rule of this grammar. Therefore, the string "aaabb" cannot be generated by the provided grammar rules.