GATE CS 2009 Q12 — Context-free Grammars and Push-down Automata

To find the language generated by the given context-free grammar, let us analyze the production rules:

1. Base Cases:

The non-recursive productions are:

• $S \to a$ (length 1, palindrome)
• $S \to b$ (length 1, palindrome)

These generate all palindromes of length $1$ over the alphabet $\{a, b\}$.

2. Recursive Steps:

The recursive productions are:

• $S \to aSa$
• $S \to bSb$

Each application of these rules wraps a previously generated string with the same character on both the left and right ends. This operation preserves the palindromic property (i.e., the string reads the same forwards and backwards) and increases the length of the string by exactly $2$.

3. Conclusion:

• Since we start with a base string of length $1$ (which is odd) and repeatedly add $2$ to the length at each step, any derived string must have an odd length.
• Because of the symmetric wrapping, every derived string is a palindrome.

Therefore, the grammar generates the set of all odd-length palindromes.

Elimination of other options:

• Option A: Incorrect, because even-length palindromes like $aa$ or $abba$ cannot be generated.
• Option C: Incorrect, because a string like $aab$ begins and ends with different symbols, and even strings like $abaa$ (which begin and end with $a$) are not palindromes and cannot be generated.
• Option D: Incorrect, because all generated strings have an odd length.