Question

What is the language $ L $ generated by the production rules $ SS \rightarrow aSa | bSb | a | b $ of the grammar over the alphabet a, b?

• A. \( L = \{ x \mid x \text{ is a string starting and ending with the same alphabet} \} \)
• B. \( L = \{ x \mid x \text{ is a string which is not a palindrome} \} \)
• C. \( L = \{ x \mid x \text{ is a string which is a palindrome of even length} \} \)
• D. \( L = \{ x \mid x \text{ is a string which is a palindrome of odd length} \} \)

Hint:

N/A

Answer 1

The Correct Option is D

The production rules \( SS \rightarrow aSa | bSb | a | b \) generate strings that are palindromes, where each string is symmetric around its center. These palindromes are of odd length because:
- The first and last characters must be the same (either 'a' or 'b').
- The recursion on \( S \) will continue until a base case of either a single 'a' or 'b' is reached, leading to strings of odd length.
Therefore, the correct answer is 4. \( L = \{ x \mid x \text{ is a string which is a palindrome of odd length} \} \).