Question

Consider the grammar $S \rightarrow aSa \mid bSb \mid a \mid b$. Which one of the following options correctly characterizes the language generated by the given grammar over the alphabet {a,b} 
 

• A. All palindromes
• B. All odd length palindromes
• C. Strings that begin and end with the same symbol
• D. All even length palindromes

Hint:

Palindromic grammars often use symmetric productions like $aSa$ or $bSb$ to preserve symmetry.

Answer 1

The Correct Option is B

Step 1: Understanding the base productions.
The productions $S \rightarrow a$ and $S \rightarrow b$ generate strings of length $1$, which are trivially palindromes and have odd length.
Step 2: Understanding recursive productions.
The productions $S \rightarrow aSa$ and $S \rightarrow bSb$ add the same symbol to both ends of the string, preserving the palindrome property.
Step 3: Analyzing string length.
Each recursive step increases the string length by $2$. Since the base strings have odd length, all derived strings will also have odd length.
Step 4: Final conclusion.
Therefore, the grammar generates exactly all odd-length palindromes over the alphabet $\{a,b\}$.