Context Free Grammar (CFG) for Language of All Palindrome(Odd Length) | CFG for all Palindrome Odd Length

L={a,b,aba,aaa,bab,bbb,aaaaa,………}

To construct a context-free grammar (CFG) for the language of all palindromes with odd length over the alphabet

{a, b}

, we need to create a CFG that generates strings where:

1.The string is a palindrome (i.e., it reads the same forwards and backwards).

2.The string has an odd length.

CFG for Odd-Length Palindromes

Here’s a CFG that generates all palindromes with odd length over the alphabet

{a, b}

1. Start Symbol: S

2. Production Rules:

Explanation of the Rules:

1. :

This rule ensures that the string is a palindrome with $a$ added to both the beginning and end. The length of the string increases by 2 for each application of this rule. It ensures that if $S$ generates an odd-length palindrome, then surrounding it with $a$ on both ends will maintain the odd length.

2. :

This rule functions similarly to the previous one but with character $b$ . It adds $b$ to both ends of the string, maintaining the palindrome property and ensuring the length is odd.

3. :

These rules handle the base cases where the palindrome consists of a single character, which is an odd-length palindrome.

Example Derivations:

For the string $aba$ :

$S Rightarrow aSa$
For $S$ :
- $S Rightarrow bSb$
- $S Rightarrow epsilon$ (for this specific derivation to show the steps, you might use an intermediate structure)
Combining these, we get: $a(b , epsilon , b) a Rightarrow aba$

For the string $abcdcba$ :

$S Rightarrow aS a$
For $S$ :
- $S Rightarrow bS b$
- Continue applying rules to build up to the full string.
Combining these, we get: $a(bcdecdcba) a$ , adjusting for exact matches.

For the string $a$ :

This CFG effectively generates all palindromes of odd length over the alphabet

{a, b}

. It ensures that the palindromes are symmetric around their center and have an odd number of characters.

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