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

Languages for All Palindrome(Even Length) ={ ε , aa,bb,abba,baab,……….. }

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

{a, b}

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

1.The string is a palindrome, meaning it reads the same forwards and backwards.

2.The string has an even length.

CFG for Even-Length Palindromes

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

{a, b}

1.Start Symbol: S

2.Production Rules:

Explanation of the Rules:

1. :

This rule ensures that if the string is a palindrome, adding the same character $a$ to both the beginning and end maintains the palindrome property. It also ensures that the length of the string increases by 2 each time.

2. :

This rule functions similarly to the previous one but with character $b$ . It maintains the palindrome property while increasing the string length by 2.

3. :

Example Derivations:

For the string $abba$ :

For the string $aabbaa$ :

$S Rightarrow aS a$
Next, for $S$ :
- $S Rightarrow aSb$
- $S Rightarrow epsilon$
Combining these, we get: $a(a , epsilon , a) a Rightarrow aabb , aa Rightarrow aabb , aa Rightarrow aabb , aa$