Context Free Grammar for Non-Palindrome | CFG for Non-Palindrome

To construct a context-free grammar (CFG) for the language of non-palindromes over the alphabet ${a, b}$, we need to create a CFG that generates strings where the property of being a palindrome is not satisfied.

A non-palindrome is any string that is not equal to its reverse. Since a CFG for palindromes generates strings that are symmetric, our CFG for non-palindromes will ensure that the strings generated do not exhibit this symmetry.

CFG for Non-Palindromes

To generate non-palindromes, we can construct a CFG that generates strings where:

1.There is a mismatch or asymmetry, ensuring the string is not a palindrome.

2.We can create strings with an imbalance in the number of characters on one side compared to the other.

Here’s a CFG that generates non-palindromes over the alphabet ${a, b}$:

1.Start Symbol: S

2.Production Rules:

Explanation of the Rules:

1. S→aS∣bS:

• These rules generate strings that start with $a$ or $b$, recursively generating more of the same type of string. This ensures the CFG can generate strings where there is an unequal distribution of characters.

2. S→aSb∣bSa:

3. X→aXb∣bXa:

These rules ensure that the strings generated are not palindromes by introducing an imbalance between the characters.

4. X→aab∣bba∣a∣b:

These base cases ensure that certain simple non-palindromic strings are included directly.

Example Derivations:

For the string $abba$:

• $X rightarrow aab$ directly generates a non-palindromic string.
• Alternatively, generating a non-palindrome like $aab$ using $X rightarrow aXb rightarrow a epsilon b rightarrow aab$, where $X$ generates a non-palindromic string in a different manner.