Context Free Grammar for Palindrome | CFG for Palindrome

November 12, 2024September 17, 2024 by Master of Computer Science

Context Free Grammar for Palindrome | CFG for Palindrome

L = { λ, a,b,aa,bb,aaa,bbb,aba,bab, ………………………..}

OR

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

{a, b}

, where a palindrome is a string that reads the same forward and backward, we need to ensure that the grammar generates all strings that exhibit this symmetric property.

CFG for Palindromes

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

{a, b}

:

Start Symbol:

S

Production Rules:

S→aSa∣bSb∣ϵ∣a∣b

Explanation of the Rules:

:

This rule ensures that if a string is a palindrome, then adding the same symbol $a$ to the beginning and end of that string will also result in a palindrome. It preserves the symmetry.

:

Similarly, this rule ensures that adding the same symbol $b$ to the beginning and end of a palindrome will also result in a palindrome.

:

This rule generates the empty string, which is trivially a palindrome.

:

These rules handle the base cases where the palindrome consists of a single character.

For the string $abba$ :

$S Rightarrow aS a$
To generate $S$ :
- $S Rightarrow bSb$
- $S Rightarrow epsilon$
Combining these, we get $a (b , epsilon , b) a Rightarrow abba$ .

For the string $a$ :

$S Rightarrow a$
This is a base case and directly results in $a$ .

For the string $bb$ :

$S Rightarrow bSb$
To generate $S$ :
- $S Rightarrow epsilon$
Combining these, we get $b , epsilon , b Rightarrow bb$ .

This CFG correctly generates all palindromes over the alphabet

{a, b}

.

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