Context Free Grammar for an Equal Number of a’s and b’s | CFG for an Equal Number of a’s and b’s

To construct a context-free grammar (CFG) for the language

L = {w mid text{w has an equal number of } atext{‘s and } btext{‘s}}

, we need a grammar that generates strings where the number of

a

‘s and

b

‘s are equal, without any specific order constraints.

Here’s a CFG that generates such strings:

Start Symbol: $S$

Production Rules:

S→aSb∣bSa∣ϵ

Explanation of the Rules:

1. :

This rule adds an $a$ to the beginning of the string and a $b$ to the end of the string, maintaining the balance of $a$ ‘s and $b$ ‘s. The string between these added $a$ and $b$ must also have an equal number of $a$ ‘s and $b$ ‘s, which is handled recursively by $S$ .

2. :

This rule is similar to the previous one but adds a $b$ to the beginning and an $a$ to the end. This also maintains the balance of $a$ ‘s and $b$ ‘s.

3. :

This rule allows for the empty string, which trivially has equal numbers of $a$ ‘s and $b$ ‘s (both counts are zero).

Example Derivations:

For the string $aabb$ :

$S Rightarrow aSb$
Now, $S$ must derive to something with equal $a$ ‘s and $b$ ‘s. Using $S rightarrow aSb$ again:

For the string $abba$ :

This CFG correctly generates all strings where the number of

a

‘s equals the number of

b

‘s.

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