Which of the following is (are) correct about the regular expression?

a a* b b* c c* d d*

1. The language for the given expression is:

$L={a^n b^n c^m d^m \mid n\ge 1,\ m\ge 1}\cup{a^n b^m c^m d^n \mid n\ge 1,\ m\ge 1}$

2. The Context Free Language for the given expression is:

$S\to AB\mid C$

$A\to aAb\mid ab$

$B\to cBd\mid cd$

$C\to aCd\mid aDd$

$D\to bDc\mid bc$

3. The language generated by this expression is equal number of $a$'s, followed by equal number of $b$'s, followed by equal number of $c$'s and followed by equal number of $d$'s.

Choose the correct answer from the options given below:

The given regular expression is $aa^*bb^*cc^dd^$, where:

• $a^*$ denotes zero or more occurrences of $a$
• $b^*$ denotes zero or more occurrences of $b$
• $c^*$ denotes zero or more occurrences of $c$
• $d^*$ denotes zero or more occurrences of $d$

1. The language generated by the regular expression

$L={a^n b^n c^m d^m \mid n \ge 1, m \ge 1}\cup{a^n b^m c^m d^n \mid n \ge 1, m \ge 1}$

consists of strings where $a$'s are followed by $b$'s, which are followed by $c$'s, and then $d$'s.

Each of these letters $a, b, c, d$ can occur one or more times in any order. For example, $abcd$, $aabcdd$, etc. So the language $L$ can generate regular expressions $aa^*bb^*cc^dd^$.

2. The Context Free Language for the given expression is:

$S \to AB \mid C$

$A \to aAb \mid ab$

$B \to cBd \mid cd$

$C \to aCd \mid aDd$

$D \to bDc \mid bc$

This is true.

For example:

$S \to AB \mid C$

$S \to abcd$

3. The language generated by this expression is equal number of $a$'s, followed by equal number of $b$'s, followed by equal number of $c$'s and followed by equal number of $d$'s.

This statement is false.

The regular expression $aa^*bb^*cc^dd^$ generates strings where each letter $a, b, c, d$ can occur one or more times, but there is no constraint that the number of occurrences of each letter must be equal.

For example, the regular expression can generate strings like $abcd$, $abccdd$, $aaaabbbccddd$, etc., where the number of occurrences of each letter is not necessarily equal.

So option C is the correct answer.