GATE CS 2011 Q.24 (CS Q.24): Regular Expressions, Finite Automata and Regular Languages

Let $P$ be a regular language and $Q$ be a context-free language such that $Q \subseteq P$ . (For example, let $P$ be the language represented by the regular expression $p^* q^*$ and $Q$ be $\{p^n q^n \mid n \in \mathbb{N}\}$ ). Then which of the following is ALWAYS regular?

$P \cap Q$

$P - Q$

$\Sigma^* - P$

$\Sigma^* - Q$

Official answer and solution

Official answer: C
From: Official GATE 2011 answer key

Solution

Analyze closure properties of regular and context-free languages under intersection, set difference, and complementation.

Let $P$ be regular and $Q$ be context-free with $Q \subseteq P$ .

Option A: $P \cap Q = Q$ (since $Q \subseteq P$ ). Because $Q$ is not necessarily regular, $P \cap Q$ is not always regular.

Option B: $P - Q$ is the difference between a regular language and a CFL. For $P = p^* q^*$ and $Q = \{p^n q^n \mid n \in \mathbb{N}\}$ , $P - Q = \{p^i q^j \mid i \ne j\}$ , which is context-free but not regular.

Option D: $\Sigma^* - Q$ is the complement of a context-free language, which in general is not even guaranteed to be context-free, let alone regular.

Option C: $\Sigma^* - P$ is the complement of the regular language $P$ . Regular languages are closed under complementation. Therefore, $\Sigma^* - P$ is ALWAYS regular.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2011 Computer Science and Information Technology (CS)
Question: 24 of 65, printed as CS Q.24
Section: Computer Science and Information Technology

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