GATE CS Regular and Context-free Languages questions

Given a language $L$, define $L^i$ as follows:$$L^0 = \{ \varepsilon \}$$$$L^i = L^{i-1} \bullet L \text{ for all } I >0$$The order of a language $L$ is defined as the smallest $k$ such that $L^k = L^{k+1}$. Consider the language $L_1 ($over alphabet $0)$ accepted by the following automaton. The order of $L_1$ is ________.

The set of all recursively enumerable languages is:

Consider the following languages. $L_1 = \{a^p \mid p \text{ is a prime number} \}$ $L_2 = \{ a^nb^mc^{2m} \mid n \geq 0, m \geq 0 \}$ $L_3 = \{a^n b^n c^{2n} \mid n \geq 0 \}$ $L_4 = \{ a^n b^n \mid n \geq 1\}$ Which of the following are CORRECT? $L_1$ is context free but not regular $L_2$ is not context free $L_3$ is not context free but …

Let $L_1, L_2$ be any two context-free languages and $R$ be any regular language. Then which of the following is/are CORRECT? $L_1 \cup L_2$ is context-free $\overline{L_1}$ is context-free $L_1 - R$ is context-free $L_1 \cap L_2$ is context-free

Consider the following types of languages: $L_{1}$: Regular, $L_{2}$: Context-free, $L_{3}$: Recursive, $L_{4}$: Recursively enumerable. Which of the following is/are TRUE ? $\overline{L_{3}} \cup L_{4}$ is recursively enumerable. $\overline{L_{2}} \cup L_{3}$ is recursive. $L^{*} _{1} \cap L_{2}$ is context-free. …

Language $L_{1}$ is defined by the grammar: $S_{1} \rightarrow a S_{1} b \mid \varepsilon$ Language $L_{2}$ is defined by the grammar: $S_{2} \rightarrow a b S_{2} \mid \varepsilon$ Consider the following statements: P: $L_{1}$ is regular Q: $L_{2}$ is regular Which one of the following is TRUE?

Which of the following is/are regular languages? $L_1: \left\{ wxw^R \mid w, x \in \{a, b\} ^* \text{ and } |w|, |x| > 0\right\}, w^R \text{ is the reverse of string } w$ $L_2: \left\{ a^nb^m \mid m \neq n \text{ and } m, n \geq 0 \right\}$ $L_3: \left\{ a^pb^qc^r \mid p, q, r \geq 0 \right\}$

Consider two decision problems $Q_1, Q_2$ such that $Q_1$ reduces in polynomial time to 3-SAT and 3-SAT reduces in polynomial time to $Q_2$. Then which one of the following is consistent with the above statement?

Which one of the following is TRUE?

Let $L_1=\{w\in\{0,1\}^*\mid w$ $\text{ has at least as many occurrences of }$ $(110)'\text{s as }$ $(011)'\text{s} \}$. Let $L_2=\{w \in\{0,1\}^*\ \mid w$ $\text{ has at least as many occurrences of }$ $(000)'\text{s as}$ $(111)'\text{s} \}$. Which one of the following is TRUE?

If $L_1\:=\{a^n \mid n\:\geq\:0\}$ and $L_2\:= \{b^n \mid n\:\geq\:0\}$ , consider $L_1.L_2$ is a regular language $L_1.L_2 = \{a^nb^n \mid n\: \geq \:0\}$ Which one of the following is CORRECT?

Let $\Sigma$ be a finite non-empty alphabet and let $2^{\Sigma^*}$ be the power set of $\Sigma^*$. Which one of the following is TRUE?

Consider the following languages. $L_1 = \left \{ 0^p1^q0^r \mid p,q,r \geq 0 \right \}$ $L_2 = \left \{ 0^p1^q0^r \mid p,q,r \geq 0, p\neq r \right \}$ Which one of the following statements is FALSE?

Which of the following statements is/are FALSE? For every non-deterministic Turing machine, there exists an equivalent deterministic Turing machine. Turing recognizable languages are closed under union and complementation. Turing decidable languages are closed under intersection and complementation. Turing recognizable languages are closed under union and …

Consider the languages $L_1 = \phi$ and $L_2 = \{a\}$. Which one of the following represents $L_1 {L_2}^* \cup {L_1}^*$ ?

Given the language $L = \left\{ab, aa, baa\right\}$, which of the following strings are in $L^{*}$? $abaabaaabaa$ $aaaabaaaa$ $baaaaabaaaab$ $baaaaabaa$

Assuming $\textsf{P} \neq \textsf{NP}$, which of the following is TRUE?

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 $\{p^nq^n \mid n \in N\})$. Then which of the following is ALWAYS regular?

Consider the languages $L1, \:L2 \:and \: L3$ as given below. $L1=\{0^p 1^q \mid p, q \in N\}, \\ L2 = \{0^p 1^q \mid p, q \in N \:and \:p=q\} \: and, \\ L3 = \{0^p 1^q 0^r \mid p, q, r \in N\: and \: p=q=r\}.$ Which of the following statements is NOT TRUE?

Let $L = L_1 \cap L_2$, where $L_1$ and $L_2$ are languages as defined below: $L_1= \left \{ a^m b^mca^nb^n \mid m,n \geq 0 \right \}$ $L_2=\left \{ a^i b^j c^k \mid i,j,k \geq 0 \right \}$ Then $L$ is