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$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline \textbf{Year}&\textbf{2021-1}&\textbf{2021-2}&\textbf{2020}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum} \\\hline\textbf{1 Mark Count}&2&3&3&2&2&2&3&3&3&2&2.5&3 \\\hline\textbf{2 Marks Count}&3&4&3&3&3&5&3&3&3&3&3.3&5 \\\hline\textbf{Total Marks}&8&11&9&8&8&12&9&9&9&\bf{8}&\bf{9.2}&\bf{12}\\\hline \end{array}}}$$

# Recent questions in Theory of Computation

0 votes
1 answer
1
Let $L=\{0^n1^n|n\ge 0\}$ be a context free language. Which of the following is correct? $\overline L$ is context free and $L^k$ is not context free for any $k\ge1$ $\overline L$ is not context free and $L^k$ is context free for any $k\ge1$ Both $\overline L$ and $L^k$ for any $k\ge1$ are context free Both $\overline L$ and $L^k$ for any $k\ge1$ are not context free
0 votes
1 answer
2
Let $\Sigma=\{a,b\}.$ Given a language $L\underline\subset \Sigma^{\ast}$ and a word $w\in\Sigma^{\ast}$, define the languages: $Extend(L,w) :=\{xw\:|\:x\in L\}$ $Shrink(L,w) :=\{x\:|\:xw\in L\}$Show that if $L$ is regular, both $Extend(L,w)$ and $Shrink(L,w)$ are regular.
5 votes
3 answers
3
Let $L \subseteq \{0,1\}^*$ be an arbitrary regular language accepted by a minimal $\text{DFA}$ with $k$ states. Which one of the following languages must necessarily be accepted by a minimal $\text{DFA}$ with $k$ states? $L-\{01\}$ $L \cup \{01\}$ $\{0,1\}^* – L$ $L \cdot L$
3 votes
3 answers
4
Let $L_1$ be a regular language and $L_2$ be a context-free language. Which of the following languages is/are context-free? $L_1 \cap \overline{L_2} \\$ $\overline{\overline{L_1} \cup \overline{L_2}} \\$ $L_1 \cup (L_2 \cup \overline{L_2}) \\$ $(L_1 \cap L_2) \cup (\overline{L_1} \cap L_2)$
1 vote
2 answers
5
Consider the following deterministic finite automaton $\text{(DFA)}$ The number of strings of length $8$ accepted by the above automaton is ___________
2 votes
2 answers
6
Suppose we want to design a synchronous circuit that processes a string of $0$'s and $1$'s. Given a string, it produces another string by replacing the first $1$ in any subsequence of consecutive $1$'s by a $0$ ... $\begin{array}{l} t=s+b \\ y=s \overline{b} \end{array}$
3 votes
1 answer
7
​​​​​​Consider the following two statements about regular languages: $S_1$: Every infinite regular language contains an undecidable language as a subset. $S_2$: Every finite language is regular. Which one of the following choices is correct? Only $S_1$ is true Only $S_2$ is true Both $S_1$ and $S_2$ are true Neither $S_1$ nor $S_2$ is true
4 votes
2 answers
8
For a string $w$, we define $w^R$ to be the reverse of $w$. For example, if $w=01101$ then $w^R=10110$. Which of the following languages is/are context-free? $\{ wxw^Rx^R \mid w,x \in \{0,1\} ^* \}$ $\{ ww^Rxx^R \mid w,x \in \{0,1\} ^* \}$ $\{ wxw^R \mid w,x \in \{0,1\} ^* \}$ $\{ wxx^Rw^R \mid w,x \in \{0,1\} ^* \}$
2 votes
1 answer
9
​​​​​​​Which of the following regular expressions represent(s) the set of all binary numbers that are divisible by three? Assume that the string $\epsilon$ is divisible by three. $(0+1(01^*0)^*1)^*$ $(0+11+10(1+00)^*01)^*$ $(0^*(1(01^*0)^*1)^*)^*$ $(0+11+11(1+00)^*00)^*$
1 vote
3 answers
10
Suppose that $L_1$ is a regular language and $L_2$ is a context-free language. Which one of the following languages is $\text{NOT}$ necessarily context-free? $L_1 \cap L_2$ $L_1 \cdot L_2$ $L_1- L_2$ $L_1\cup L_2$
3 votes
3 answers
11
Let $\langle M \rangle$ denote an encoding of an automaton $M$. Suppose that $\Sigma = \{0,1\}$. Which of the following languages is/are $\text{NOT}$ recursive? $L= \{ \langle M \rangle \mid M$ is a $\text{DFA}$ such that $L(M)=\emptyset \}$ $L= \{ \langle M \rangle \mid M$ is ... $L(M)=\emptyset \}$ $L= \{ \langle M \rangle \mid M$ is a $\text{PDA}$ such that $L(M)=\Sigma ^* \}$
0 votes
1 answer
12
Consider the following language: $L= \{ w \in \{0,1\}^* \mid w \text{ ends with the substring } 011 \}$ Which one of the following deterministic finite automata accepts $L?$
3 votes
3 answers
13
For a Turing machine $M$, $\langle M \rangle$ denotes an encoding of $M$ ... and $L_2$ are decidable $L_1$ is decidable and $L_2$ is undecidable $L_1$ is undecidable and $L_2$ is decidable Both $L_1$ and $L_2$ are undecidable
3 votes
2 answers
14
In a pushdown automaton $P=(Q, \Sigma, \Gamma, \delta, q_0, F)$, a transition of the form, where $p,q \in Q$, $a \in \Sigma \cup \{ \epsilon \}$, and $X,Y \in \Gamma \cup \{ \epsilon \}$, represents $(q,Y) \in \delta(p,a,X).$ Consider the ... $\Gamma = \{ \#, A\}$. The number of strings of length $100$ accepted by the above pushdown automaton is ___________
1 vote
2 answers
15
Consider $L=L_1 \cap L_2$ where $L_1 = \{ 0^m 1^m 20^n 1^n \mid m,n \geq 0 \}$ $L_2 = \{0^m1^n2^k \mid m,n,k \geq 0 \}$ Then, the language $L$ is Recursively enumerable but not context free Regular Context free but not regular Not recursive
0 votes
2 answers
16
Let $L_1$ and $L_2$ be languages over $\Sigma = \{a,b\}$ represented by the regular expressions $(a^* +b)^*$ and $(a+b)^*$ respectively. Which of the following is true with respect to the two languages? $L_1 \subset L_2$ $L_2 \subset L_1$ $L_1 = L_2$ $L_1 \cap L_2 = \phi$
1 vote
2 answers
17
Which of the following statements is true? The union of two context free languages is context free The intersection of two context free languages is context free The complement of a context free language is context free If a language is context free, it can always be accepted by a deterministic pushdown automaton
1 vote
1 answer
18
Let $G_1$ and $G_2$ be arbitrary context free languages and $R$ an arbitrary regular language. Consider the following problems: Is $L(G_1)=L(G_2)$? Is $L(G_2) \leq L(G_1)$? Is $L(G_1)=R$? Which of the problems are undecidable? Choose the correct answer from the options given below: $(a)$ only $(b)$ only $(a)$ and $(b)$ only $(a)$, $(b)$ and $(c)$
0 votes
2 answers
19
Consider the following languages: $L_1=\{a^{\grave{z}^z} \mid \grave{Z} \text{ is an integer} \}$ $L_2=\{a^{z\grave{z}} \mid \grave{Z} \geq 0\}$ $L_3=\{ \omega \omega \mid \omega \epsilon \{a,b\}^*\}$ Which of the languages is(are) regular? Choose the correct answer from the options given below: $L_1$ and $L_2$ only $L_1$ and $L_3$ only $L_1$ only $L_2$ only
0 votes
2 answers
20
Which of the following grammars is(are) ambiguous? $s \rightarrow ss \mid asb \mid bsa \mid \lambda$ $s \rightarrow asbs \mid bsas \mid \lambda$ $s \rightarrow aAB \\ A \rightarrow bBb \\ B \rightarrow A \mid \lambda \text{ where } \lambda \text{ denotes empty string}$ Choose the correct answer from the options given below: $(a)$ and $(c)$ only $(b)$ only $(b)$ and $(c)$ only $(a)$ and $(b)$ only