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Regular expressions and finite automata, Context-free grammars and push-down automata, Regular and context-free languages, Pumping lemma, Turing machines and undecidability.

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

Previous GATE Questions in Theory of Computation

13 votes
4 answers
1
If $L$ is a regular language over $\Sigma = \{a,b\} $, which one of the following languages is NOT regular? $L.L^R = \{xy \mid x \in L , y^R \in L\}$ $\{ww^R \mid w \in L \}$ $\text{Prefix } (L) = \{x \in \Sigma^* \mid \exists y \in \Sigma^* $such that$ \ xy \in L\}$ $\text{Suffix }(L) = \{y \in \Sigma^* \mid \exists x \in \Sigma^* $such that$ \ xy \in L\}$
asked Feb 7, 2019 in Theory of Computation Arjun 4.6k views
22 votes
6 answers
2
For $\Sigma = \{a ,b \}$, let us consider the regular language $L=\{x \mid x = a^{2+3k} \text{ or } x=b^{10+12k}, k \geq 0\}$. Which one of the following can be a pumping length (the constant guaranteed by the pumping lemma) for $L$ ? $3$ $5$ $9$ $24$
asked Feb 7, 2019 in Theory of Computation Arjun 10.3k views
12 votes
3 answers
3
Which one of the following languages over $\Sigma=\{a, b\}$ is NOT context-free? $\{ww^R \mid w \in \{a, b\}^*\}$ $\{wa^nb^nw^R \mid w \in \{a,b\}^*, n \geq 0\}$ $\{wa^nw^Rb^n \mid w \in \{a,b\}^* , n \geq 0\}$ $\{ a^nb^i \mid i \in \{n, 3n, 5n\}, n \geq 0\}$
asked Feb 7, 2019 in Theory of Computation Arjun 3.9k views
21 votes
3 answers
4
Consider the following sets: S1: Set of all recursively enumerable languages over the alphabet $\{0, 1\}$ S2: Set of all syntactically valid C programs S3: Set of all languages over the alphabet $\{0,1\}$ S4: Set of all non-regular languages over the alphabet $\{ 0,1 \}$ Which of the above sets are uncountable? S1 and S2 S3 and S4 S2 and S3 S1 and S4
asked Feb 7, 2019 in Theory of Computation Arjun 4.1k views
19 votes
3 answers
5
Let $\Sigma$ be the set of all bijections from $\{1, \dots , 5\}$ to $\{1, \dots , 5 \}$, where $id$ denotes the identity function, i.e. $id(j)=j, \forall j$. Let $\circ$ ... $L=\{x \in \Sigma^* \mid \pi (x) =id\}$. The minimum number of states in any DFA accepting $L$ is _______
asked Feb 7, 2019 in Theory of Computation Arjun 6.4k views
49 votes
5 answers
6
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 O) accepted by the following automaton. The order of $L_1$ is ____
asked Feb 14, 2018 in Theory of Computation gatecse 8.2k views
31 votes
10 answers
7
Consider the following languages: $\{a^mb^nc^pd^q \mid m+p=n+q, \text{ where } m, n, p, q \geq 0 \}$ $\{a^mb^nc^pd^q \mid m=n \text{ and }p=q, \text{ where } m, n, p, q \geq 0 \}$ ... Which of the above languages are context-free? I and IV only I and II only II and III only II and IV only
asked Feb 14, 2018 in Theory of Computation gatecse 7.2k views
17 votes
3 answers
8
Consider the following problems. $L(G)$ denotes the language generated by a grammar $G$. L(M) denotes the language accepted by a machine $M$. For an unrestricted grammar $G$ and a string $w$, whether $w \in L(G)$ Given a Turing machine $M$, ... following statement is correct? Only I and II are undecidable Only II is undecidable Only II and IV are undecidable Only I, II and III are undecidable
asked Feb 14, 2018 in Theory of Computation gatecse 5.9k views
17 votes
5 answers
9
The set of all recursively enumerable languages is: closed under complementation closed under intersection a subset of the set of all recursive languages an uncountable set
asked Feb 14, 2018 in Theory of Computation gatecse 3.9k views
13 votes
2 answers
10
Let $N$ be an NFA with $n$ states. Let $k$ be the number of states of a minimal DFA which is equivalent to $N$. Which one of the following is necessarily true? $k \geq 2^n$ $k \geq n$ $k \leq n^2$ $k \leq 2^n$
asked Feb 14, 2018 in Theory of Computation gatecse 3k views
5 votes
0 answers
11
https://gateoverflow.in/118290/gate2017-1-10 There is a confusion between option B and C It cannot be option B because the variable c has no power , its always taken as 1 which is not what the grammar says Option C - should be the ans i feel please help
asked Oct 24, 2017 in Theory of Computation A_i_$_h 206 views
6 votes
2 answers
12
Let $L=\left \{ w\in\{0,1\}^* | \text{number of occurances of }(110)=\text{number of occurances of}(011) \right \}$ What is $L$? I think $L$ is regular . Regular expression is -: $L=\left \{ 0^{*}+1^{*}+\left ( \left ( \varepsilon +0+1 \right ) \left ( \varepsilon +0+1 \right ) \right ) + 0^{*}\left ( 0110 \right )^*0^{*}+1^* \left ( 11011 \right )^{*}1^{*} \right \}$
asked Oct 9, 2017 in Theory of Computation sourav. 860 views
27 votes
5 answers
13
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 ... $L_2$ is not context free $L_3$ is not context free but recursive $L_4$ is deterministic context free I, II and IV only II and III only I and IV only III and IV only
asked Feb 14, 2017 in Theory of Computation Madhav 4k views
27 votes
4 answers
14
Let $L(R)$ be the language represented by regular expression $R$. Let $L(G)$ be the language generated by a context free grammar $G$. Let $L(M)$ be the language accepted by a Turing machine $M$. Which of the following decision problems are undecidable? Given a regular expression $R$ and a ... $w$, is $w \in L(M)$? I and IV only II and III only II, III and IV only III and IV only
asked Feb 14, 2017 in Theory of Computation Madhav 3.7k views
46 votes
10 answers
15
Let $\delta$ denote the transition function and $\widehat{\delta}$ denote the extended transition function of the $\epsilon$-NFA whose transition table is given below: $\begin{array}{|c|c|c|c|}\hline \delta & \text{$\epsilon$} & \text{$a$} & \text{$ ... $\widehat{\delta}(q_2, aba)$ is $\emptyset$ $\{q_0, q_1, q_3\}$ $\{q_0, q_1, q_2\}$ $\{q_0, q_2, q_3 \}$
asked Feb 14, 2017 in Theory of Computation Arjun 8.8k views
60 votes
3 answers
16
Let $A$ and $B$ be finite alphabets and let $\#$ be a symbol outside both $A$ and $B$. Let $f$ be a total function from $A^{*}$ to $B^{*}$. We say $f$ is computable if there exists a Turing machine $M$ which given an input $x \in A^{*}$ ... . If $f$ is computable then $L_{f}$ is recursive, but not conversely. If $f$ is computable then $L_{f}$ is recursively enumerable, but not conversely.
asked Feb 14, 2017 in Theory of Computation Arjun 8.3k views
31 votes
7 answers
17
Consider the following languages over the alphabet $\sum = \left \{ a, b, c \right \}$. Let $L_{1} = \left \{ a^{n}b^{n}c^{m}|m,n \geq 0 \right \}$ and $L_{2} = \left \{ a^{m}b^{n}c^{n}|m,n \geq 0 \right \}$. Which of the following are context-free languages? $L_{1} \cup L_{2}$ $L_{1} \cap L_{2}$ I only II only I and II Neither I nor II
asked Feb 14, 2017 in Theory of Computation Arjun 5.4k views
36 votes
8 answers
18
Consider the context-free grammars over the alphabet $\left \{ a, b, c \right \}$ given below. $S$ and $T$ are non-terminals. $G_{1}:S\rightarrow aSb \mid T, T \rightarrow cT \mid \epsilon$ $G_{2}:S\rightarrow bSa \mid T, T \rightarrow cT \mid \epsilon$ The language $L\left ( G_{1} \right )\cap L(G_{2})$ is Finite Not finite but regular Context-Free but not regular Recursive but not context-free
asked Feb 14, 2017 in Theory of Computation Arjun 4.9k views
31 votes
6 answers
19
If $G$ is a grammar with productions $S\rightarrow SaS\mid aSb\mid bSa\mid SS\mid\epsilon$ where $S$ is the start variable, then which one of the following strings is not generated by $G$? $abab$ $aaab$ $abbaa$ $babba$
asked Feb 14, 2017 in Theory of Computation Arjun 3.6k views
32 votes
11 answers
20
Consider the language $L$ given by the regular expression $(a+b)^{*} b (a+b)$ over the alphabet $\{a,b\}$. The smallest number of states needed in a deterministic finite-state automaton (DFA) accepting $L$ is ___________ .
asked Feb 14, 2017 in Theory of Computation Arjun 9k views
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