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

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2024-1} & \textbf{2024-2} & \textbf{2023} & \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} &1&1&3& 2 &2&3& 1&2&3
\\\hline\textbf{2 Marks Count} &2&3&3& 3 &3&4& 2&3&4
\\\hline\textbf{Total Marks} &5&7&9& 8 &8&11&\bf{5}&\bf{8}&\bf{11}\\\hline
\end{array}}}$$

Recent questions in Theory of Computation

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CFL and regular language
Consider a regular language R and a context free language C. Let the PDA that recognizes C be called P=(QP,∑,Γ,δP,q0P,FP), and the DFA that reconginzes R be (QR,∑,δR,q0R,FR).

Consider a regular language R and a context free language C. Let the PDA that recognizes C be called P=(QP,∑,Γ,δP,q0P,FP), and the DFA that reconginzes R be (QR...

Vedantthakkar

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Vedantthakkar asked Feb 24

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Pumping Lemma
Use the Pumping Lemma to show that the following languages over Σ={�,�}Σ={a,b} are not regular. In each case, carefully describe the string that will be pumped and explain why pumping it leads to a contradiction. {aaabnan∣n≥0} {ww∣w∈Σ∗}

Use the Pumping Lemma to show that the following languages over Σ={�,�}Σ={a,b} are not regular. In each case, carefully describe the string that will be pumped and ...

jg662

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jg662 asked Feb 22

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Prove that the following languages are not regular. You may use the pumping lemma and the closure of the class of regular languages under union, intersection, and complement. a. {0"1"0"| m, n 2 0} b. (0"1"| m ‡ n) c. (w| w € {0,1)* is not a palindrome) d. (wtw| w,t € (0,1)*)

krishan_rathi

172 views

krishan_rathi asked Feb 21

3 votes

2 answers

GATE CSE 2024 | Set 2 | Question: 12
Which one of the following regular expressions is equivalent to the language accepted by the $\text{DFA}$ given below? $0^{*} 1\left(0+10^{*} 1\right)^{*}$ $0^{*}\left(10^{*} 11\right)^{*} 0^{*}$ $0^{*} 1\left(010^{*} 1\right)^{*} 0^{*}$ $0\left(1+0^{*} 10^{*} 1\right)^{*} 0^{*}$

Which one of the following regular expressions is equivalent to the language accepted by the $\text{DFA}$ given below?$0^{*} 1\left(0+10^{*} 1\right)^{*}$$0^{...

Arjun

2.9k views

Arjun asked Feb 16

1 votes

2 answers

GATE CSE 2024 | Set 2 | Question: 31
Let $\text{M}$ be the $5$-state $\text{NFA}$ with $\epsilon$-transitions shown in the diagram below. Which one of the following regular expressions represents the language accepted by $\text{M}$? $(00)^{*}+1(11)^{*}$ $0^{*}+\left(1+0(00)^{*}\right)(11)^{*}$ $(00)^{*}+\left(1+(00)^{*}\right)(11)^{*}$ $0^{+}+1(11)^{*}+0(11)^{*}$

Let $\text{M}$ be the $5$-state $\text{NFA}$ with $\epsilon$-transitions shown in the diagram below.Which one of the following regular expressions represents the la...

Arjun

2.7k views

Arjun asked Feb 16

1 votes

3 answers

GATE CSE 2024 | Set 2 | Question: 52
Let $L_{1}$ be the language represented by the regular expression $b^{*} a b^{*}\left(a b^{*} a b^{*}\right)^{*}$ and $L_{2}=\left\{w \in(a+b)^{*}|| w \mid \leq 4\right\}$, where $|w|$ denotes the length of string $w$. The number of strings in $L_{2}$ which are also in $L_{1}$ is _________.

Let $L_{1}$ be the language represented by the regular expression $b^{*} a b^{*}\left(a b^{*} a b^{*}\right)^{*}$ and $L_{2}=\left\{w \in(a+b)^{*}|| w \mid \leq 4\right\}...

Arjun

2.4k views

Arjun asked Feb 16

1 votes

1 answer

GATE CSE 2024 | Set 1 | Question: 13
Let $L_1, L_2$ be two regular languages and $L_3$ a language which is not regular. Which of the following statements is/are always TRUE? $L_1=L_2$ if and only if $L_1 \cap \overline{L_2}=\phi$ $L_1 \cup L_3$ is not regular $\overline{L_3}$ is not regular $\overline{L_1} \cup \overline{L_2}$ is regular

Let $L_1, L_2$ be two regular languages and $L_3$ a language which is not regular.Which of the following statements is/are always TRUE?$L_1=L_2$ if and only if $L_1 \c...

Arjun

3.0k views

Arjun asked Feb 16

1 votes

1 answer

GATE CSE 2024 | Set 1 | Question: 40
Consider the $5$ -state $\text{DFA}$. $M$ accepting the language $L(M) \subset(0+1)^{*}$ shown below. For any string $w \in(0+1)^*$ let $n_0(w)$ be the number of $0^{\prime} s$ in $w$ and $n_1(w)$ be the number of 1 's in $w$. ... $4$ are distinguishable in $M$ States $2$ and $5$ are distinguishable in $M$ Any string $w$ with $n_0(w)=n_1(w)$ is in $L(M)$

Consider the $5$ -state $\text{DFA}$. $M$ accepting the language $L(M) \subset(0+1)^{*}$ shown below. For any string $w \in(0+1)^*$ let $n_0(w)$ be the number of $0^{\...

Arjun

2.8k views

Arjun asked Feb 16

2 votes

3 answers

GATE CSE 2024 | Set 1 | Question: 51
Consider the following two regular expressions over the alphabet $\{0,1\}$ : $r= 0^{*}+1^{*}$ $s = 01^{*} + 10^{*}$ The total number of strings of length less than or equal to $5$, which are neither in $r$ nor in $s$, is ________.

Consider the following two regular expressions over the alphabet $\{0,1\}$ :$r= 0^{*}+1^{*}$$s = 01^{*} + 10^{*}$The total number of strings of length less than or equal ...

Arjun

2.2k views

Arjun asked Feb 16

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DFA construction
Each of the following languages is the complement of a simpler language. In each part, construct a DFA for the simpler language, then use it to give the state diagram of a DFA for the language given. In all parts, Σ = {a, b}. 1- {w| w does not contain the substring ab} 2- {w| ... neither the substrings ab nor ba} 4- {w| w is any string not in $a^{*}\cup b^{*}$ } ( ∪ is the union )

Each of the following languages is the complement of a simpler language.In each part, construct a DFA for the simpler language, then use it to give the state diagram of a...

rania

182 views

rania asked Feb 14

0 votes

1 answer

State diagram
Give state diagrams of DFAs recognizing the following languages. In all parts, the alphabet is {0,1}. 1- {w| w starts with 0 and has odd length, or starts with 1 and has even length}

Give state diagrams of DFAs recognizing the following languages. In all parts, the alphabet is {0,1}.1- {w| w starts with 0 and has odd length, or starts with 1 and has e...

rania

198 views

rania asked Feb 14

4 votes

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GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 31
DeMorgan's Laws ensure that Closure under intersection and complementation imply closure under union. Closure under intersection and union imply closure under complementation. Closure under union and complementation imply closure ... Closure under any two of union, intersection, and complementation implies closure under all three.

DeMorgan’s Laws ensure thatClosure under intersection and complementation imply closure under union.Closure under intersection and union imply closure under complementa...

GO Classes

347 views

GO Classes asked Feb 5

3 votes

1 answer

GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 32
Which of the following sets has the greatest cardinality? The set of real numbers R The set of all functions from R to {0,1} The set of all finite subsets of natural numbers The set of all finite-length binary strings

Which of the following sets has the greatest cardinality?The set of real numbers RThe set of all functions from R to {0,1}The set of all finite subsets of natural numbers...

GO Classes

512 views

GO Classes asked Feb 5

3 votes

3 answers

GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 35
Which of the following strings are a member of the language described by the regular expression $\left(a^* {b} {a}^* b a^* b {a}^*\right)^*$ $b b b b$ $bbaaabb$ $bbaaabbbabb$ $b b a b b b a b$

Which of the following strings are a member of the language described by the regular expression $\left(a^* {b} {a}^* b a^* b {a}^*\right)^*$$b b b b$$bbaaabb$$bbaaabbbabb...

GO Classes

637 views

GO Classes asked Feb 5

5 votes

1 answer

GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 64
Which of the following languages are Turing-recognizable? A. $\{\langle M\rangle \mid M$ is a (deterministic) Turing machine and $M$ accepts 010$\}$. B. $\{\langle M\rangle \mid M$ is a nondeterministic Turing machine and $M$ accepts 010 ... $\left\{\langle M\rangle \mid M\right.$ is a Turing machine and $\left.L(M)=\Sigma^*\right\}$.

Which of the following languages are Turing-recognizable?A. $\{\langle M\rangle \mid M$ is a (deterministic) Turing machine and $M$ accepts 010$\}$.B. $\{\langle M\rangle...

GO Classes

583 views

GO Classes asked Feb 5

4 votes

0 answers

GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 65
Which of the following statements are true for every language $\mathrm{L} \subseteq\{0,1\}^*?$ $L^{\star}$ is infinite. $L$ is accepted by some DFA if and only if $L$ is accepted by some NFA. If $\mathrm{L}$ is the ... $\mathrm{L}$ is undecidable. If $\mathrm{L}$ is the union of two decidable languages, then $\mathrm{L}$ is decidable.

Which of the following statements are true for every language $\mathrm{L} \subseteq\{0,1\}^*?$$L^{\star}$ is infinite.$L$ is accepted by some DFA if and only if $L$ is ac...

GO Classes

649 views

GO Classes asked Feb 5

6 votes

1 answer

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 23
Which of the following statements is/are false? If a context-free grammar $\mathrm{G}$ is in Chomsky's normal form, then $\mathrm{G}$ is not ambiguous. For every number $n$ ... s a $10$-state NFA that accepts $\text{L}$ then there's a $100$-state DFA that accepts $\mathrm{L}$.

Which of the following statements is/are false?If a context-free grammar $\mathrm{G}$ is in Chomsky's normal form, then $\mathrm{G}$ is not ambiguous.For every number $n$...

GO Classes

682 views

GO Classes asked Jan 28

6 votes

1 answer

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 24
Let $\mathrm{M}$ be a $8$-state Deterministic Finite Automaton over the alphabet $\{a, b, c\}$. If the language accepted by $\text{M}$ i.e. $\text{L(M)}$ is finite, then the maximum possible cardinality of $\text{L(M)}$ is __________.

Let $\mathrm{M}$ be a $8$-state Deterministic Finite Automaton over the alphabet $\{a, b, c\}$. If the language accepted by $\text{M}$ i.e. $\text{L(M)}$ is finite, then ...

GO Classes

606 views

GO Classes asked Jan 28

5 votes

1 answer

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 55
Recall that a Turing machine $\text{T}$ can be represented or 'coded' by an integer $m$. Let us write 'the $m$ th Turing machine' to mean the Turing machine coded by $m$. Which of the following sets is not ... machine halts on the input $m$. The set of $n$ such that all Turing machines halt on the input $n$.

Recall that a Turing machine $\text{T}$ can be represented or 'coded' by an integer $m$. Let us write 'the $m$ th Turing machine' to mean the Turing machine coded by $m$....

GO Classes

579 views

GO Classes asked Jan 28

7 votes

1 answer

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 56
For a string $x=x_1 \cdots x_n \in \Sigma^*$, where $\Sigma$ is any alphabet and $x_1, \ldots, x_n \in \Sigma$, we write $x^{\uparrow m}=x^m$ (that is, the usual power of strings) and $x^{\downarrow m}=x_1^m \cdots x_n^m$. For empty ... $\left\{(a b c)^{\downarrow n} \mid n \geq 0\right\}$

For a string $x=x_1 \cdots x_n \in \Sigma^*$, where $\Sigma$ is any alphabet and $x_1, \ldots, x_n \in \Sigma$, we write $x^{\uparrow m}=x^m$ (that is, the usual power of...

GO Classes

511 views

GO Classes asked Jan 28

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