Theory of Computation

Question 1
Consider the following languages:

\begin{aligned} L_1&= \{ ww|w \in \{a,b \}^* \} \\ L_2&= \{a^nb^nc^m | m,n \geq 0 \} \\ L_3 &= \{a^mb^nc^n|m,n \geq 0 \} \end{aligned}

Which of the following statements is/are FALSE?
MSQ
A
L_1 is not context-free but L_2 and L_3 are deterministic context-free.
B
Neither L_1 nor L_2 is context-free.
C
L_2,L_3 and L_2 \cap L_3 all are context-free.
D
Neither L_1 nor its complement is context-free.
GATE CSE 2022      Context Free Language
Question 2
Consider the following languages:

\begin{aligned} L_1&= \{a^n wa^n|w \in \{a,b \}^* \} \\ L_2&= \{wxw^R | w,x \in \{a,b \}^*, |w|,|x| \gt 0 \} \end{aligned}

Note that w^R is the reversal of the string w. Which of the following is/are TRUE?
MSQ
A
L_1 and L_2 are regular.
B
L_1 and L_2 are context-free.
C
L_1 is regular and L_2 is context-free.
D
L_1 and L_2 are context-free but not regular.
GATE CSE 2022      Context Free Language
Question 3
Which of the following is/are undecidable?
MSQ
A
Given two Turing machines M1 and M2 , decide if L(M1 ) = L(M2 ).
B
Given a Turing machine M , decide if L(M ) is regular.
C
Given a Turing machine M , decide if M accepts all strings.
D
Given a Turing machine M , decide if M takes more than 1073 steps on every string.
GATE CSE 2022      Turing Machine
Question 4
Which of the following statements is/are TRUE?
MSQ
A
Every subset of a recursively enumerable language is recursive.
B
If a language L and its complement \bar{L} are both recursively enumerable, then L must be recursive.
C
Complement of a context-free language must be recursive.
D
If L_1 and L_2 are regular, then L_1 \cap L_2 must be deterministic context-free.
GATE CSE 2022      Recursive Language
Question 5
Which one of the following regular expressions correctly represents the language of the finite automaton given below?

A
ab*bab* + ba*aba*
B
(ab*b)*ab* + (ba*a)*ba*
C
(ab*b + ba*a)*(a* + b*)
D
(ba*a + ab*b)*(ab* + ba*)
GATE CSE 2022      Finite Automata
Question 6
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.
[MSQ]
A
(0+1(01^*0)^*1)^*
B
(0+11+10(1+00)^*01)^*
C
(0^*(1(01^*0)^*1)^*)^*
D
(0+11+11(1+00)^*00)^*
GATE CSE 2021 SET-2      Regular Expression
Question 7
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?
[MSQ]
A
\{ wxw^Rx^R \mid w,x \in \{0,1\} ^* \}
B
\{ ww^Rxx^R \mid w,x \in \{0,1\} ^* \}
C
\{ wxw^R \mid w,x \in \{0,1\} ^* \}
D
\{ wxx^Rw^R \mid w,x \in \{0,1\} ^* \}
GATE CSE 2021 SET-2      Context Free Language
Question 8
Consider the following two statements about regular languages:

S1: Every infinite regular language contains an undecidable language as a subset.
S2: Every finite language is regular.

Which one of the following choices is correct?
A
Only S1 is true
B
Only S2 is true
C
Both S1 and S2 are true
D
Neither S1 nor S2 is true
GATE CSE 2021 SET-2      Regular Language
Question 9
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. Consider the following example.

\begin{array}{ll} \text{Input sequence:} & 00100011000011100 \\ \text{Output sequence:} & 00000001000001100 \end{array}

A Mealy Machine is a state machine where both the next state and the output are functions of the present state and the current input.
The above mentioned circuit can be designed as a two-state Mealy machine. The states in the Mealy machine can be represented using Boolean values 0 and 1. We denote the current state, the next state, the next incoming bit, and the output bit of the Mealy machine by the variables s, t, b and y respectively.
Assume the initial state of the Mealy machine is 0.

What are the Boolean expressions corresponding to t and y in terms of s and b?
A
t=s+b
y=sb
B
t=b
y=sb
C
t=b
y=s\bar{b}
D
t=s+b
y=s\bar{b}
GATE CSE 2021 SET-2      Finite Automata
Question 10
Consider the following deterministic finite automaton (DFA)

The number of strings of length 8 accepted by the above automaton is _________
A
32
B
256
C
64
D
512
GATE CSE 2021 SET-2      Finite Automata


There are 10 questions to complete.

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