Theory of Computation

Question 1
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 2
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 3
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 4
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 5
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
Question 6
Let L_1 be a regular language and L_2 be a context-free language. Which of the following languages is/are context-free?
[MSQ]
A
L_1 \cap \overline{L_2}
B
\overline{\overline{L_1} \cup \overline{L_2}}
C
L_1 \cup (L_2 \cup \overline{L_2})
D
(L_1 \cap L_2) \cup (\overline{L_1} \cap L_2)
GATE CSE 2021 SET-2      Context Free Language
Question 7
Let L \subseteq \{0,1\}^* be an arbitrary regular language accepted by a minimal DFA with k states. Which one of the following languages must necessarily be accepted by a minimal DFA with k states?
A
L - \{01\}
B
L \cup \{01\}
C
\{0,1\}^* -L
D
L \cdot L
GATE CSE 2021 SET-2      Regular Language
Question 8
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 \} ,\;X,Y, \in \Gamma \cup \{ \epsilon \}, represents

(q,Y) \in \delta(p,a,X)

Consider the following pushdown automaton over the input alphabet \Sigma = \{a,b\} and stack alphabet \Gamma = \{ \#, A\}.

The number of strings of length 100 accepted by the above pushdown automaton is ___________
A
50
B
49
C
101
D
100
GATE CSE 2021 SET-1      Push-down Automata
Question 9
For a Turing machine M, < M > denotes an encoding of M. Consider the following two languages.

\begin{array}{ll} L1 = \{ \langle M \rangle \mid M \text{ takes more than } 2021 \text{ steps on all inputs} \} \\ L2 = \{ \langle M \rangle \mid M\text{ takes more than } 2021 \text{ steps on some input} \} \end{array}

Which one of the following options is correct?
A
Both L1 and L2 are decidable.
B
L1 is decidable and L2 is undecidable.
C
L1 is undecidable and L2 is decidable.
D
Both L1 and L2 are undecidable.
GATE CSE 2021 SET-1      Turing Machine
Question 10
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?

A
A
B
B
C
C
D
D
GATE CSE 2021 SET-1      Finite Automata


There are 10 questions to complete.

Leave a Comment

Like this FREE website? Please share it among all your friends and join the campaign of FREE Education to ALL.