1
GATE CSE 2025 Set 2
Numerical
+2
-0

Let $\Sigma=\{1,2,3,4\}$ For $x \in \Sigma^*$, let prod $(x)$ be the product of symbols in $x$ modulo 7 . We take $\operatorname{prod}(\varepsilon)=1$, where $\varepsilon$ is the null string.

For example, $\operatorname{prod}(124)=(1 \times 2 \times 4) \bmod 7=1$.

Define $L=\left\{x \in \Sigma^{\star} \mid \operatorname{prod}(x)=2\right\}$.

The number of states in a minimum state DFA for $L$ is _________ (Answer in integer)

Your input ____
2
GATE CSE 2025 Set 1
MCQ (Single Correct Answer)
+2
-0

Consider the following two languages over the alphabet $\{a, b\}$ :

$$\begin{aligned} & L_1=\left\{\alpha \beta \alpha \mid \alpha \in\{a, b\}^{+} \text {AND } \beta \in\{a, b\}^{+}\right\} \\ & L_2=\left\{\alpha \beta \alpha \mid \alpha \in\{a\}^{+} \text {AND } \beta \in\{a, b\}^{+}\right\} \end{aligned}$$

Which ONE of the following statements is CORRECT?

A
Both $L_1$ and $L_2$ are regular languages.
B
$L_1$ is a regular language but $L_2$ is not a regular language.
C
$L_1$ is not a regular language but $L_2$ is a regular language.
D
Neither $L_1$ nor $L_2$ is a regular language.
3
GATE CSE 2025 Set 1
MCQ (More than One Correct Answer)
+2
-0

Consider the following deterministic finite automaton (DFA) defined over the alphabet, $\Sigma=\{a, b\}$. Identify which of the following language(s) is/are accepted by the given DFA.

GATE CSE 2025 Set 1 Theory of Computation - Finite Automata and Regular Language Question 3 English

A
The set of all strings containing an even number of $b$ 's.
B
  The set of all strings containing the pattern bab.
C
The set of all strings ending with the pattern bab.
D
The set of all strings not containing the pattern aba.
4
GATE CSE 2025 Set 1
Numerical
+2
-0

Consider a finite state machine (FSM) with one input $X$ and one output $f$, represented by the given state transition table. The minimum number of states required to realize this FSM is ________ . (Answer in integer)

Present state Next state Output $f$
$X = 0$ $X = 1$ $X = 0$ $X = 1$
A F B 0 0
B D C 0 0
C F E 0 0
D G A 1 0
E D C 0 0
F F B 1 1
G G H 0 1
H G A 1 0

Your input ____
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