Consider the 5-state DFA 𝑀 accepting the language 𝐿(𝑀) ⊂ (0 + 1) &lowast; shown below. For any string 𝑤 ∈ (0 + 1) &lowast; let 𝑛0(𝑤) be the number of 0 ′ 𝑠 in 𝑤 and 𝑛1(𝑤) be the number of 1′𝑠 in 𝑤. .Which of the following statements is/are FALSE? [GATE 2024||SET-1 MSQ|| 2-mark]

States 2 and 5 are distinguishable in 𝑀

States 2 and 4 are distinguishable in 𝑀

Any string 𝑤 with 𝑛0 (𝑤) = 𝑛1(𝑤) is in 𝐿(𝑀)

Consider the two lists List I and List II given below:List 1List 2(i) Context free languages(a) Closed under union(ii) Recursive languages(b) Not closed under complementation(iii) Regular languages(c) Closed under intersectionFor matching of items in List I with those in List II, which of the following option(s) is/are CORRECT? [GATE 2025||SET-2 MSQ|| 1-mark]

(i) – (b), (ii) – (c), and (iii) – (a)

(i) – (a), (ii) – (b), and (iii) – (c)

(i) – (a), (ii) – (c), and (iii) – (b)

GATE||Regular Language and Finite Automata|| Pyq (2010 to2025)

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Question 1

Consider the 5-state DFA 𝑀 accepting the language 𝐿(𝑀) ⊂ (0 + 1) ^∗shown below. For any string 𝑤 ∈ (0 + 1) ^∗ let 𝑛₀(𝑤) be the number of 0 ′ 𝑠 in 𝑤 and 𝑛₁(𝑤) be the number of 1′𝑠 in 𝑤.

Which of the following statements is/are FALSE?

[GATE 2024||SET-1 MSQ|| 2-mark]

States 2 and 4 are distinguishable in 𝑀
States 3 and 4 are distinguishable in 𝑀
States 2 and 5 are distinguishable in 𝑀
Any string 𝑤 with 𝑛₀ (𝑤) = 𝑛₁(𝑤) is in 𝐿(𝑀)

Question 2

Let Σ = {𝑎, 𝑏, 𝑐}. For 𝑥 ∈ Σ ∗ , and 𝛼 ∈ Σ, let #𝛼(𝑥) denote the number of occurrences of 𝛼 in 𝑥. Which one or more of the following option(s) define(s) regular language(s)?

[GATE 2025||SET-2 MSQ|| 2-mark]

{𝑎 ^𝑚𝑏^𝑛 | 𝑚, 𝑛 ≥ 0}
{𝑎, 𝑏} ∗ ∩ {𝑎^𝑚𝑏^𝑛 𝑐^𝑚−𝑛 | 𝑚 ≥ 𝑛 ≥ 0}
{𝑤 | 𝑤 ∈ {𝑎, 𝑏} ∗ , #_𝑎 (𝑤) ≡ 2 (mod 7), and #_𝑏 (𝑤) ≡ 3 (mod 9)}
{𝑤 | 𝑤 ∈ {𝑎, 𝑏} ∗ , #_𝑎 (𝑤) ≡ 2 (mod 7), and #_𝑎 (𝑤) = #𝑏(𝑤)}

Question 3

Consider the two lists List I and List II given below:

List 1	List 2
(i) Context free languages	(a) Closed under union
(ii) Recursive languages	(b) Not closed under complementation
(iii) Regular languages	(c) Closed under intersection

For matching of items in List I with those in List II, which of the following option(s) is/are CORRECT?

[GATE 2025||SET-2 MSQ|| 1-mark]

(i) – (a), (ii) – (b), and (iii) – (c)
(i) – (b), (ii) – (a), and (iii) – (c)
(i) – (b), (ii) – (c), and (iii) – (a)
(i) – (a), (ii) – (c), and (iii) – (b)

Question 4

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)

Screenshot_2025-09-01_121813-removebg-preview — .

[GATE 2025||SET-1 NAT|| 2-mark]

Question 5

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

[GATE 2025||SET-2 MSQ|| 2-mark]

The set of all strings containing an even number of 𝑏’s.
The set of all strings containing the pattern 𝑏𝑎𝑏.
The set of all strings ending with the pattern 𝑏𝑎𝑏.
The set of all strings not containing the pattern 𝑎𝑏𝑎.

Question 6

Consider the following two languages over the alphabet {𝑎, 𝑏}:
𝐿1 = { 𝛼𝛽𝛼 | 𝛼 ∈ {𝑎, 𝑏}⁺ AND 𝛽 ∈ {𝑎, 𝑏}⁺ }
𝐿2 = { 𝛼𝛽𝛼 | 𝛼 ∈ {𝑎}⁺ AND 𝛽 ∈ {𝑎, 𝑏}⁺ }

Which ONE of the following statements is CORRECT?

[GATE 2025||SET-1 MCQ|| 2-mark]

Both 𝐿1 and 𝐿2 are regular languages.
𝐿1 is a regular language but 𝐿2 is not a regular language.
𝐿1 is not a regular language but 𝐿2 is a regular language.
Neither 𝐿1 nor 𝐿2 is a regular language.

Question 7

Let 𝑆 be the set of all ternary strings defined over the alphabet {𝑎, 𝑏, 𝑐}. Consider all strings in 𝑆 that contain at least one occurrence of two consecutive symbols, that is, “aa”, “bb” or “cc”. The number of such strings of length 5 that are possible is _______. (Answer in integer)

[GATE 2025||SET-1 NAT|| 1-mark]

Question 8

A regular language 𝐿 is accepted by a non-deterministic finite automaton (NFA) with 𝑛 states. Which of the following statement(s) is/are FALSE?

[GATE 2025||SET-1 MSQ|| 1-mark]

𝐿 may have an accepting NFA with < 𝑛 states.
𝐿 may have an accepting DFA with < 𝑛 states.
There exists a DFA with ≤ 2^𝑛 states that accepts 𝐿.
Every DFA that accepts 𝐿 has > 2^𝑛 states.

Question 9

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 _________

[GATE 2024||SET-1 NAT|| 2-mark]

Question 10

Let Σ = {1,2,3,4}. For 𝑥 ∈ Σ ∗ , let 𝑝𝑟𝑜𝑑(𝑥) be the product of symbols in 𝑥 modulo 7. We take 𝑝𝑟𝑜𝑑(𝜖) = 1, where 𝜖 is the null string.
For example, 𝑝𝑟𝑜𝑑(124) = (1 × 2 × 4) mod 7 = 1.
Define 𝐿 = {𝑥 ∈ Σ ∗ | 𝑝𝑟𝑜𝑑(𝑥) = 2}.
The number of states in a minimum state DFA for 𝐿 is ___________. (Answer in integer)

[GATE 2025||SET-2 NAT|| 2-mark]

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