GATE | GATE-CS-2014-(Set-1) | Question 24

Which one of the following is TRUE? 

(A)

The language L = {aⁿbⁿ | n ≥ 0} is regular.

(B)

 The language L = {aⁿ| n is prime} is regular.

(C)

The language L = {w|w has 3k + 1 b’s for some k ∈N with Σ = {a, b}} is regular. 

(D)

The language L = {ww | we Σ* with Σ = {0,1}} is regular.

Answer: (C)

Explanation:

(A) L = {a n b n |n >= 0} is not regular because there does not exists a finite automaton that can derive this grammar. Intuitively, finite automaton has finite memory, hence it can’t track number of as. It is a standard CFL though. (B) L = {a n b n |n is prime} is again not regular because there is no way to remember/check if current n is prime or not. Hence, no finite automaton exists to derive this grammar, thus it is not regular. (C) L = {w|w has 3k+1 bs} is a regular language because k is a fixed constant and we can easily emulate L as a ∗ ba ∗ …..ba ∗ such that there are exactly 3k + 1 bs and a ∗ s surrounding each b in the grammar. 

 (D) L = {ww| w ∈ Σ ∗ } is again not a regular grammar, infact it is not even a CFG. There is no way to remember and derive double word using finite automaton. Hence, correct answer would be (C). 

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