GATE | GATE CS 2019 | Question 65

If L is a regular language over ∑ = {a, b}, which one of the following languages is NOT regular?
(A) L⋅ L^R {xy ⏐ x ∈ L, y^R∈ L}
(B) Suffix (L) = {y ∈ ∑* ⏐ ∃x ∈ ∑* such that xy ∈ L}
(C) Prefix (L) = {x ∈ ∑* ⏐ ∃y ∈ ∑* such that xy ∈ L}
(D) {ww^R ⏐ w ∈ L}


Answer: (D)

Explanation: Regular languages closed under reversal, concatenation, prefix(L), and suffix(L) property. So, languages given in option (A), (B), and (C) are regular.

But language L = {ww^R ⏐ w ∈ L} is infinite and not regular because it involves string matching and we can increase in length indefinitely and then finite automata will run out of memory, so it require stack. Hence, it is context-free but not regular.

Option (D) is correct.

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