If 𝐿 is a regular language over Σ = {𝑎, 𝑏}, which one of the following languages is NOT regular ?

Q. If 𝐿 is a regular language over Σ = {𝑎, 𝑏}, which one of the following languages is NOT regular ?

(A) 𝐿 ⋅ 𝐿^𝑅 = {𝑥𝑦 | 𝑥 ∈ 𝐿, 𝑦^𝑅 ∈ 𝐿}

(B) {𝑤𝑤^𝑅 | 𝑤 ∈ 𝐿}

(C) Prefix (𝐿) = {𝑥 ∈ 𝛴^∗|∃𝑦 ∈ 𝛴^∗ such that 𝑥𝑦 ∈ 𝐿}

(D) Suffix (𝐿) = {𝑦 ∈ 𝛴^∗|∃𝑥 ∈ 𝛴^∗ such that 𝑥𝑦 ∈ 𝐿}

Ans:{𝑤𝑤^𝑅 | 𝑤 ∈ 𝐿}

Solution:

language L = {wwR ⏐ 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.