GATE | GATE CS 2019 | Question 40

Which one of the following languages over ∑ = {a, b} is NOT context-free?
(A) {anbi ⏐ i ∈ {n, 3n, 5n}, n≥ 0}
(B) {wanwRbn ⏐ w ∈ {a, b}*, n≥ 0}
(C) {wwR ⏐ w ∈ {a, b}*}
(D) {wan bn wR ⏐ w ∈ {a, b}*, n≥ 0}

Answer: (B)

Explanation: (A): It is CFL, because we have union of three CFLs, and CFLs are closed under union property.

{anbi ⏐ i ∈ {n, 3n, 5n}, n≥ 0} 
= anbn ∪ anb3n ∪ anb5n 

We can identify strings of these language using only one stack.

(B): It is not CFL, because you can identify strings of given langauge using only one stack, you need atleast 2 stacks. There is alternate string matching which is not possible using only one stack. It is context sensitive language but not content free.

(C): It is very well known CFL. It is language of palindrome over only two alphabets which can be recognize by using only one stack.

(D): It is also CFL, since we can first push w, then a’s, b’s pop with a’s and wR pops with the w. So PDA can accept the language.

Option (B) is correct.

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