GATE | GATE CS 2021 | Set 2 | Question 51

For a string w, we define w^R to be the reverse of w. For example, if w = 01101 then w^R = 10110.

Which of the following languages is/are context-free?
(A) {wxw^Rx^R ∣ w,x∈{0,1}*}
(B) {ww^Rxx^R ∣ w,x∈{0,1}*}
(C) {wxw^R ∣ w,x∈{0,1}*}
(D) {wxx^Rw^R ∣ w,x∈{0,1}*}


Answer: (B) (C) (D)

Explanation: Option A: L={w x w^R x^R | w, x ∈ {0,1}* }
This is not CFL as if we push “w” then “x” then we cannot match w^R with “w” as top of stack contains x.

Option B: L={w w^R x x^R | w, x ∈ {0,1}* }
This is CFL. We non deterministically guess the middle of the string. So we push “w” then match with w^R and again push x and match with x^R

Option C: L={w x x^R w^R | w, x ∈ {0,1}* }
This is also CFL. We non deterministically guess the middle of the string. So we push “w” then push x and then match with x^R and again match with w^R

Option D: L={w x w^R | w, x ∈ {0,1}* }
This is a regular language (hence CFL). In this language every string start and end with same symbol (as x can expand).

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