Context free languages and Push-down automata

Question 91

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?

• {wxw^Rx^R ∣ w,x∈{0,1}*}
• {ww^Rxx^R ∣ w,x∈{0,1}*}
• {wxw^R ∣ w,x∈{0,1}*}
• {wxx^Rw^R ∣ w,x∈{0,1}*}

Question 92

Consider the following languages.

L₁ = {aⁱ b^j c^k | i = j, k ≥ 1}
L₁ = {aⁱ b^j | j = 2i, i ≥ 0}
Which of the following is true?

• L₁ is not a CFL but L₂ is
• L₁ ∩ L₂ = ∅ and L₁ is non-regular
• L₁ ∪ L₂ is not a CFL but L₂ is
• There is a 4-state PDA that accepts L₁, but there is no DPDA that accepts L₂

Question 93

Which of following statement(s) is/are not correct? (I) Languages generated by the grammar S→aSa ∣ aa is not regular. (II) Languages generated by the grammar S→aSb ∣ aa is not regular. (III) Languages generated by the grammar S→S1|S3, S1→aS1c |S2|λ, S2→aS2b|λ, S3→aS3b|S4| λ, S4→bS4c|λ is {a^nb^mc^k | k = |n - m|, n≥0, m≥0, k≥0}. (IV) Languages generated by the grammar S→S1S3, S1→aS1c |S2|λ, S2→aS2b|λ, S3→aS3b|S4| λ, S4→bS4c|λ is {a^nb^mc^k | k = |n - m|, n≥0, m≥0, k≥0}.

• Only (II), (IV)

• Only (I), (III), (IV)

• Only (I), (II), (III)

• All of the above