Context free languages and Push-down automata

Question 61

Given the following statements : (A) A class of languages that is closed under union and complementation has to be closed under intersection. (B) A class of languages that is closed under union and intersection has to be closed under complementation. Which of the following options is correct ?

• Both (A) and (B) are false.
• Both (A) and (B) are true.
• (A) is true, (B) is false.
• (A) is false, (B) is true.

Question 62

Let G = (V, T, S, P) be a context-free grammar such that every one of its productions is of the form A → v, with |v| = K > 1. The derivation tree for any W ∈ L(G) has a height h such that

• logK|W| ≤ h ≤ logK((|W|-1)/k-1)
• logK|W| ≤ h ≤ logK(K|W|)
• logK|W| ≤ h ≤ K logK|W|
• logK|W| ≤ h ≤ ((|W|-1)/k-1)

Question 63

Given the following two languages : L1 = {aⁿ bⁿ | n ≥ 0, n ≠ 100} L2 = {w ∈ {a, b, c}*| n_a(w) = n_b(w) = n_c(w)} Which of the following options is correct ?

• Both L₁ and L₂ are not context free language
• Both L₁ and L₂ are context free language.
• L₁ is context free language, L₂ is not context free language.
• L₁ is not context free language, L₂ is context free language.

Question 64

What is the number of steps required to derive the string ((() ()) ())

S → SS
S → (S)
S → ε

• 10
• 15
• 12
• 16

Question 65

Consider the following grammar.

S -> AB
A -> a
A -> BaB
B -> bbA

Which of the following statements is FALSE?

• The length of every string produced by this grammar is even
• No string produced by this grammar has three consecutive a\'s
• The length of substring produced by B is always odd
• No string produced by this grammar has four consecutive b\'s

Question 66

Let G = (V, T, S, P) be a context-free grammar such that every one of its productions is of the form A → ν, with |ν| = k > 1. The derivation tree for any string W ∈ L (G) has a height such that

• h<((|W| - 1)/k-1)
  
• log_k|W| < h
  
• log_k|W| < h < ((|W| - 1)/k-1)
  
• log_k|W| <= h <= ((|W| - 1)/k-1)

Question 67

Let L = {0ⁿ1ⁿ|n ≥ 0} be a context free language. Which of the following is correct ?

• L` is context free and L<sup>k</sup> is not context free for any k ≥ 1.
• L` is not context free and L^k is not context free for any k ≥ 1.
• Both L` and L<sup>k</sup> is for any k ≥ 1 are context free.
• Both L` and L^k is for any k ≥ 1 are not context free.

Question 68

Given a Turing Machine M = ({q₀, q₁, q₂, q₃}, {a, b}, {a, b, B}, δ, B, {q₃}) Where δ is a transition function defined as δ(q₀, a) = (q₁, a, R) δ(q₁, b) = (q₂, b, R) δ(q₂, a) = (q₂, a, R) δ(q₃, b) = (q₃, b, R) The language L(M) accepted by the Turing Machine is given as:

• aa*b
• abab
• aba*b
• aba*

Question 69

The context free grammar given by S → XYX X → aX|bX|λ Y → bbb generates the language which is defined by regular expression:

• (a + b)*bbb
• abbb(a + b)*
• (a + b)*(bbb)(a + b)*
• (a + b)(bbb)(a + b)*

Question 70

Match the following:

• (1)
• (2)
• (3)
• (4)