Context free languages and Push-down automata

Question 31

Consider a CFG with the following productions. S → AA | B A → 0A | A0 | 1 B → 0B00 | 1 S is the start symbol, A and B are non-terminals and 0 and 1 are the terminals. The language generated by this grammar is

• {0ⁿ 10²ⁿ | n ≥ 1}
• {0ⁱ 10^j 10^k | i, j, k ≥ 0} ∪ {0ⁿ 10²ⁿ | n ≥ l}
• {0ⁱ 10^j | i, j ≥ 0} ∪ {0ⁿ 10²ⁿ | n ≥ l}
• The set of all strings over {0, 1} containing at least two 0\'s
• None of the above

Question 32

A CFG G is given with the following productions where S is the start symbol, A is a non-terminal and a and b are terminals.
S→aS∣A
A→aAb∣bAa∣ϵ
Which of the following strings is generated by the grammar above?

• aabbaba
• aabaaba
• abababb
• aabbaab

Question 33

A CFG G is given with the following productions where S is the start symbol, A is a non-terminal and a and b are terminals.
S→aS∣A
A→aAb∣bAa∣ϵ
For the correct answer in Q75, how many steps are required to derive the string and how many parse trees are there?

• 6 and 1
• 6 and 2
• 7 and 2
• 4 and 2

Question 34

Consider 2 scenarios:

C1: For DFA (ϕ, Ʃ, δ, qo, F),
         if F = ϕ, then L = Ʃ*
C2: For NFA (ϕ, Ʃ, δ, qo, F),
         if F = ϕ, then L = Ʃ*
Where F = Final states set
ϕ = Total states set

Choose the correct option ?

• Both are true
• Both are False
• C1 is true, C2 is false
• C1 is false, C2 is true

Question 35

Let G be the CFG, l be the number of left most derivations, r be the number of right most derivations and P be the number of parse trees. Assume l , r and P are computed for a particular string. For a given CFG ‘G’ and given string ‘w’, what is the relation between l , P , r ?

• l ≤ P ≥ r
• l = P = r
• l ≥ P ≤ r
• none of these

Question 36

Given the following two statements : A. L = {w|n_a(w) = n_b(w)} is deterministic context free language, but not linear. B. L = {aⁿ bⁿ} ∪ {aⁿ b²ⁿ} is linear, but not deterministic context free language. 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 37

If L1 is a context free language and L2 as a regular language, which of the following is/are False? a.  L1-L2 is not context free b.  L1 ∩ L2 is context free c.  ~L1 is context free d.  ~L2 is regular

• Only b
• Only c
• Both a and c
• Both b and c

Question 38

[5 Marks question] Show that the language L = { xcx | x∈ {0,1}* and c is a terminal symbol } is not context free, c is not 0 or 1.

Question 39

Regarding the power of recognition of languages, which of the following statement is false?

• The non-deterministic finite state automata are equivalent to deterministic finite state automata.
• Non-deterministic pushdown automata is equivalent to deterministic pushdown automata.
• Non-deterministic Turing machines are equivalent to deterministic Turing machines.
• Multi-tape Turing machines are equivalent to Single-tape Turing machines.

Question 40

Let M = ({q₀, q₁}, {0, 1}, {z₀, x}, δ, q₀, z₀, ∅) be a pushdown automaton where δ is given by δ(q₀, 1, z₀) = {(q₀, xz₀)} δ(q₀, ε, z₀) = {(q₀, ε)} δ(q₀, 1, X) = {(q₀, XX)} δ(q₁, 1, X) = {(q₁, ε)} δ(q₀, 0, X) = {(q₁, X)} δ(q₀, 0, z₀) = {(q₀, z₀)}

1. What is the language accepted by this PDA by empty stack?
2. Describe informally the working of the PDA