Context free languages and Push-down automata

Question 1

Consider the following languages. 

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 Which one of the following statements is FALSE?

• L2 is context-free.

• L1 intersection L2 is context-free.

• Complement of L2 is recursive.

• Complement of L1 is context-free but not regular.

Question 2

Which of the following pairs have DIFFERENT expressive power?

• Deterministic finite automata(DFA) and Non-deterministic finite automata(NFA)
• Deterministic push down automata(DPDA)and Non-deterministic push down automata(NPDA)
• Deterministic single-tape Turing machine and Non-deterministic single-tape Turing machine
• Single-tape Turing machine and multi-tape Turing machine

Question 3

Let P be a regular language and Q be context-free language such that Q ⊆ P. (For example, let P be the language represented by the regular expression p*q* and Q be {pⁿ qⁿ  | n ∈ N}). Then which of the following is ALWAYS regular? 

(A) P ∩ Q 

(B) P - Q 

(C) ∑* - P 

(D) ∑* - Q

• A

• B

• C

• D

Question 4

Consider the language L1,L2,L3 as given below. L1={[Tex]0^{p}1^{q}[/Tex] | p,q [Tex]\\in[/Tex] N} L2={[Tex]0^{p}1^{q}[/Tex] | p,q [Tex]\\in[/Tex] N and p=q} L3={[Tex]0^{p}1^{q}0^{r}[/Tex] | p,q,r [Tex]\\in[/Tex] N and p=q=r} Which of the following statements is NOT TRUE?

• Push Down Automata (PDA) can be used to recognize L1 and L2
• L1 is a regular language
• All the three languages are context free
• Turing machine can be used to recognize all the three languages

Question 5

Consider the languages - 
L1 = {0ⁱ1^j | i != j}. 
L2 = {0ⁱ1^j | i = j}. 
L3 = {0ⁱ1^j | i = 2j+1}. 
L4 = {0ⁱ1^j | i != 2j}.

• Only L2 is context free

• Only L2 and L3 are context free

• Only L1 and L2 are context free

• All are context free

Question 6

S -> aSa|bSb|a|b; The language generated by the above grammar over the alphabet {a,b} is the set of

• All palindromes

• All odd length palindromes.

• Strings that begin and end with the same symbol

• All even length palindromes

Question 7

Let L = L1∩L2, where L1 and L2 are languages as defined below:

L1 = {a^m b^m caⁿ bⁿ | m, n >= 0}

L2 = {aⁱ b^j c^k | i, j, k >= 0}

Then L is

• Not recursive

• Regular

• Context free but not regular

• Recursively enumerable but not context free.

Question 8

The language L= {0ⁱ21ⁱ | i≥0 } over the alphabet {0,1, 2} is:

• not recursive

• is recursive and is a deterministic CFL.

• is a regular language.

• is not a deterministic CFL but a CFL.

Question 9

Consider the CFG with {S,A,B) as the non-terminal alphabet, {a,b) as the terminal alphabet, S as the start symbol and the following set of production rules

S --> aB        S --> bA
B --> b         A --> a
B --> bS        A --> aS
B --> aBB       A --> bAA

Which of the following strings is generated by the grammar?

• aaaabb
• aabbbb
• aabbab
• abbbba

Question 10

For the correct answer strings to below grammar, how many derivation trees are there?

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• 1

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