Recursively enumerable sets and Turing machines

Question 1

Which of the following statements is/are FALSE?

1. For every non-deterministic Turing machine, 
   there exists an equivalent deterministic Turing machine.
2. Turing recognizable languages are closed under union 
   and complementation.
3. Turing decidable languages are closed under intersection 
   and complementation.
4. Turing recognizable languages are closed under union 
   and intersection. 

• 1 and 4 only

• 1 and 3 only

• 2 only

• 3 only

Question 2

Let L1 be a recursive language. Let L2 and L3 be languages that are recursively enumerable but not recursive. Which of the following statements is not necessarily true?

• L2 – L1 is recursively enumerable.
• L1 – L3 is recursively enumerable
• L2 ∩ L1 is recursively enumerable
• L2 ∪ L1 is recursively enumerable

Question 3

Which of the following is true for the language

• It is not accepted by a Turing Machine
• It is regular but not context-free
• It is context-free but not regular
• It is neither regular nor context-free, but accepted by a Turing machine

Question 4

If L and L\' are recursively enumerable, then L is

• regular
• context-free
• context-sensitive
• recursive

Question 5

Let L be a language and L\' be its complement. Which one of the following is NOT a viable possibility?

• Neither L nor L\' is recursively enumerable (r.e.).
• One of L and L\' is r.e. but not recursive; the other is not r.e.
• Both L and L\' are r.e. but not recursive.
• Both L and L\' are recursive

Question 6

Let A ≤_m B denotes that language A is mapping reducible (also known as many-to-one reducible) to language B. Which one of the following is FALSE?

• If A ≤_m B and B is recursive then A is recursive.
• If A ≤_m B and A is undecidable then B is undecidable.
• If A ≤_m B and B is recursively enumerable then A is recursively enumerable.
• If A ≤_m B  and B is not recursively enumerable then A is not recursively enumerable.

Question 7

For S ∈ (0 + 1) * let d(s) denote the decimal value of s (e.g. d(101) = 5). Let L = {s ∈ (0 + 1)* d(s)mod5 = 2 and d(s)mod7 != 4}. Which one of the following statements is true?

• L is recursively enumerable, but not recursive
• L is recursive, but not context-free
• L is context-free, but not regular
• L is regular

Question 8

Let L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?

L1\' --> Complement of L1
L2\' --> Complement of L2 

• L1\' is recursive and L2\' is recursively enumer­able
• L1\' is recursive and L2\' is not recursively enumerable
• L1\' and L2\' are recursively enumerable
• L1\' is recursively enumerable and L2\' is recursive

Question 9

L1 is a recursively enumerable language over Σ. An algorithm A effectively enumerates its words as w1, w2, w3, ... Define another language L2 over Σ Union {#} as {wi # wj : wi, wj ∈ L1, i < j}. Here # is a new symbol. Consider the following assertions.

S1 :  L1 is recursive implies L2 is recursive
S2 : L2 is recursive implies L1 is recursive 

Which of the following statements is true ?

• Both S1 and S2 are true
• S1 is true but S2 is not necessarily true
• S2 is true but S1 is not necessarily true
• Neither is necessarily true

Question 10

Nobody knows yet if P = NP. Consider the language L defined as follows : Which of the following statements is true ?

• L is recursive
• L is recursively enumerable but not recursive
• L is not recursively enumerable
• Whether L is recursive or not will be known after we find out if P = NP