Recursively enumerable sets and Turing machines

Question 11

A single tape Turing Machine M has two states q0 and q1, of which q0 is the starting state. The tape alphabet of M is {0, 1, B} and its input alphabet is {0, 1}. The symbol B is the blank symbol used to indicate end of an input string. The transition function of M is described in the following table

	0	1	B
q0	q1, 1, R	q1, 1, R	Halt
q1	q1, 1, R	q0, 1, L	q0, B, L

  The table is interpreted as illustrated below. The entry (q1, 1, R) in row q0 and column 1 signifies that if M is in state q0 and reads 1 on the current tape square, then it writes 1 on the same tape square, moves its tape head one position to the right and transitions to state q1. Which of the following statements is true about M ?

• M does not halt on any string in (0 + 1)+
• M does not halt on any string in (00 + 1)*
• M halts on all string ending in a 0
• M halts on all string ending in a 1

Question 12

Define languages L0 and L1 as follows :

L0 = {< M, w, 0 > | M halts on w}
L1 = {< M, w, 1 > | M does not halts on w} 

Here < M, w, i > is a triplet, whose first component. M is an encoding of a Turing Machine, second component, w, is a string, and third component, i, is a bit. Let L = L0 ∪ L1. Which of the following is true ?

• L is recursively enumerable, but L\' is not
• L\' is recursively enumerable, but L is not
• Both L and L\' are recursive
• Neither L nor L\' is recursively enumerable

Question 13

Which of the following is true?

• The complement of a recursive language is recursive.
• The complement of a recursively enumerable language is recursively enumerable.
• The complement of a recursive language is either recursive or recursively enumerable.
• The complement of a context-free language is context-free.

Question 14

For any two languages L1 and L2 such that L1 is context free and L2 is recursively enumerable but not recursive, which of the following is/are necessarily true?

1. L1\' (complement of L1) is recursive 
2. L2\' (complement of L2) is recursive
3. L1\' is context-free 
4. L1\' ∪ L2 is recursively enumerable 

• 1 only
• 3 only
• 3 and 4 only
• 1 and 4 only

Question 15

Let X be a recursive language and Y be a recursively enumerable but not recursive language. Let W and Z be two languages such that Y\' reduces to W, and Z reduces to X\' (reduction means the standard many-one reduction). Which one of the following statements is TRUE

• W can be recursively enumerable and Z is recursive.
• W an be recursive and Z is recursively enumerable.
• W is not recursively enumerable and Z is recursive.
• W is not recursively enumerable and Z is not recursive

Question 16

Consider the following types of languages:

L1 Regular,
L2: Context-free,
L3: Recursive,
L4: Recursively enumerable. 

Which of the following is/are TRUE?

I.   L3\' U L4 is recursively enumerable
II.  L2  U L3 is recursive
III. L1* U L2 is context-free
IV.  L1 U L2\' is context-free 

• I only
• I and III only
• I and IV only
• I, II and III only

Question 17

A language L is called Turing-decidable (or just decidable), if there exists a Turing Machine M such that on input x, M accepts if x ∈ L, and M rejects otherwise. L is called undecidable if it is not decidable. Which of following option is false?

• The class of decidable languages is closed under complement.
• The class of decidable languages is closed under union
• The class of decidable languages is closed under intersection
• None of these

Question 18

The set of all recursively enumerable languages is

• closed under complementation.
• closed under intersection.
• a subset of the set of all recursive languages.
• an uncountable set.

Question 19

Let L = {a^p | p is a prime}. Then which of the following is true?

• 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 20

Which of the following statements is not correct?

• Every recursive language is recursively enumerable.
• L = {0ⁿ1ⁿ 0ⁿ │n=1, 2 , 3, ....} is recursively enumerable.
• Recursive languages are closed under intersection.
• Recursive languages are not closed under intersection.