RELs , Turing Machine & Closure properties PYQ QUIZ GATE CS

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Question 1

Which of the following problems are decidable?

gatecs2012automata2

  • 1, 2, 3, 4

  • 1, 2

  • 2, 3, 4

  • 3, 4

Question 2

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 3

Consider the following languages. 

 

 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 4

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 5

Let L1 be a regular language and L2 be a context-free language. Which of the following languages is/are context-free?

  • L1∩L2'

  • (L1'∪L2')'

  • L1∪(L2∪L2')

  • (L1∩L2)∪(L1∩L2)

Question 6

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 7

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 8

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

  • regular

  • context-free

  • context-sensitive

  • recursive

Question 9

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 10

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.

There are 23 questions to complete.

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