Undecidability

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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

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Question 2

Which of the following is/are undecidable?

gatecs2013.15

  • 3 only

  • 3 and 4 only

  • 1, 2 and 3 only

  • 2 and 3 only

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Question 3

Which of the following are decidable? 

I. Whether the intersection of two regular languages is infinite
II. Whether a given context-free language is regular
III. Whether two push-down automata accept the same language
IV. Whether a given grammar is context-free

  • I and II

  • I and IV

  • II and III

  • II and IV

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Question 4

Which of the following problems is undecidable?

  • Membership problem for CFGs

  • Ambiguity problem for CFGs.

  • Finiteness problem for FSAs.

  • Equivalence problem for FSAs.

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Question 5

Let be the encoding of a Turing machine as a string over ∑= {0, 1}.  Let L = { |M is a Turing machine that accepts a string of length 2014 }. Then, L is

  • decidable and recursively enumerable

  • undecidable but recursively enumerable

  • undecidable and not recursively enumerable

  • decidable but not recursively enumerable

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Question 6

Which of the following problems is undecidable?

  • Deciding if a given context-free grammar is ambiguous.

  • Deciding if a given string is generated by a given context-free grammar.

  • Deciding if the language generated by a given context-free grammar is empty.

  • Deciding if the language generated by a given context-free grammar is finite.

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Question 7

Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?

  • P3 is decidable if P1 is reducible to P3

  • P3 is undecidable if P3 is reducible to P2

  • P3 is undecidable if P2 is reducible to P3

  • P3 is decidable if P3 is reducible to P2's complement

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Question 8

Consider two languages L1 and L2 each on the alphabet ∑. Let f : ∑ → ∑ be a polynomial time computable bijection such that (∀ x) [x ∈ L1 if f(x) ∈ L2]. Further, let f-1 be also polynomial time computable. Which of the following CANNOT be true?

  • L1 ∈ P and L2 is finite

  • L1 ∈ NP and L2 ∈ P

  • L1 is undecidable and L2 is decidable

  • L1 is recursively enumerable and L2 is recursive

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Question 9

Consider the following problem X. 
  Given a Turing machine M over the input alphabet Σ, any  state q of M And a word w∈Σ*, does the computation of M  on w visit the state q?
Which of the following statements about X is correct?
  • X is decidable
  • X is undecidable but partially decidable
  • X is undecidable and not even partially decidable
  • X is not a decision problem
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Question 10

Consider the following decision problems:

(P1) Does a given finite state machine accept a given string (P2) Does a given context free grammar generate an infinite       number of strings

Which of the following statements is true?

  • Both (P1) and (P2) are decidable

  • Neither (P1) nor (P2) are decidable

  • Only (P1) is decidable

  • Only (P2) is decidable

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There are 27 questions to complete.

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