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 problems is/are decidable problem(s) (recursively enumerable) on a Turing machine M? (a) G is a CFG with L(G)=∅ (b) There exist two TMs M​1​ and M2​ such that L(M) ⊆{L(M​1​)UL(M​2​)}= language of all TMs (c) M is a TM that accepts w using a most 2​|w|​ cells of tape
  • a and b only
  • a only
  • a, b and c
  • c only
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Question 4

​​​​​​Consider the following two statements about regular languages:

  • S1: Every infinite regular language contains an undecidable language as a subset.
  • S2: Every finite language is regular.

Which one of the following choices is correct?

  • Only S1 is true

  • Only S2 is true

  • Both S1 and S2 are true

  • Neither S1 nor S2 is true

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Question 5

Consider the following sets, where n≥2:

  • S1: Set of all n×n matrices with entries from the set {a,b,c}
  • S2: Set of all functions from the set {0,1,2 ... ,n2−1} to the set {0,1,2}

Which of the following choice(s) is/are correct?

  • There does not exist a bijection from S1 to S2

  • There exists a surjection from S1 to S2

  • There exists a bijection from S1 to S2

  • There does not exist an injection from S1 to S2

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Question 6

For a Turing machine M, ⟨M⟩ denotes an encoding of M. Consider the following two languages.

  • L1 = { ⟨M⟩ ∣ M takes more than 2021 steps on all inputs }
  • L2 = { ⟨M⟩ ∣ M takes more than 2021 steps on some input }
    • Which one of the following options is correct?

  • Both L1 and L2 are decidable

  • L1 is decidable and L2 is undecidable

  • L1 is undecidable and L2 is decidable

  • Both L1 and L2 are undecidable

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Question 7

Let ⟨M⟩ denote an encoding of an automaton M. Suppose that Σ={0,1}. Which of the following languages is/are NOT recursive?

  • L = {⟨M⟩ ∣ M is a DFA such that L(M)=∅}

  • L = {⟨M⟩ ∣ M is a DFA such that L(M)=Σ*}

  • L = {⟨M⟩ ∣ M is a PDA such that L(M)=∅}

  • L = {⟨M⟩ ∣ M is a PDA such that L(M)=Σ*}

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Question 8

Which of the following languages are undecidable? Note that ⟨M⟩ indicates encoding of the Turing machine M.

  • L1 = { ⟨M⟩ ∣ L(M)=∅ }
  • L2 = { ⟨M,w,q⟩ ∣ M on input w reaches state q in exactly 100 steps }
  • L3 = { ⟨M⟩ ∣ L(M) is not recursive }
  • L4 = { ⟨M⟩ ∣ L(M) contains at least 21 members }

  • L1, L3, and L4 only

  • L1 and L3 only

  • L2 and L3 only

  • L2, L3, and L4 only

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Question 9

Which of the following statements is false?
  • Every context-sensitive language is recursive.
  • The set of all languages that are not recursively enumerable is countable.
  • The family of recursively enumerable languages is closed under union.
  • The families of recursively enumerable and recursive languages are closed under reversal.
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Question 10

Which of the following problems is undecidable?
  • To determine if two finite automata are equivalent
  • Membership problem for context free grammar
  • Finiteness problem for finite automata
  • Ambiguity problem for context free grammar
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