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

    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

    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

    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

    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

    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

    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*}

    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

    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.

    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

    There are 27 questions to complete.

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