Undecidability

Question 21

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 22

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 23

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 24

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 25

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 26

​​​​​​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 27

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