Undecidability (original) (raw)

Which of the following problems are decidable?

Which of the following is/are undecidable?

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

​​​​​​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?

• Neither S1 nor S2 is true

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

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

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*}

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 }

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.

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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