## NPTEL Theory of Computation Week 8 Assignment Answers 2024

1. The class NP is defined as the class of problems that

- Can be solved in polynomial time by a nondeterministic Turing Machine
- Cannot be solved in polynomial time by a deterministic Turing Machine
- Can be solved in polynomial time by a deterministic Turing Machine
- Cannot be solved in polynomial time by a nondeterministic Turing Machine

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2. The class coNP is defined as

- {L∣L¯¯∈NP}
- {L∣L∉NP}
- {L¯¯∣L⊆L′∈NP}
- {L¯¯∣L∉NP}

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3. L∈NP if there exists constants c>0 and k≥0, and a polynomial time Turing Machine V, such that

- ∃x∈Σ
^{∗},x∈L⟺∀y∈Σ^{∗}such that |y|≤c⋅|x|^{k}and V(x,y)=1 - ∀x∈Σ
^{∗},x∈L⟺∃y∈Σ^{∗ }such that |y|≤c⋅|x|^{k}and V(x,y)=1 - ∀x∈Σ
^{∗},x∈L⟺∀y∈Σ^{∗}such that |y|≤c⋅|x|^{k}and V(x,y)=1 - ∃x∈Σ
^{∗},x∈L⟺∃y∈Σ^{∗}such that |y|≤c⋅|x|^{k}and V(x,y)=1

Answer :-

4. L∈coNP if there exists constants c>0 and k≥0, and a polynomial time Turing Machine V, such that

- ∃x∈Σ
^{∗},x∈L⟺∀y∈Σ^{∗ }such that |y|≤c⋅|x|^{k}and V(x,y)=1 - ∀x∈Σ
^{∗},x∈L⟺∃y∈Σ^{∗}such that |y|≤c⋅|x|^{k }and V(x,y)=1 - ∀x∈Σ
^{∗},x∈L⟺∀y∈Σ^{∗}such that |y|≤c⋅|x|^{k}and V(x,y)=1 - ∃x∈Σ
^{∗},x∈L⟺∃y∈Σ^{∗}such that |y|≤c⋅|x|^{k}and V(x,y)=1

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5. Which of the following problems is not known to be in the complexity class P?

- Deciding whether a natural number is even
- Deciding whether a boolean formula is satisfiable
- Deciding whether a graph is connected
- Deciding whether a binary string has even number of 1’s

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6. Which of the following problems is known to be in the complexity class coNP?

- Deciding whether a natural number is prime
- Deciding whether a boolean formula is satisfiable
- Deciding whether a graph has a Hamiltonian path
- Deciding whether two graphs are isomorphic

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7. Let L be a language in the complexity class P. Which of the following is not known?

- L∈NP
- L∈coNP
- L∈NP∩coNP
- L∈NP-complete

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8. If L∈NTIME(f(n)), then

- L∈DTIME(f(n))
- L∈DTIME(logf(n))
- L∈DTIME(f(n)
^{2}) - L∈DTIME(2
^{O(f(n))})

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9. The class NP is not known to be closed under which of the following operations?

- Union
- Intersection
- Concatenation
- Complementation

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10. If coNP ⊆ NP then which of the following is true?

- NP=coNP
- P=NP
- P≠NP
- NP≠coNP

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11. The class NP-hard is defined as the set of

- Problems which are reducible in polynomial time to some NP complete problem
- Problems which are reducible in polynomial space to some NP complete problem
- Problems to which all problems in NP are reducible in polynomial time
- Problems to which all problems in NP are reducible in polynomial space

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12. If a decision problem A is polynomial time reducible to another decision problem B, then which of the following statements is necessarily true?

- If A is NP-hard then B is also NP-hard
- If B is NP-hard then A is also NP-hard
- If A is in NP then B is also in NP
- If A is NP-complete then B is also NP-complete

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