Co-NP

Definition

co-NP is the complexity class that contains decision problems for which a 'no' instance can be verified by a deterministic polynomial-time algorithm given the right certificate or witness. In simpler terms, if a problem is in co-NP, it means that if the answer is 'no', there exists a proof that can be checked quickly to confirm that the answer is indeed 'no'. This concept is tightly connected to NP-completeness because co-NP consists of problems that are complements of those in NP, and understanding these classes helps in classifying the hardness of computational problems.

5 Must Know Facts For Your Next Test

1. The relationship between NP and co-NP is an important open question in computer science; it is not known whether they are equal or if one is strictly larger than the other.
2. If a problem is in co-NP, its complement is in NP, meaning if you can efficiently verify 'yes' instances for one, you can efficiently verify 'no' instances for the other.
3. Many well-known problems, such as the complement of SAT (Boolean satisfiability), belong to co-NP.
4. The class co-NP is often viewed as equally important as NP, especially when discussing the limits of efficient computation and proof verification.
5. Some researchers believe that proving NP != co-NP would have significant implications for cryptography and computational theory.

Review Questions

• How does co-NP relate to NP and why is this relationship significant?
  • co-NP relates to NP as it contains the complements of the decision problems found in NP. This relationship is significant because it helps us understand the boundaries of computational complexity and the nature of verifiable problems. If we can prove that NP is not equal to co-NP, it would imply fundamental limitations on what can be efficiently verified or solved, impacting various fields such as cryptography.
• Discuss an example of a problem that belongs to co-NP and explain why it fits this classification.
  • An example of a problem in co-NP is the complement of the SAT problem, known as UNSAT. The SAT problem asks if there exists an assignment to variables that makes a Boolean formula true, while UNSAT asks if no such assignment exists. For UNSAT, if we claim that a formula is unsatisfiable, we can provide a proof showing that all possible assignments lead to false outcomes. This verification process can be done efficiently, fitting the definition of co-NP.
• Evaluate the implications of discovering that NP equals co-NP on our understanding of computational theory.
  • If NP were proven to equal co-NP, it would suggest that every problem for which we can verify 'yes' answers efficiently also allows for efficient verification of 'no' answers. This would radically change our understanding of complexity theory, indicating that many hard problems could potentially be solved quickly. Additionally, it would have profound impacts on cryptography, as many systems rely on the difficulty of certain problems believed to be in NP but not in P.

Related terms

NP: NP is the complexity class that includes decision problems for which a 'yes' instance can be verified in polynomial time given a certificate or witness.

NP-complete: NP-complete problems are a subset of NP problems that are as hard as any problem in NP; if any NP-complete problem can be solved in polynomial time, then every problem in NP can also be solved in polynomial time.

P: P is the class of decision problems that can be solved in polynomial time by a deterministic Turing machine.