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    Claiming NP ∩ coNP membership conflates the existence of ... — Carmelics
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    Challenges→FACTORIZATION is in NP ∩ coNP simultaneously

    Claiming NP ∩ coNP membership conflates the existence of efficient verification with the existence of succinct, universally accepted certificate schemes, which for FACTORIZATION remains an open constructive question.

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    Key Terms

    Constructive question(mathematics and logic)
    A problem asking whether we can actually *build* or *create* something (like a proof or method), not just prove it theoretically exists.
    Efficient verification(computational complexity)
    The ability to check whether an answer is correct quickly, without having to solve the whole problem from scratch.
    FACTORIZATION(Used as an example of a problem in NP ∩ coNP not currently known to be in P.)
    The computational problem: given ⟨n,m⟩, does n have a factor d with 1 < d ≤ m?
    NP (nondeterministic polynomial time)(Major complexity class based on nondeterministic model)
    The union over all natural numbers k of NTIME(n^k); the class of languages decidable by a nondeterministic Turing machine in polynomial time.
    NP ∩ coNP

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    The class of problems that possess polynomial-sized certificates for demonstrating both membership and non-membership. Includes all problems in P, and also some problems not currently known to be feasibly decidable.
    Succinct certificate(computer science and mathematical proofs)
    A short, compact piece of proof or evidence that convinces someone your answer is right—ideally much shorter than the answer itself.
    Universally accepted(logic and mathematics)
    Agreed upon by everyone as legitimate and trustworthy, with no disagreement about its validity.
    coNP (Complement of NP)(as used in computational complexity theory)
    A category of computer problems where disproving an answer is fast—basically the 'opposite' version of NP problems.

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    FACTORIZATION is in NP ∩ coNP simultaneously

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