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    FACTORIZATION is in NP ∩ coNP. — Carmelics
    Home/Modality & Possibility
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    Connected to 2 discussions

    Supports→If FACTORIZATION is not in P and NP ≠ coNP, then there exist natural mathematical problems that are not feasibly decidable but also not NP-complete.
    Supports→If FACTORIZATION is not in P, then a positive answer to Open Question 2 (NP ∩ coNP ≠ NP) would entail that there are natural mathematical problems which are not feasibly decidable but also not NP-complete.

    FACTORIZATION is in NP ∩ coNP.

    Modality & PossibilityTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.A divisor d with 1 < d ≤ m that divides n serves as a polynomial certificate for membership of ⟨n,m⟩ in FACTORIZATION.
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    • 2.A prime factorization of n in which no prime factor is less than m serves as a polynomial certificate for membership of ⟨n,m⟩ in the complement of FACTORIZATION.
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    • 3.Primality of individual factors can be verified in polynomial time by the AKS algorithm.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The certificate for coNP membership presupposes unique prime factorization, which is a non-trivial mathematical theorem, not a logical given.
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    • 2.If the Fundamental Theorem of Arithmetic requires proof, then the coNP certificate argument embeds an unacknowledged mathematical assumption that could fail in alternative number-theoretic frameworks.
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    • 3.Wittgenstein's rule-following considerations suggest that 'divides evenly' as a verification procedure smuggles in an infinite normative commitment not capturable by finite polynomial-time computation alone.
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    Reason against 2 of 2
    ?
    • 1.The AKS primality certificate verifies primality of individual factors, but the coNP certificate requires verifying the *completeness* of the factorization, a globally quantified claim not reducible to local factor checks.
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    • 2.Verifying that no prime factor smaller than m exists requires either exhaustive search or trust in the completeness of the presented factorization, reintroducing the original computational difficulty in disguised form.
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    Modality & PossibilityTruth & Knowledge

    Connections

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    Related

    A divisor d with 1 < d ≤ m that divides n serves as a polynomial certificate for...A positive answer to Open Question 2 implies that problems in NP ∩ coNP are not ...A prime factorization of n in which no prime factor is less than m serves as a p...If FACTORIZATION is not in P and NP ≠ coNP, then there exist natural mathematica...
    +10 moreShow less
    If FACTORIZATION is not in P, it is not feasibly decidable.If FACTORIZATION is not in P, then a positive answer to Open Question 2 (NP ∩ co...If NP ≠ coNP, then problems in NP ∩ coNP are not NP-complete.If the Fundamental Theorem of Arithmetic requires proof, then the coNP certifica...Our current inability to find an efficient factorization algorithm is indicative...Primality of individual factors can be verified in polynomial time by the AKS al...The AKS primality certificate verifies primality of individual factors, but the ...

    Similar

    FACTORIZATION is in NP ∩ coNP98%Problems in NP ∩ coNP are in coNP by definition.93%FACTORIZATION is in NP ∩ coNP simultaneously90%Problems in NP ∩ coNP are unlikely to be NP-complete.81%

    Source

    AI-extracted
    SEP: computational-complexity
    View source passageHide passage
    [21] It also follows from the transitivity of \(\leq_P\) that the existence of a polynomial time algorithm for even one \(\textbf{NP}\)-complete problem would entail the existence of polynomial time algorithms for all problems in \(\textbf{NP}\). The existence of such an algorithm would thus run strongly counter to expectation in virtue of the extensive effort which has been devoted to finding efficient solutions for particular \(\textbf{NP}\)-complete problems such as \(\sc{INTEGER}\ \sc{PROGRA

    Details

    Type
    premise
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit
    The certificate for coNP membership presupposes unique prime factorization, whic...
    Verifying that no prime factor smaller than m exists requires either exhaustive ...
    Wittgenstein's rule-following considerations suggest that 'divides evenly' as a ...