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    FACTORIZATION is in coNP — Carmelics
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    Supports→FACTORIZATION is in NP ∩ coNP simultaneously

    FACTORIZATION is in coNP

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

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    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.Membership of ⟨n,m⟩ in the complement of FACTORIZATION can be demonstrated by exhibiting a prime factorization of n in which no factor is less than m
      ?

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    • 2.Prime factorizations are unique, so a single factorization certificate is sufficient
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    • 3.The primality of individual factors in the factorization can be verified in polynomial time using the AKS algorithm
      ?

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    Reason for 2 of 2
    ?
    • 1.Membership of ⟨n,m⟩ in the complement of FACTORIZATION can be certified by exhibiting a prime factorization of n in which no factor is less than m
      ?

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    • 2.The 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 AKS verification of primality operates on the bit-length of each factor, but the product of these lengths can grow super-polynomially relative to the input encoding of n.
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    • 2.A coNP certificate must be verifiable in polynomial time in the length of the original input ⟨n,m⟩, not in the length of auxiliary components of the certificate.
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    • 3.Therefore the supporting argument conflates polynomial-time verification of certificate components with polynomial-time verification of the certificate as a whole.
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    Reason against 2 of 2
    ?
    • 1.Uniqueness of prime factorization (the Fundamental Theorem of Arithmetic) guarantees a single valid certificate exists, but does not guarantee that certificate is polynomially sized relative to the binary encoding of n.
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    • 2.A coNP witness must be both unique-enough to be convincing and polynomially bounded in size; the prime factorization of n can require exponentially many bits relative to log n when n has many small factors.
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    • 3.Argument 2's appeal to uniqueness addresses soundness of the certificate but is silent on the length constraint that defines membership in coNP.
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    Topics

    Truth & KnowledgeProof of definition segments

    Connections

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    Related

    A coNP certificate must be verifiable in polynomial time in the length of the or...A coNP witness must be both unique-enough to be convincing and polynomially boun...Argument 2's appeal to uniqueness addresses soundness of the certificate but is ...FACTORIZATION is in NP
    +9 moreShow less
    FACTORIZATION is in NP ∩ coNP simultaneouslyMembership of ⟨n,m⟩ in the complement of FACTORIZATION can be certified by exhib...Membership of ⟨n,m⟩ in the complement of FACTORIZATION can be demonstrated by ex...Prime factorizations are unique, so a single factorization certificate is suffic...The AKS verification of primality operates on the bit-length of each factor, but...The primality of individual factors can be verified in polynomial time by the AK...The primality of individual factors in the factorization can be verified in poly...

    Similar

    SAT is in NP100%BHP is in NP100%PRIMES is in P100%FACTORIZATION is in NP100%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    For on the one hand, a number \(1 \lt d \leq m\) which divides \(n\) serves as a certificate for the membership of \(\langle n,m \rangle\) in \(\sc{FACTORIZATION}\). And on the other hand, in order to demonstrate the membership of \(\langle n,m \rangle\) in \(\overline{\sc{FACTORIZATION}}\), it suffices to exhibit a prime factorization of \(n\) in which no factor is less than \(m\). , falsifying valuations in the case of \(\sc{VALIDITY}\)) and because the primality of the individual factors can
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Therefore the supporting argument conflates polynomial-time verification of cert...
    Uniqueness of prime factorization (the Fundamental Theorem of Arithmetic) guaran...
    Type
    claim
    Perspectives
    4 (2 for, 2 against)
    Edits
    1 edit