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

    Truth & Knowledge
    Overall Strength:30%
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.FACTORIZATION is in NP
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    • 2.FACTORIZATION is in coNP
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Membership proofs for NP require a verifiable witness, but FACTORIZATION's complement requires certifying the *absence* of non-trivial factors, which standard NP witnesses cannot directly encode.
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    • 2.Pratt's primality certificates establish coNP membership for PRIMES only because of deep number-theoretic structure; analogous certificates for composite numbers with *all* factor pairs are not obviously constructible in polynomial-length form.
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    • 3.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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    Reason against 2 of 2
    ?
    • 1.The intersection NP ∩ coNP is a syntactic complexity class defined relative to a fixed computational model; FACTORIZATION as a decision problem (does n have a factor ≤ k?) differs structurally from the search problem, and membership claims must specify which formulation is at issue.
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    • 2.Hartmanis and Hopcroft's framework for structural complexity treats class membership as a property of formal languages, not of mathematical problems per se, so asserting FACTORIZATION 'is in' NP ∩ coNP without specifying the encoding and decision variant commits a use-mention conflation.
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    Connections

    1 topic

    Proof of definition segments2 linked

    Related

    Claiming NP ∩ coNP membership conflates the existence of efficient verification ...FACTORIZATION is in NPFACTORIZATION is in coNPHartmanis and Hopcroft's framework for structural complexity treats class member...
    +3 moreShow less
    Membership proofs for NP require a verifiable witness, but FACTORIZATION's compl...Pratt's primality certificates establish coNP membership for PRIMES only because...The intersection NP ∩ coNP is a syntactic complexity class defined relative to a...

    Similar

    FACTORIZATION is in NP ∩ coNP90%FACTORIZATION is in NP ∩ coNP.90%Problems in NP ∩ coNP are in coNP by definition.86%Problems in NP ∩ coNP are unlikely to be NP-complete.75%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    It is easy to see that \(\textbf{NP} \cap \textbf{coNP}\) includes all problems in \(\textbf{P}\). But this class also contains some problems which are not currently known to be feasibly decidable. An important example is \(\sc{FACTORIZATION}\) (as defined in Section 1.1). 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 member
    Extraction notes

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

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit