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    P ≠ NP if and only if there exists a class of ordered str... — Carmelics
    Home/Philosophy of Language
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    P ≠ NP if and only if there exists a class of ordered structures definable in existential second-order logic that is not definable by any formula of FO(LFP).

    Modality & PossibilityPhilosophy of Language
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.NP is captured over ordered structures by SO∃.
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    • 2.P is captured over ordered structures by FO(LFP).
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    • 3.If P = NP then every property expressible in SO∃ would be expressible in FO(LFP), and conversely.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Descriptive complexity results like Fagin's theorem presuppose fixed finite ordered structures, but the P vs NP problem ranges over all computational models regardless of representational assumptions.
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    • 2.The logical characterization via SO∃ and FO(LFP) is order-sensitive: without a built-in linear order, FO(LFP) fails to capture P, undermining the biconditional's claim to equivalence with the full complexity-theoretic question.
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    • 3.Immerman and Vardi's theorem requires that order be present in the structure, so the logical reformulation captures only a restricted variant of P vs NP, not the general problem as standardly posed.
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    Reason against 2 of 2
    ?
    • 1.The biconditional conflates provability within a descriptive complexity framework with the truth of an independent computational conjecture, committing a use-mention error about mathematical equivalence.
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    • 2.Razborov and Rudich's natural proofs barrier suggests that any combinatorial or logical property definably separating P from NP would itself be constructible by efficient algorithms, creating a self-undermining circularity in logical separation arguments.
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    Topics

    Philosophy of LanguageModality & Possibility

    Connections

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    Proof of definition segments2 linked

    Related

    Descriptive complexity results like Fagin's theorem presuppose fixed finite orde...If P = NP then every property expressible in SO∃ would be expressible in FO(LFP)...Immerman and Vardi's theorem requires that order be present in the structure, so...NP is captured over ordered structures by SO∃.
    +4 moreShow less
    P is captured over ordered structures by FO(LFP).Razborov and Rudich's natural proofs barrier suggests that any combinatorial or ...The biconditional conflates provability within a descriptive complexity framewor...The logical characterization via SO∃ and FO(LFP) is order-sensitive: without a b...

    Similar

    P ≠ NP if and only if there exists a class of ordered structures defin...100%P ≠ NP if and only if there exists a class of ordered structures defin...100%P ≠ NP if and only if there exists a class of ordered structures defin...100%NP is captured by existential second-order logic (SO∃) over ordered st...86%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    The availability of such characterizations is often taken to provide additional evidence for the mathematical robustness of classes like \(\textbf{NP}\). 2 generalizes to provide a characterization of the classes which comprise the Polynomial Hierarchy. For instance, the logics \(\Sigma^1_i\) and \(\Pi^1_i\) uniformly capture the complexity classes \(\Sigma^P_i\) and \(\Pi^P_i\) (where \(\mathsf{SO}\exists = \Sigma^1_1\) ). e. full second-order logic) captures \(\textbf{PH}\) itself. On the othe
    Extraction notes

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

    Details

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