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

    Modality & PossibilityTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.NP is captured by existential second-order logic (SO∃) over ordered structures
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    • 2.P is captured by FO(LFP) over ordered structures
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    • 3.A separation between P and NP would therefore manifest as a class of structures expressible in SO∃ but not in FO(LFP)
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The Immerman-Vardi theorem assumes linear orders on structures, but natural computational problems lack canonical orderings.
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    • 2.Order-invariant definability in FO(LFP) diverges from ordered FO(LFP) definability, making the logical capture of P order-sensitive.
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    • 3.A biconditional linking P≠NP to a purely logical separation inherits the undecidability of its antecedent without reducing it.
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    Reason against 2 of 2
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    • 1.Descriptive complexity equivalences are representation-theoretic, not metaphysically transparent: they characterize complexity classes only relative to encoding conventions.
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    • 2.Fagin's theorem and Immerman-Vardi establish co-extensionality of classes, not identity of properties, so the biconditional is weaker than it appears.
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    Related

    A biconditional linking P≠NP to a purely logical separation inherits the undecid...A separation between P and NP would therefore manifest as a class of structures ...Descriptive complexity equivalences are representation-theoretic, not metaphysic...Fagin's theorem and Immerman-Vardi establish co-extensionality of classes, not i...
    +4 moreShow less
    NP is captured by existential second-order logic (SO∃) over ordered structuresOrder-invariant definability in FO(LFP) diverges from ordered FO(LFP) definabili...P is captured by FO(LFP) over ordered structuresThe Immerman-Vardi theorem assumes linear orders on structures, but natural comp...

    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...87%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    The logic \(\textsf{SO}(\texttt{LFP})\) and \(\textsf{SO}(\texttt{TC})\) are defined analogously by adding these operators to \(\textsf{SO}\) and allowing them to apply to formulas containing second-order variables. e. models \(\mathcal{A}\) for structures interpreting \(\leq\) as a linear order on \(A\)). Immerman (1999, p. 3 as “increas[ing] our intuition that polynomial time is a class whose fundamental nature goes beyond the machine models with which it is usually defined”. e. \(\textbf{P} \
    Extraction notes

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

    Details

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