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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that 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).

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 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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      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reason for 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 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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