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    Carmelics

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    Made withinDC&Austin
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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 which is not definable by a 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.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 for 2 of 2
    ?
    • 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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    Reasons Against

    1 perspective
    Reason against
    ?
    • 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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