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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
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    Inverse View

    It is not the case that First-order logic FO captures only the very weak complexity class AC^0 and cannot express properties in stronger classes such as P without extensions.

    ?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.FO equivalence to AC^0 depends critically on the presence of a linear order predicate; over unordered structures, FO captures only a strict subset of AC^0.
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    • 2.The claim conflates expressive power over ordered structures with expressive power simpliciter, smuggling in a non-trivial assumption about structural representation.
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    • 3.Descriptive complexity results are sensitive to the choice of encoding, so the boundary between FO and stronger classes is not a fact about logic alone but about logic-plus-structure.
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    Reason for 2 of 2
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    • 1.Immerman and Vardi's theorem shows FO(LFP) captures P over ordered structures, demonstrating that FO with least fixed-point extension does reach P without abandoning first-order syntax.
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    • 2.The claim that FO 'cannot express properties in P without extensions' trivially conflates the base logic with its natural and well-motivated closure operations, which logicians since Kleene have treated as intrinsic to logical expressibility.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.AC^0 consists of languages decidable by polynomial-size circuits of constant depth.
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    • 2.It can be shown that FO is equivalent in expressive power to AC^0 over ordered structures.
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    • 3.Properties such as PARITY, which require counting, are not definable in FO(LFP) without an ordering predicate, illustrating that FO lacks sufficient expressive capacity.
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