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

    It is not the case that 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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    Reasons For

    1 perspective
    Reason for
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    • 1.The distinction conflates expressiveness with definability; AC^0 itself is invariant to presentation order, not order-dependent.
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    • 2.Many AC^0-complete problems (like parity on unordered sets) have straightforward polynomial-time solutions without lexical ordering.
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    • 3.The claim assumes FO inadequacy on unordered structures without distinguishing between natural query properties and their representations.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Linear order enables FO to encode binary counters and threshold predicates, matching AC^0's parallel counting capabilities.
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    • 2.Unordered structures prevent FO from expressing majority functions and symmetric predicates requiring global comparisons.
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    • 3.This aligns with proven results: FO+LO captures exactly AC^0, while FO alone on unordered domains has strictly lower expressiveness.
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