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    FO equivalence to AC^0 depends critically on the presence... — Carmelics
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    Challenges→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.

    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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    1 reason for
    1 reason against

    Reasons For

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    Reason for
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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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    Reasons Against

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    Reason against
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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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    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    First-order logic FO captures only the very weak complexity class AC^0 and canno...Linear order enables FO to encode binary counters and threshold predicates, matc...Many AC^0-complete problems (like parity on unordered sets) have straightforward...The claim assumes FO inadequacy on unordered structures without distinguishing b...
    +3 moreShow less
    The distinction conflates expressiveness with definability; AC^0 itself is invar...This aligns with proven results: FO+LO captures exactly AC^0, while FO alone on ...Unordered structures prevent FO from expressing majority functions and symmetric...

    Details

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