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    First-order logic FO captures only the very weak complexi... — Carmelics
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    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.

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Related

    AC^0 consists of languages decidable by polynomial-size circuits of constant dep...Descriptive complexity results are sensitive to the choice of encoding, so the b...FO equivalence to AC^0 depends critically on the presence of a linear order pred...Immerman and Vardi's theorem shows FO(LFP) captures P over ordered structures, d...
    +4 moreShow less
    It can be shown that FO is equivalent in expressive power to AC^0 over ordered s...Properties such as PARITY, which require counting, are not definable in FO(LFP) ...The claim conflates expressive power over ordered structures with expressive pow...The claim that FO 'cannot express properties in P without extensions' trivially ...

    Similar

    First-order logic (FO) cannot capture complexity classes above AC⁰ wit...93%First-order logic (FO) alone is insufficient to characterize complexit...92%First-order logic captures only AC⁰, the class of languages decidable ...84%First-order logic is the strongest logic where Löwenheim-Skolem holds ...82%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
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    The availability of such characterizations is often taken to provide additional evidence for the mathematical robustness of classes like \(\textbf{NP}\). 2 generalizes to provide a characterization of the classes which comprise the Polynomial Hierarchy. For instance, the logics \(\Sigma^1_i\) and \(\Pi^1_i\) uniformly capture the complexity classes \(\Sigma^P_i\) and \(\Pi^P_i\) (where \(\mathsf{SO}\exists = \Sigma^1_1\) ). e. full second-order logic) captures \(\textbf{PH}\) itself. On the othe
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit