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    Home/Original/inverse
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    Inverse View

    It is not the case that NP is captured by the logic SO-exists (second-order existential logic)

    ?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.Fagin's theorem holds only over finite structures, but NP is typically defined over problems with inputs of arbitrary size including infinite domains.
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    • 2.The restriction to finite model theory means SO-exists captures NP only under a non-standard semantics that smuggles in finiteness as a hidden assumption.
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    • 3.A characterization that requires domain finiteness is not a full logical characterization of NP but a characterization of a restricted fragment of it.
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    Reason for 2 of 2
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    • 1.The claim conflates syntactic expressibility in SO-exists with semantic equivalence to NP, but expressibility results depend on the chosen encoding of problem instances as relational structures.
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      Think about whether this reason is strong or weak

    • 2.Immerman and Vardi showed PTIME corresponds to least fixed-point logic only on ordered structures, revealing that logical captures are encoding-relative, not intrinsic to the complexity class.
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    • 3.If the logical capture of NP is sensitive to how inputs are encoded as structures, then SO-exists does not characterize NP simpliciter but only NP-relative-to-a-chosen-representation.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.For every problem in NP there exists a signature and a formula in SO-exists that defines the problem's instances as finite structures
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      Think about whether this reason is strong or weak

    • 2.SO-exists provides a machine-independent characterization of NP that does not reference a specific model of computation
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