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    NP is captured by the logic SO-exists (second-order exist... — Carmelics
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    Home/Truth & Knowledge
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    NP is captured by the logic SO-exists (second-order existential logic)

    Proof of definition segments
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    • 2.SO-exists provides a machine-independent characterization of NP that does not reference a specific model of computation
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    Reasons Against

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

    Truth & KnowledgeProof of definition segments

    Connections

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    Philosophy of Language1 linked

    Related

    A characterization that requires domain finiteness is not a full logical charact...Fagin's theorem holds only over finite structures, but NP is typically defined o...For every problem in NP there exists a signature and a formula in SO-exists that...If the logical capture of NP is sensitive to how inputs are encoded as structure...
    +4 moreShow less
    Immerman and Vardi showed PTIME corresponds to least fixed-point logic only on o...SO-exists provides a machine-independent characterization of NP that does not re...The claim conflates syntactic expressibility in SO-exists with semantic equivale...The restriction to finite model theory means SO-exists captures NP only under a ...

    Similar

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    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    for all propositional formulas \(\phi\), \(\phi \in \sc{VALID}\) if and only if \(\vdash_{\mathcal{P}_i} \phi\) for \(i \in \{1,2,3\}\). In the context of complexity theory, it is convenient to reformulate the definition of a proof system as a mapping \(\mathcal{P}: \{0,1\}^* \rightarrow \sc{VALID}\) whose domain consist of all binary string and whose range is the class of all valid formulas. Recall, for instance, that a Hilbert derivation is a finite sequences of formulas \(\psi_1,\ldots,\psi_
    Extraction notes

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

    Details

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