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    P equals coP — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
    HistoryEditSee Inverse

    P equals coP

    Modality & PossibilityTruth & Knowledge
    ?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.P is closed under complementation
      ?

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    • 2.coC is defined as the class of problems whose complements are in class C
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The closure of P under complementation establishes coP ⊆ P, but does not establish that every problem in coP has a polynomial-time algorithm for its positive instances.
      ?

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    • 2.Conflating syntactic closure properties with semantic decidability equivocates between computational and logical notions of symmetry, a distinction central to complexity-theoretic separations.
      ?

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    Reason against 2 of 2
    ?
    • 1.The oracle separation results of Baker, Gill, and Solovay (1975) demonstrate that P=coP holds relative to some oracles and fails relative to others, undermining any purely structural proof.
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    • 2.If the equality P=coP were trivially entailed by closure properties alone, it would collapse the polynomial hierarchy in ways inconsistent with widely accepted complexity-theoretic conjectures such as PH being infinite.
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    Topics

    Modality & PossibilityTruth & Knowledge

    Connections

    1 topic

    Philosophy of Language1 linked

    Related

    Conflating syntactic closure properties with semantic decidability equivocates b...If the equality P=coP were trivially entailed by closure properties alone, it wo...P is closed under complementationThe closure of P under complementation establishes coP ⊆ P, but does not establi...
    +2 moreShow less
    The oracle separation results of Baker, Gill, and Solovay (1975) demonstrate tha...coC is defined as the class of problems whose complements are in class C

    Similar

    PSPACE equals NPSPACE100%PSPACE equals NPSPACE (PSPACE = NPSPACE)95%If the F equals the G, and the F is H, then the G is H88%BHP is in P only if P equals NP84%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    In contrast to the non-deterministic Turing machine model \(\mathfrak{N}\), the acceptance and rejection conventions for deterministic models of computation such as \(\mathfrak{T}\) are symmetric. In other words, for a deterministic machine \(T\) to either accept or reject an input \(x\), it is both necessary and sufficient that there exist a single halting computation \(C_0(x),\ldots,C_n(x)\). The output of the machine is then determined by whether \(C_n(x)\) is an accepting or rejecting config
    Extraction notes

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

    Details

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