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    Home/Original/inverse
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    Inverse View

    It is not the case that Complexity classes like P are closed under complementation

    ?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.The symmetry between acceptance and rejection in P assumes a fixed encoding, but complementation requires that 'non-membership' is itself a decidable property.
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    • 2.Complexity theorists like Ladner and Lynch distinguish between syntactic and semantic closure properties, and P's closure under complement is a syntactic artifact of determinism, not a universal logical truth about decidability.
      ?

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    • 3.The claim conflates computational convention with logical necessity: flipping accept/reject states produces a correct complement machine only if the original machine halts on all inputs, which is a substantive assumption about total functions.
      ?

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    Reason for 2 of 2
    ?
    • 1.Oracle-relative results (Baker, Gill, Solovay 1975) demonstrate that structural properties like closure under complement can fail or hold contingently depending on the computational environment.
      ?

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    • 2.If closure under complementation were a deep logical truth grounded in determinism's symmetry, it would be invariant across all relativizations, but oracle constructions show this closure is not invariant in analogous nondeterministic settings precisely because symmetry breaks.
      ?

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    Reasons Against

    1 perspective
    Reason against
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    • 1.For a deterministic machine, acceptance and rejection conventions are symmetric
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    • 2.For a deterministic machine T to accept or reject an input x, it is both necessary and sufficient that there exist a single halting computation
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    • 3.The output is determined by whether the final configuration is accepting or rejecting
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