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    If closure under complementation were a deep logical trut... — Carmelics
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    Challenges→Complexity classes like P are closed under complementation

    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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    Key Terms

    Invariant(the statement contrasts this with context-shifting)
    Something that stays the same and doesn't change across different situations or contexts.
    Nondeterministic(in discussions of free will and determinism)
    Describing a situation where outcomes are not completely fixed or predetermined—there's genuine randomness or unpredictability involved.
    Symmetry(one of the three shared logical properties between the two comparisons)
    A logical property meaning if A relates to B in some way, then B relates to A the same way—like if swan A is the same color as swan B, then swan B is the same color as swan A.
    closure under complementation(logic and set theory)
    A mathematical property where if you have a set of things, and you take the opposite or 'complement' of any member, that opposite is also in the set.

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    determinism(Discussion of classical mechanics, quantum mechanics, and general relativity)
    A property of physical theories concerning whether the laws governing a system fully fix future (and past) states given present conditions; admits of degrees ('fall only a bit short')
    grounded in(whether distinctness or identity is explained by intrinsic features)
    To be explained by or to have its reason or basis in something else—like how a tree being wet is grounded in (explained by) recent rain.
    logical truth(Bolzano's formal definition, inapplicable to purely logical propositions)
    A proposition is logically true if all of its i^s-variants (substitutions of extra-logical ideas) are true
    oracle constructions(theoretical computer science and logic)
    A technique in computer science where you imagine a magical helper that can instantly answer certain hard questions, used to prove what is and isn't possible.
    relativizations(logic and computer science)
    Different versions or perspectives of a logical system where you adjust the basic assumptions or context.
    symmetry breaks(physics and mathematics)
    A situation where a balanced or uniform property suddenly stops applying—things that worked the same way before now work differently.

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    Complexity classes like P are closed under complementation

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