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    Home/Original/inverse
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    Inverse View

    It is not the case that The collapse of ≤_m and ≤_T would entail that every Turing-complete set is also many-one complete, yet Post's construction of simple sets demonstrates Turing-complete sets lacking many-one completeness.

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    Reasons For

    1 perspective
    Reason for
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    • 1.The claim conflates 'collapse of relations' with 'every instance follows the rule'—reducibilities can differ without contradicting Post's results.
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    • 2.Simple sets being Turing-complete doesn't entail they lack many-one completeness; Post proved they're not m-complete, which is consistent.
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    • 3.The argument assumes any collapse would force uniform reduction types, but local distinctions can persist even if reducibilities asymptotically align.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Post's simple sets are Turing-complete but lack many-one completeness, establishing that ≤_T and ≤_m are genuinely distinct notions.
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    • 2.If ≤_m and ≤_T collapsed, all sets in the same Turing degree would share many-one degree, contradicting Post's construction.
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    • 3.The existence of counterexamples (simple sets) is logically sufficient to refute claims about universal equivalence of reduction notions.
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