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    A non-constructive proof that polynomial algorithms exist... — Carmelics
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    Challenges→A polynomial time algorithm for any single NP-complete problem would entail the existence of polynomial time algorithms for all problems in NP.

    A non-constructive proof that polynomial algorithms exist for all NP problems need not yield any actual algorithm, undermining the computational significance of the entailment.

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    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Non-constructive proofs (e.g., via contradiction or probabilistic methods) can establish existence without providing execution mechanisms or complexity bounds.
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    • 2.Computational significance requires not just theoretical existence but practical extractability—knowing an algorithm exists differs from using it.
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    • 3.A proof that P=NP via non-constructive means leaves open whether the algorithm is implementable within any human or physical timescale.
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    Reasons Against

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    Reason against
    ?
    • 1.Any rigorous mathematical proof of P=NP implicitly contains algorithmic content extractable in principle, even if proof technique obscures it initially.
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    • 2.Computational significance isn't undermined by proof method—the entailment that algorithms exist is what matters for theoretical computer science, not implementability.
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    • 3.Historical precedent shows non-constructive existence proofs later yielded explicit algorithms; the limitation is epistemological, not ontological.
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    Connections

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    Proof of definition segments1 linkedModality & Possibility1 linked

    Related

    A polynomial time algorithm for any single NP-complete problem would entail the ...A proof that P=NP via non-constructive means leaves open whether the algorithm i...Any rigorous mathematical proof of P=NP implicitly contains algorithmic content ...Computational significance isn't undermined by proof method—the entailment that ...
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    Computational significance requires not just theoretical existence but practical...Historical precedent shows non-constructive existence proofs later yielded expli...Non-constructive proofs (e.g., via contradiction or probabilistic methods) can e...

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