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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Perspectives
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    Home/Original/inverse
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    Inverse View

    It is not the case that A polynomial time algorithm for any single NP-complete problem would entail the existence of polynomial time algorithms for all problems in NP.

    ?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 existential quantifier in 'would entail the existence of' conflates constructive and non-constructive existence, a distinction central to intuitionist mathematics.
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    • 2.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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    • 3.Brouwer and Dummett's anti-realism requires that mathematical existence claims be backed by constructive witness, not merely classical logical consequence.
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    Reason for 2 of 2
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    • 1.Polynomial-time reducibility preserves tractability only if the composition of reduction and solving algorithm remains within polynomial bounds uniformly across all inputs.
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    • 2.Buss and Cook's work on bounded arithmetic reveals that the metatheoretic reasoning validating transitivity of reductions may itself require proof-theoretic resources exceeding those formalizable in weak systems.
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    • 3.If the background logic used to establish the entailment is proof-theoretically stronger than the system modeling computation, the modal force of 'would entail' is theory-relative, not absolute.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.The polynomial-time reducibility relation is transitive.
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    • 2.NP-complete problems are defined such that every problem in NP is polynomial-time reducible to them.
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    Strongest counterpoint
    Explore the most compelling reason on the other side.