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    A polynomial time algorithm for any single NP-complete pr... — Carmelics
    Home/Modality & Possibility
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    A polynomial time algorithm for any single NP-complete problem would entail the existence of polynomial time algorithms for all problems in NP.

    Modality & PossibilityProof of definition segments
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Topics

    Modality & PossibilityProof of definition segments

    Connections

    1 linked claim · 1 topic

    Truth & Knowledge3 linked
    A polynomial time algorithm for any one NP-complete problem would entail polynom...

    Related

    A non-constructive proof that polynomial algorithms exist for all NP problems ne...A polynomial time algorithm for any one NP-complete problem would entail polynom...Brouwer and Dummett's anti-realism requires that mathematical existence claims b...Buss and Cook's work on bounded arithmetic reveals that the metatheoretic reason...
    +5 moreShow less
    If the background logic used to establish the entailment is proof-theoretically ...NP-complete problems are defined such that every problem in NP is polynomial-tim...Polynomial-time reducibility preserves tractability only if the composition of r...

    Similar

    The existence of a polynomial time algorithm for any single NP-complet...99%A polynomial time algorithm for any one NP-complete problem would enta...98%If any single NP-complete problem has a polynomial time algorithm, the...96%If any NP-complete problem has a polynomial time algorithm, then all p...94%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    It also follows from the transitivity of \(\leq_P\) that the existence of a polynomial time algorithm for even one \(\textbf{NP}\)-complete problem would entail the existence of polynomial time algorithms for all problems in \(\textbf{NP}\). The existence of such an algorithm would thus run strongly counter to expectation in virtue of the extensive effort which has been devoted to finding efficient solutions for particular \(\textbf{NP}\)-complete problems such as \(\sc{INTEGER}\ \sc{PROGRAMMING
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    The existential quantifier in 'would entail the existence of' conflates construc...
    The polynomial-time reducibility relation is transitive.
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit