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

    Modality & PossibilityTruth & Knowledge
    ?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 existence of a polynomial-time algorithm is a modal claim about computability, not a demonstrated constructive fact, and modal existence does not transfer across reductions without preserving algorithmic structure.
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    • 2.Polynomial-time reducibility preserves decision-problem solvability in principle, but the reduction itself may introduce constant or hidden complexity factors that render the composed algorithm impractical even if formally polynomial.
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    • 3.Hartmanis and Stearns's foundational work shows complexity classes are defined relative to machine models, so 'polynomial time' is not a robust absolute property immune to variation in computational substrate.
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    Reason against 2 of 2
    ?
    • 1.The inference from 'a polynomial algorithm exists for one NP-complete problem' to 'polynomial algorithms exist for all NP problems' treats existence as a transferable property, but Kripkean possible-worlds semantics demands we specify in which worlds and under which interpretations such algorithms exist.
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    • 2.The supporting argument's transitivity premise holds for decision problems under standard Turing reductions, but Ladner's theorem demonstrates that if P≠NP there exist problems in NP neither in P nor NP-complete, complicating the universality of the entailment.
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    Topics

    Modality & PossibilityTruth & Knowledge

    Connections

    1 linked claim · 2 topics

    Proof of definition segments2 linkedSkepticism1 linked
    The existence of a polynomial time algorithm for any NP-complete problem would i...

    Related

    Hartmanis and Stearns's foundational work shows complexity classes are defined r...NP-complete problems are defined such that every problem in NP is polynomial-tim...Polynomial-time reducibility preserves decision-problem solvability in principle...The existence of a polynomial time algorithm for any NP-complete problem would i...
    +4 moreShow less
    The existence of a polynomial-time algorithm is a modal claim about computabilit...

    Similar

    A polynomial time algorithm for any single NP-complete problem would e...99%A polynomial time algorithm for any one NP-complete problem would enta...96%The existence of a polynomial time algorithm for any NP-complete probl...96%If any single NP-complete problem has a polynomial time algorithm, the...95%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
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    [21] 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{PROGRA
    Extraction notes

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

    Details

    The inference from 'a polynomial algorithm exists for one NP-complete problem' t...
    The polynomial-time reducibility relation is transitive.
    The supporting argument's transitivity premise holds for decision problems under...
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit