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    If any NP-complete problem has a polynomial time algorith... — Carmelics
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    Home/Modality & Possibility
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    If any NP-complete problem has a polynomial time algorithm, then all problems in NP have polynomial time algorithms

    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.Polynomial-time reducibility ≤_P is transitive
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    • 2.Every problem in NP is polynomial-time reducible to any NP-complete problem
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The claim presupposes a fixed, language-independent notion of 'polynomial time' that ignores relativization results (Baker-Gill-Solovay 1975).
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    • 2.Oracles exist relative to which P=NP and others relative to which P≠NP, showing the claim's proof must transcend relativizable methods.
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    • 3.No currently known proof technique for complexity separations is non-relativizing, so the conditional claim lacks a coherent proof-theoretic foundation.
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    Reason against 2 of 2
    ?
    • 1.Reduction-based arguments assume that computational problems have determinate, mind-independent identity conditions across different encodings.
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    • 2.Wittgensteinian and Kripkean considerations about rule-following suggest that 'same problem' across reductions is interpretation-relative, not intrinsic.
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    • 3.If problem identity is encoding-relative, polynomial-time reducibility ≤_P does not transitively preserve the property of being 'the same computational task'.
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    Modality & PossibilityTruth & Knowledge

    Related

    Every problem in NP is polynomial-time reducible to any NP-complete problemIf problem identity is encoding-relative, polynomial-time reducibility ≤_P does ...No currently known proof technique for complexity separations is non-relativizin...Oracles exist relative to which P=NP and others relative to which P≠NP, showing ...
    +4 moreShow less
    Polynomial-time reducibility ≤_P is transitiveReduction-based arguments assume that computational problems have determinate, m...The claim presupposes a fixed, language-independent notion of 'polynomial time' ...Wittgensteinian and Kripkean considerations about rule-following suggest that 's...

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    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
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    The graph \(G_{\phi}\) for the formula \((p_1 \vee p_2 \vee p_3) \wedge (\neg p_1 \vee p_2 \vee \neg p_3) \wedge (p_1 \vee \neg p_2 \vee \neg p_3)\). A reduction of \(3\text{-}\sc{SAT}\) to \(\sc{INDEPENDENT}\ \sc{SET}\) can now be described as follows: Let \(\phi\) be a \(3\text{-}\sc{CNF}\) formula consisting of \(n\) clauses as depicted above. We construct a graph \(G_{\phi} = \langle V,E \rangle\) consisting of \(n\)-triangles \(T_1,\ldots,T_n\) such that the nodes of \(T_i\) are respect
    Extraction notes

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

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit