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    Resolution is not polynomially bounded. — Carmelics
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    Home/Modality & Possibility
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    Resolution is not polynomially bounded.

    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.Any resolution proof of PHP_n must have size at least exponential in n (Haken's result).
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    • 2.A proof system is polynomially bounded only if all tautologies have proofs of size polynomial in the size of the tautology.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Haken's 1985 proof relies on a specific syntactic formalization of PHP_n; alternative encodings may yield structurally distinct tautologies with shorter proofs.
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    • 2.The polynomial boundedness criterion is encoding-relative, so no single tautology family establishes system-wide unprovability across all representations.
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    Reason against 2 of 2
    ?
    • 1.Cook and Reckhow's framework presupposes a classical notion of proof size; non-classical proof-theoretic accounts (e.g., deep inference) dissolve the exponential lower bound.
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    • 2.Resolution's complexity profile is a property of the formalism, not of the underlying logical truths, so the claim conflates proof-system limitations with logical necessity.
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    Related

    A proof system is polynomially bounded only if all tautologies have proofs of si...Any resolution proof of PHP_n must have size at least exponential in n (Haken's ...Cook and Reckhow's framework presupposes a classical notion of proof size; non-c...Haken's 1985 proof relies on a specific syntactic formalization of PHP_n; altern...
    +2 moreShow less
    Resolution's complexity profile is a property of the formalism, not of the under...The polynomial boundedness criterion is encoding-relative, so no single tautolog...

    Similar

    Resolution is not polynomially bounded as a proof system90%The failure of polynomial boundedness has not been proven for most fam...82%No proof system has yet been shown to be polynomially bounded81%Squaring a polynomial still yields a polynomial, so NPSPACE does not e...78%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
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    Haken showed that any resolution proof of \(\text{PHP}_n\) must have size at least exponential in \(n\). From this it follows that resolution is not polynomially bounded. However, it was later shown by Buss (1987) that the system \(\mathcal{P}_1\) (and hence also systems like \(\mathcal{P}_2\), \(\mathcal{P}_3\) which can be shown to efficiently simulate \(\mathcal{P}_1\)) do admit proofs of \(\text{PHP}_n\) which are of size polynomial in \(n\). One subsequent direction of research in proof co
    Extraction notes

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    Details

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