Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Diagonalization cannot be used to separate P and NP. — Carmelics
    Home/Modality & Possibility
    HistoryEditSee Inverse

    Diagonalization cannot be used to separate P and 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.Baker, Gill, and Solovay (1975) established oracles A and B such that P^A = NP^A and P^B ≠ NP^B
      ?

      Think about whether this reason is strong or weak

    • 2.A proof of P ≠ NP based on diagonalization would relativize to both oracle A and oracle B
      ?

      Think about whether this reason is strong or weak

    • 3.No single diagonalization argument can simultaneously yield P^A = NP^A and P^B ≠ NP^B
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Relativization barriers show only that *relativizing* diagonalization fails, not that all diagonalization-based techniques are exhausted.
      ?

      Think about whether this reason is strong or weak

    • 2.Non-relativizing proof techniques (e.g., arithmetization underlying IP=PSPACE) demonstrate that diagonal-style separation methods can escape oracle limitations.
      ?

      Think about whether this reason is strong or weak

    • 3.Therefore the BGS result eliminates a proof *strategy*, not the broader family of diagonalization-inspired arguments that exploit circuit lower bounds or non-uniform complexity.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.The BGS oracle argument presupposes that a valid P≠NP proof must be syntactically relativizable, but this is a methodological assumption, not a logical necessity.
      ?

      Think about whether this reason is strong or weak

    • 2.Razborov and Rudich's 'natural proofs' barrier is a distinct obstruction, confirming that relativization alone does not fully characterize the space of impossible proof strategies.
      ?

      Think about whether this reason is strong or weak

    • 3.Claiming diagonalization *cannot* separate P and NP conflates one well-defined barrier (relativization) with an exhaustive prohibition, thereby overstating what BGS actually proved.
      ?

      Think about whether this reason is strong or weak

    Sign in or register to share your perspective on this statement.

    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.

    Topics

    Modality & PossibilityTruth & Knowledge

    Related

    A proof of P ≠ NP based on diagonalization would relativize to both oracle A and...Baker, Gill, and Solovay (1975) established oracles A and B such that P^A = NP^A...Claiming diagonalization *cannot* separate P and NP conflates one well-defined b...No single diagonalization argument can simultaneously yield P^A = NP^A and P^B ≠...
    +5 moreShow less
    Non-relativizing proof techniques (e.g., arithmetization underlying IP=PSPACE) d...Razborov and Rudich's 'natural proofs' barrier is a distinct obstruction, confir...Relativization barriers show only that *relativizing* diagonalization fails, not...The BGS oracle argument presupposes that a valid P≠NP proof must be syntacticall...Therefore the BGS result eliminates a proof *strategy*, not the broader family o...

    Similar

    Diagonalization cannot be used to separate P from NP.99%There are few known candidates for separating BPP from P85%Measurements of P and Q are incompatible.83%Under AFA, ℘* and ℘* are distinct82%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    But in a letter to von Neumann, Gödel (1956) observed that if we were able to efficiently decide \(n\text{-}\sc{PROVABILITY}_{\mathsf{T}}\), then this would already have enormous significance for mathematical practice. For note that it seems plausible to assume that no human mathematician will ever be able to comprehend a proof containing 100 million symbols (\(\approx 25000\) pages). If we were able to efficiently check if \(\phi \in n\text{-}\sc{PROVABILITY}_{\mathsf{T}}\) (say for \(n = 10^8\
    Extraction notes

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

    Details

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