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    If P = NP, then finding a satisfying valuation for a prop... — Carmelics
    Home/Modality & Possibility
    HistoryEditSee Inverse

    If P = NP, then finding a satisfying valuation for a propositional formula is no harder than constructing its truth table, and factoring a natural number would be no more difficult than verifying a given factorization.

    Truth & 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.P is the class of problems decidable efficiently.
      ?

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    • 2.NP is the class of problems verifiable efficiently given a certificate.
      ?

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    • 3.If P = NP, then the difficulty of deciding and verifying coincide (up to a polynomial factor) for all problems in NP.
      ?

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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Computational complexity classes are defined relative to models of computation, and their equivalence is model-dependent, not an absolute metaphysical fact.
      ?

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    • 2.The claim conflates extensional equivalence of complexity classes with identity of epistemic difficulty, which are distinct notions even under P = NP.
      ?

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    • 3.Even if P = NP, the witnessing polynomial reductions may have constants or degrees rendering them practically intractable, preserving real cognitive asymmetry.
      ?

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    Reason against 2 of 2
    ?
    • 1.Kripke's distinction between epistemic and metaphysical possibility implies that P = NP being mathematically true does not collapse the epistemic gap between search and verification.
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    • 2.The asymmetry between finding and checking reflects a constitutive feature of bounded rationality that survives any formal complexity-theoretic collapse.
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    Topics

    Modality & PossibilityTruth & Knowledge

    Key Terms

    P = NP(computational complexity theory)
    The hypothesis that for all problems in NP, the difficulty of deciding membership and verifying membership coincide up to a polynomial factor
    Satisfying valuation(in formal logic and problem-solving)
    An assignment of true or false values to variables that makes a logical statement work out as true.
    factoring a natural number(as used in mathematics and cryptography)
    Breaking a whole number down into its prime number building blocks—for example, factoring 15 means finding that it equals 3 × 5.
    propositional formula(the object that a Turing machine needs to evaluate)
    A mathematical expression made of logical statements (like 'A AND B' or 'C OR NOT D') that can be either true or false.
    truth table(as used in logic)
    A chart that lists every possible combination of true/false values for a logical statement and shows what the result is in each case.
    verifying a factorization(as used in mathematics)
    Checking that a proposed set of prime factors actually multiply together to equal the original number.

    Connections

    1 topic

    Proof of definition segments2 linked

    Related

    Computational complexity classes are defined relative to models of computation, ...Even if P = NP, the witnessing polynomial reductions may have constants or degre...If P = NP, then the difficulty of deciding and verifying coincide (up to a polyn...Kripke's distinction between epistemic and metaphysical possibility implies that...

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    1 On the significance of \(\textbf{P} \neq \textbf{NP}\)? The appreciation of complexity theory outside of theoretical computer science is largely due to the notoriety of open questions such as 1–4. \) – has attracted the greatest attention. e. the Millennium Problems (Cook 2006). g. (Sipser 1992), (Fortnow 2009), and (Fortnow 2013). \) will prove to have far reaching practical and theoretical consequences outside of computer science. Perhaps the most significant of these revolves around the pos
    Extraction notes

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

    Details

    +4 moreShow less
    NP is the class of problems verifiable efficiently given a certificate.P is the class of problems decidable efficiently.The asymmetry between finding and checking reflects a constitutive feature of bo...The claim conflates extensional equivalence of complexity classes with identity ...

    Similar

    If P = NP, then finding a satisfying valuation for a propositional for...94%If P equals NP, finding a satisfying valuation for a propositional for...92%If P equals NP, factoring a natural number would be no more difficult ...85%Finding a satisfying valuation for a propositional formula is an NP pr...84%
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit