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ShowingImportancePrizesTopicP versus NP problem
The open problem in computational complexity theory, due to Cook and Levin, asking whether P — the class of decision problems solvable in polynomial time — equals NP, the class whose proposed solutions are verifiable in polynomial time. Claims about whether P equals NP, purported proofs or disproofs, proof barriers, and consequences of either resolution belong here.
Stephen Cook and Leonid Levin independently formulated the P versus NP problem in 1971.
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Computational complexity theoryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Cook's 1971 paper introduced NP-completeness and posed the P versus NP question.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Computational complexity theoryNP-complete problemsimportance · settledImportance 0.12, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Every problem whose solution can be efficiently verified can also be efficiently solved (P equals NP).
UnsupportedNo credible evidence found, though the claim is not contradicted.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Computational complexity theoryimportance · majorImportance 0.80, from 0 to 1 · major: real consequence within a domain, actively argued. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
No claimed proof that P equals NP or that P differs from NP has withstood expert scrutiny.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Computational complexity theoryimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Relativizing, natural, and algebrizing proof techniques cannot resolve whether P equals NP.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Computational complexity theoryComplexity-theoretic proof barriersimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
If P equals NP, the polynomial hierarchy collapses to P.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Computational complexity theoryPolynomial hierarchyimportance · settledImportance 0.12, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
No polynomial-time algorithm is known for any NP-complete problem.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Computational complexity theoryNP-complete problemsimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Most theoretical computer scientists believe that P is not equal to NP.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Computational complexity theoryExpert opinion surveysimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
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