trailEvery problem whose solution can be efficiently verified can also be efficiently solved (P equals NP).
Absence of any proven separation· for
No superpolynomial lower bound has been proven for any NP problem in a general model of computation.
atomic
Why the question remains open· neutral
Relativizing, natural, and algebrizing proof techniques cannot resolve whether P equals NP.
atomic
No claimed proof that P equals NP or that P differs from NP has withstood expert scrutiny.
atomic
Heuristic case for P ≠ NP· against
No polynomial-time algorithm is known for any NP-complete problem.
atomic
If P equals NP, the polynomial hierarchy collapses to P.
atomic
Cryptographic one-way functions exist.
atomic
+1 more
assumed
Boolean satisfiability is NP-complete.
claim · 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 →claim page ↗︎
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 →credence 0.07Credence, from 0 to 1: the Steward's probability that the claim, as stated, is true. Stated only where a single number is an honest summary; normative and evaluative claims usually carry none.constitution →
Nothing in the graph builds on this claim yet.
this rests on ↓
claimA box is a claim: a single proposition the graph assesses, with its own page and map. Click any claim to centre the map on it.constitution →argumentA pill is an argument: one line of reasoning stating how the claims beneath it combine to bear on the claim above it, for or against. Arguments are not destinations; click their claims to explore.constitution →✓verifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →↑supportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →⇄contestedCredible evidence or argument exists on multiple sides.constitution →○unsupportedNo credible evidence found, though the claim is not contradicted.constitution →✕contradictedAvailable evidence weighs against the claim.constitution →?unknownInsufficient information to assess.constitution →◌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 →verified factopen questionvalue premisetheoremprizeA double ring marks a claim with a live prize: an amount in owls offered for a machine-checked proof or disproof of its published formal statement. The amount is in the preview and on the claim page; a prize changes nothing about how the claim is assessed or how important it is judged to be.prize rules →⊢machine-checked⊢ marks a claim whose published formal statement has a machine-checked proof or disproof. The checker confirms the proof; the verdict beside it is still the steward's judgment of the claim as worded.prize rules →supportsthis provides evidence for the parentsteward instructions →contradictsthis argues against the parentsteward instructions →assumesbackground the parent's framing takes as givensteward instructions →requiresa load-bearing premise: the parent is false without itsteward instructions →Fig. Detail falls off with distance; every claim is an address.