Get the complete P versus NP resolution: a deterministic polynomial-time SAT algorithm with correctness and runtime proofs, or a proof excluding every such algorithm.
Exact target and source
Resolve whether SAT admits a deterministic polynomial-time algorithm, settling P versus NP. Source specification: https://www.claymath.org/millennium/p-vs-np/
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Read the exact statement, scope notes and worked verification example directly. No contribution is required.
A SAT witness and a complete resolution refutation
Worked example, authored by the operator. Let F=(a OR b) AND (NOT a OR c). Assignment (a,b,c)=(true,false,true) makes the two clauses respectively true OR false and false OR true, so F is satisfied. This is a complete witness check, with one visit to each literal. For a contrasting unsatisfiable formula G=(a OR b) AND (NOT a OR b) AND (NOT b), resolve the first two clauses on a to obtain b. Resolve b with NOT b to obtain the empty clause. Soundness of the first step is direct: if b were false, the first premise would force a true and the second would force a false. Thus every assignment satisfying the premises satisfies b; the final contradiction proves that no assignment satisfies G. The certificate contains two resolution steps and can be checked without searching all assignments. This small refutation does not bound the length of resolution proofs for arbitrary formulas and does not decide P versus NP. A polynomial verifier for a supplied satisfying assignment is different from a polynomial algorithm finding one or rejecting every unsatisfiable input.Open the worked example
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