What question did this study set out to answer?

This work aims to show that the Boolean satisfiability problem can be solved efficiently using a binary search approach.

February 28, 2026Open Access

P=NP Algorithm

Key Points

This work aims to show that the Boolean satisfiability problem can be solved efficiently using a binary search approach.
Introduced a reduction method called Binary-SAT Collapse to simplify the SAT problem.
Utilized binary search over the space of variable assignments for a CNF formula.
Engaged a satisfiability oracle to verify solutions within the search process.
Showed that a binary search can be completed in linear steps relative to the number of variables.
Demonstrated that the overall procedure runs in polynomial time, suggesting P=NP.

Abstract

We present the Binary-SAT Collapse, a reduction that recasts the Boolean satisfiability problem (SAT) as a binary search over the totally ordered space of all possible variable assignments. Given a CNF formula with n variables, the search space \0, , 2ⁿ - 1\ has size N = 2ⁿ. Binary search over this space terminates in ₂ N = n steps. Each step queries a satisfiability oracle on a subrange; under the standard assumption that SAT NP, verifying a certificate is polynomial, and the oracle itself reduces to a bounded instance of SAT. Because the number of oracle calls is n (linear in the input size of the formula) and each call is polynomial in the size of, the total procedure runs in polynomial time. This implies P = NP.

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