What question did this study set out to answer?

The aim is to provide a formal proof that P equals NP by showing a reduction from SAT to a new class of functions.

February 5, 2026Open Access

P = NP: Resolution of NP-Complete Problems in O(log n)

Key Points

The aim is to provide a formal proof that P equals NP by showing a reduction from SAT to a new class of functions.
Established a polynomial-time reduction from SAT to Problem A.
Defined Problem A as a class of monotonically ascending deterministic functions.
Demonstrated that Problem A is NP-complete and can be solved in O(log N) time using binary search.
Utilized transitivity of Karp reductions for broader conclusions.
Concluded that all NP-complete problems are solvable in logarithmic time.
Provided a formal proof establishing P = NP in computational theory.

Abstract

This paper presents a formal proof for P = NP by establishing a polynomial-time reduction from the Boolean Satisfiability Problem SAT to a newly defined class of monotonically ascending deterministic functions (Problem A). We demonstrate that Problem A is NP-complete and resolvable in O (\ N) time via binary search. By transitivity of Karp reductions, we conclude that all NP-complete problems are solvable in logarithmic time.

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