Nearly all $k$-SAT functions are unate

TL;DR

This paper proves that almost all $k$-SAT functions become monotone after variable negations, confirming a long-standing conjecture and revealing the typical structure of these Boolean functions.

Contribution

It establishes that nearly all $k$-SAT functions are unate, resolving a conjecture from 2003 and advancing understanding of Boolean function properties.

Findings

Over 99% of $k$-SAT functions are unate as $n$ grows large

Confirms the conjecture by Bollobás, Brightwell, and Leader from 2003

Provides insight into the typical structure of $k$-SAT functions

Abstract

We prove that $1-o(1)$ fraction of all $k$-SAT functions on $n$ Boolean variables are unate (i.e., monotone after first negating some variables), for any fixed positive integer $k$ and as $n \to \infty$. This resolves a conjecture by Bollob\'as, Brightwell, and Leader from 2003.