The number of 3-SAT functions

TL;DR

This paper proves that the number of 3-SAT functions on n variables asymptotically equals 2 raised to the power of n plus the binomial coefficient of n choose 3, confirming a conjecture for k=3.

Contribution

It establishes the asymptotic count of 3-SAT functions, confirming a conjecture about the growth rate of functions definable by k-SAT formulas.

Findings

G_3(n) is asymptotic to 2^{n+binom(n,3)}

Confirms the conjecture for k=3 about the logarithm of G_k(n)

Provides a strong form of the conjecture for fixed k

Abstract

With $G_k(n)$ the number of functions of $n$ boolean variables definable by $k$-SAT formulae, we prove that $G_3(n)$ is asymptotic to $2^{n+\binom{n}{3}}$. This is a strong form of the case $k=3$ of a conjecture of Bollob\'as, Brightwell and Leader stating that for fixed $k$, $\log_2 G_k(n)\sim \binom{n}{k}$.