On the Number of 2-SAT Functions

TL;DR

This paper provides an alternative proof for the asymptotic count of 2-SAT definable boolean functions, linking it to the enumeration of certain triangle-free graphs, thereby advancing understanding of 2-SAT function complexity.

Contribution

It offers a new proof of the known asymptotic number of 2-SAT functions and connects this to the enumeration of odd-blue-triangle-free graphs.

Findings

Number of 2-SAT functions is approximately 2^{(n+1 choose 2)}

Asymptotics of odd-blue-triangle-free graphs are characterized

Provides an alternative proof to a known conjecture

Abstract

We give an alternative proof of a conjecture of Bollob\'as, Brightwell and Leader, first proved by Peter Allen, stating that the number of boolean functions definable by 2-SAT formulae is $(1+o(1))2^{\binom{n+1}{2}}$. One step in the proof determines the asymptotics of the number of "odd-blue-triangle-free" graphs on $n$ vertices.