Counting inequivalent monotone Boolean functions

TL;DR

This paper introduces a new method to count inequivalent monotone Boolean functions by analyzing their profiles, successfully computing the count for 7 variables, which was previously unknown.

Contribution

The paper presents a novel strategy to count inequivalent MBFs by partitioning the problem based on function profiles, extending the known counts to 7 variables.

Findings

Number of inequivalent MBFs in 7 variables: 490,013,148

Breakdown of counting method based on function profiles

Extension of previous counts from up to 6 variables

Abstract

Monotone Boolean functions (MBFs) are Boolean functions $f: {0,1}^n \rightarrow {0,1}$ satisfying the monotonicity condition $x \leq y \Rightarrow f(x) \leq f(y)$ for any $x,y \in {0,1}^n$. The number of MBFs in n variables is known as the $n$th Dedekind number. It is a longstanding computational challenge to determine these numbers exactly - these values are only known for $n$ at most 8. Two monotone Boolean functions are inequivalent if one can be obtained from the other by renaming the variables. The number of inequivalent MBFs in $n$ variables was known only for up to $n = 6$. In this paper we propose a strategy to count inequivalent MBF's by breaking the calculation into parts based on the profiles of these functions. As a result we are able to compute the number of inequivalent MBFs in 7 variables. The number obtained is 490013148.