Counting inequivalent monotone Boolean functions

Abstract

Monotone Boolean functions (MBFs) are Boolean functions f:{0,1}n→{0,1} satisfying the monotonicity condition x≤y⇒f(x)≤f(y) for any x,y∈{0,1}n. The number of MBFs in n variables is known as the nth 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 equivalent if one can be obtained from the other by permuting 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 MBFs 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.