The Number of Nonequivalent Monotone Boolean Functions of 8 Variables

A Boolean function $f:\{0,1\}^{n}\mapsto \{0,1\}$ is a monotone Boolean function (MBF) of $n$ variables if for each pair of vectors $x,y\in \{0,1\}^{n}$ from $x\leqslant y$ follows $f(x)\leqslant f(y)$ . Two MBFs are considered equivalent if one of them can be obtained from the other by permuting the input variables. Let $d_{n}$ be the number of MBFs of $n$ variables (which is known as Dedekind number) and let $r_{n}$ be a number of non-equivalent MBFs of $n$ variables. The numbers $d_{n}$ and $r_{n}$ have been so far calculated for $n\leqslant 8$ , and $n\leqslant 7$ , respectively. This paper presents the calculation of $r_{8}=1 392 195 548 889 993 358$ . Determining Dedekind numbers and $r_{n}$ is a long-standing problem.