The Number of Nonequivalent Monotone Boolean Functions of 8 Variables

A Boolean function f : {0, 1}(n) bar right arrow {0, 1} is a monotone Boolean function (MBF) of n variables if for each pair of vectors x, y is an element of {0, 1}(n) from x <= y follows f (x) <= 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 <= 8, and n <= 7, respectively. This paper presents the calculation of r(8) = 1392195548889993358. Determining Dedekind numbers and r(n) is a long-standing problem.