Invertible Boolean Functions

Abstract

A Boolean function has an inverse when every output is the result of one and only one input. There are 2n! Boolean functions of n variables which have an inverse. Equivalence classes of these functions are sets of equivalent functions in the sense that they are identical under a group operation on the input and output variables. This paper counts through five variables the number of equivalence classes of invertible Boolean functions under the group operation of complementation, permutation, and complementation and permutation, linear transformations and affine transformations. Lower bounds are given which experimentally give an asymptotic approximation. A representative function is given of each of the 52 classes of invertible Boolean functions of three variables under complementation and permutation. These are divided into three types of classes, 21 self-inverting functions, three functions have an inverse in the same class and 14 pairs of functions, each function of the pair in a different class. The four representative functions under the affine transformation are self-invertible.