Most Complex Boolean Functions Detected by the Specialized Normal Form

It is well known that Exclusive Sum-Of-Products (ESOP) expressions for Boolean functions require on average the smallest number of cubes. Thus, a simple complexity measure for a Boolean function is the number of cubes in its simplest ESOP. It will be shown that this structure-oriented measure of the complexity can be improved by a unique complexity measure which is based on the function. Thus, it is suggested to detect all most complex Boolean functions more precisely by means of the Specialized Normal Form (SNF). The SNF is a unique (canonical) ESOP representation of a Boolean function. In this paper properties of the most complex Boolean functions are studied. Adjacency graphs of the SNF will be used to calculate minimal ESOPs as well as to detect special properties of most complex Boolean functions. These properties affect the procedure of finding exact minimal ESOPs. A solution is given that overcomes these observed problems. Using the SNF, the number of most complex Boolean functions was found. A recursive algorithm will be given that calculates each most complex Boolean function for a given number of variables.