Average Values of Quantities Appearing in Multiple Output Boolean Minimization

Abstract

In connection with the problem of two-level minimization of systems of Boolean functions, formulas are obtained for the following statistical quantities: average number of k cubes, prime k cubes, and essential k cubes of a system of Boolean functions. The parameters appearing in the formulas are the number of variables, the number of functions of the system, and the number of ``one'' vertices of each function. Numerical evaluations are given. Increasing by one the number of variables n of a system of m functions roughly results in multiplying the average numbers of cubes and prime cubes by a factor of about 2.2 to 2.3. The ratio of the average numbers of essential cubes and prime cubes rapidly decreases increasing n or m, so that the minimization algorithms, which obtain the essential cubes before the prime cubes, seem statistically unsuitable to solve ``large'' minimization problems. The average occupation memory occurring in Quine, McCluskey, and Bartee algorithms is also evaluated. Its rate of increase with the number of variables is about 2.5.