Realization of Boolean Polynomials Based on Incidence Matrices

Abstract

This paper describes a general algebraic method of finding minimum contact networks for any given Boolean polynomial. Solutions obtained by this method may in general be any kind of connection with any number of contacts for each variable. Furthermore, any practical requirements such as series-parallel cases usually found in most electronic devices, and single contact for specified variables, can all be considered in the calculation, if necessary. A routine algorithm on incidence matrices will automatically yield any ingenious connections. The node-branch and branch loop incidence matrices which were revealed by G. R. Kirchhoff 1847, are adopted as unknown, especially those of modulo 2 which were elaborated by O. Veblen in 1916 are mostly used. However, simultaneous use of modulo zero (or infinity), 2 and other integers was found useful for combinatorial consideration in the multi-contact case. General Galois fields are already used in switching theory by Moisil and his Rumanian group.