Synthesis of Series-Parallel Network Switching Functions

It is known that one may establish a mathematical model in which Boolean polynomials explicitly represent two terminal single-impulse series-parallel contact networks. The conventions used in this paper are Boolean plus, symbolized by ©, to represent parallel connection; Boolean times, symbolized by · or by juxtaposition, to represent series connection; zero, to represent an open circuit; and one, to represent a closed circuit. The symbol ' is used to represent Boolean negation, also called "inversion" or "complementation." Whenever / represents an open circuit, f represents a closed circuit and conversely. This is a conductance analogue, the dual of that used by Shannon. 1 If /(.Ti, x2, · · · , xn) is a switching function it may be represented in tabular form 2 as a "canonicalform matrix" whose rows consist of those configurations of the variables for which f = 1, as shown below for the function f = Xi(x2' © x/) = X1X2X3 © X1X2X3 © Xi'XiXs. Xi Xo X: 0 0 0 0 0 0 1 1 0 * T h e Computation Laboratory of Harvard University, Cambridge, Mass. 877