Application of Boolean algebra to switching circuit design and to error detection

Abstract

A solution is sought to the general problem of simplifying switching circuits that have more than one output. The mathematical treatment of the problem applies only to circuits that may be represented by “polynomials” in Boolean algebra. It is shown that certain parts of the multiple output problem for such circuits may be reduced to a single output problem whose inputs are equal in number to the sum of the numbers of inputs and outputs in the original problem. A particularly simple reduction may be effected in the case of two outputs. Various techniques are described for simplifying Boolean expressions, called “+ polynomials,” in which the operation “exclusive or” appears between terms. The methods described are particularly suitable for use with an automatic computer, and have been tested on the Illiac. An unexpected metric relationship is shown to exist between the members of certain classes of “+ polynomials” called “nets.” This relationship may be used for constructing error-detecting codes, provided the number of bits in the code is a power of two.