Boolean matrices applied to sequential circuit theory and threshold logics

This paper treats the application of Boolean matrix theory to problems in sequential circuit theory and threshold logics. First, the solution to the Boolean matrix equation is reviewed. This is extended to equations in which one of the matrices is "unitary," or column-permuting. It is then shown how a transformation of Boolean variables may be considered as multiplication by a unitary matrix. This concept is applied to a synchronous recursive circuit whose output is a function of two sets of variables: a set of initial conditions and a set of time-changing variables. The problem of how the circuitry can be designed given the sequence of output functions is solved through Boolean matrices. A threshold-logic circuit may be considered as a set of Boolean functions, one function for each value of the threshold; as such, the circuit is analyzed by means of Boolean matrix theory. Alternatively, threshold circuits are analyzed by means of Post multivalued-logic matrices. In addition to a multivalued-logic matrix to transform the variables of a circuit, a matrix is introduced which will accomplish the effect of a change in threshold.