A majority or threshold element is a device with a finite number of inputs and a single output. The output, Fnis a Boolean function of the n binary valued variables (x1, x2, ..., xn) applied to the inputs. The analysis of a threshold element is the determination of the output Boolean function, Fnwhen the input variables and their multiplicities are specified. The multiplicity of an input variable xiis the number of inputs to which the variable is applied. The fundamental problem in analysis is the determination of Fnwhen the number of inputs is moderately large. An algorithm is given which permits a simple straightforward determination of Fnin reduced, disjunctive form.

A large number of Boolean functions may be realized as the output of a single threshold element; however, not all functions are so realizable. Those Fnwhich are, are said to belong to the class R. No easily applicable criteria have been formulated which enable an unequivocal determination to be made of whether a given Fnϵ R. There are, however, certain transformations allowable on those Fnϵ R which exhibit a closure property, that is, the transformation results in an Fpϵ R. There are also certain properties of the form of Fnϵ R which can be recognized easily. A knowledge of these permissible transformations and observations of the form of a given Fnwill, in many instances, allow for the desired. determination. This paper, for the most part, examines those permissible transformations and properties of form.

The synthesis of a threshold element which realizes a desired Boolean function as its output may be accomplished by solving a basic set of linear inequalities. The determination of this basic set is discussed with the aid of a geometric interpretation of such realizable functions on the vertices of a unit N cube. A solution may be obtained to these inequalities by linear programming. In this paper, a method of synthesis which lends itself to rapid and simple hand computation is given.