A direct method of determining the maximum compatibility sets of an incompletely specified flow table of a sequential machine is presented. Subsets of pairwise incompatibles are utilized to decompose the set of all states in a step-by-step process into the maximum compatibles in a few steps. The method is simpler and faster than previously reported tabular, algebraic, and graphical techniques.

A new approach for the realization of multithreshold threshold elements is presented. The procedure is based on the fact that the excitations at contradictory vertices of the switching function must be unequal. The weights of the multithreshold element, in general, satisfy simple relations of the form U·W = 0, where U=(u1, u2, ... , un) and W=(w1, w2, ... , wn) such that ui∈{1,0, -1,}, i=1,2, ... n, and W∈In. Comparison of the excitations E(Xi) = W·Xiand E(Xj) = W·XjM at TRUE, and FALSE vertices Xiand Xj, respectively, for all specified vertices reusult in some inequalities of the form U·W≠0. Subsets of the remaining set of weight expressions U·W that are compatible are then determined, i.e., no linear combination of some or all of these expressions results in an expression Ui·W such that Ui·W≠0 and independent of each other. Each expression of each of these subsets is then equated to zero, and simple relations between weights are established. These are then used to find the weights vectors W's. The threshold vector T for each W is next established. From the set of weight-threshold vectors (W, T) the desired solution is determined by some minimality criterion. An example has been worked out by hand and an algorithm is given for systematic synthesis procedure.

Abstract—A switching function which is 2-summable is not linearly separable (a threshold function). A test for 2-summability is given which uses the decimal numbers corresponding to the standard products (minterms) of a function, a slide rule like device, and a preformed chart. The device may be used with functions of up to six variables.

Recently, Fischler and Tannenbaum2indicated a counter example to Theorem 2 of this author's short note.1Theorem 2 (as well as the other theorems in the note) is a statement of the sufficiency conditions only. This means that if the conditions given in Theorem 2 are satisfied, then it is always possible to find a minimal realization for five positive transitions. It does not say that if the conditions given in Theorem 2 are not fulfilled, it is not possible to obtain such a network. The counterexample is such a case. The author regrets the use of the phrase "necessary and sufficient conditions" in the abstract and elsewhere in the note.1This should be read as "sufficient conditions only."

The problem of minimal multilevel synthesis of symmetric switching functions with threshold logic elements is studied. A weight matrix representation of a threshold logic network is given. A number of theorems are presented which give necessary and sufficient conditions for minimal synthesis of symmetric functions with a maximum of eight positive transitions. In each case the minimal network is specified. An example has been worked out.