For the Hadamard matrix, a modified factorization procedure is developed that is likely as economical in storage requirements and in the number of computational operations as the conventional fast Hadamard transform. Using this specific factoring method, the procedure for obtaining the fast Hadamard transform may be interpreted as operations on an H diagram. The H diagram was originally derived by Marihugh and Anderson [1] to provide a graphical representation for logic functions.

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.

This note presents a method for identifying total or partial symmetry of switching functions based on the application of the principle of residue test by numerical methods. The invariance of a switching function under a single interchange of two variables can be readily detected from the equality of some of the residues of expansion about these two variables. This procedure of detecting invariance is directly applied for the identification of total or partial symmetry of a switching function whose variables of symmetry may be either all unprimed (or all primed), mixed, or of multiform nature.

This paper deals with the realization of the combinatorial logic network of sequential machines with threshold logic elements, with a view to one- level realization. The state assignment problem is already a difficult one, and here it is further complicated by the restriction that the logical gates to be used are all threshold logic elements. This problem is quite important and is worthy of intensive study because, although threshold logic element as a general gate is well known, its value is not fully appreciated unless its applicability in various situations is demonstrated.

In this paper we shall study the problem of nonlinear separation. As usual, a Boolean function F of n binary variables, xl,..., xn, xiΣ{1,0}, i=1,..., n, will be represented by a set of vertices C in the n-dimensional Euclidean space Enwhere each vertex has n binary-valued components, {1,0}.

In this note the successive-higher-ordering method for testing and realization of threshold functions is applied to the realization of a threshold function F, such that it will contain a given function F1, which may or may not be a threshold function, and such that it will be contained in F1+Fø, where Føis the function representing the don't care vertices. Before the application of the successive-higher-ordering method, the given functions F1and F0= (F1+Fø) are first changed into unate functions by the successive positivizing of the functions with respect to the variables, one at a time. Some theorems relating to this formation of unate functions are presented. A systematic procedure for testing and realization is developed. An example is given for illustration.

In this paper, a modification or generalization of Sheng's secondary ordering method for testing and realization of threshold functions is presented. Instead of assigning integral values to the incremental weights according to secondary ordering, a search for successively higher ordering is made, and incremental weights of higher orders are successively substituted back into the inequalities until finally no more higher ordering can be found. If the given function is a threshold function, it will turn out that the sum of the coefficients of all the terms on the left side of each of the inequalities will be greater than the sum of the coefficients of all the terms on the right side. Then a minimal integral assignment can be made by assigning unity to every incremental weight of any order appearing in the final set of inequalities. If the given function is not a threshold function, a contradiction will be revealed. Some theorems are proved to justify the method. A complete procedure for testing and realization is given. An example is worked out in detail to illustrate this method.

Threshold logic elements, although essentially used as gates in determinis,~ie combinatorial circuits, can be applied ~o switching circuits with disere~,e random inputs ~c~ generate discrete random variables of any probability distribution, i.e. to serve as a discrete ]probability transformer.In this paper some properties of threshold functions related to the realization of a probability trattsformer are studied.The principle of using threshold logic elements for probability transformation, is presented and the techniques for realization are developed.Examples are given for illustration. Int'roductionThis research was supported in

No abstract available.

A graphical interpretation of the realization of symmetric Boolean functions with threshold logic elements is presented, from which a systematic synthesis method is developed. Theoretically, symmetric functions of any number of variables can be realized. Examples are given to show that, practically, there is no difficulty at all in realizing symmetric Boolean functions of as many as thirty or forty variables. Some theorems related to graphical interpretation and realization are presented, and the lower and upper bounds of the number of threshold logic elements required for the realization of symmetric functions are also derived from the graphical point of view.

This paper deals with the problem of compound synthesis of threshold-logic network, or of realizing a general Boolean function with a number of threshold-logic elements. Some basic theorems concerning the expression of a general Boolean function as a combination of a number of threshold functions are presented. The general structure of a network of m threshold-logic elements is analyzed in light of the basic theorems and properties of threshold functions. The properties of isobaric threshold functions are discussed and are utilized for compound synthesis. Methods for decomposing a function into a sum or product of unate functions and into a sum or product of threshold functions are developed. A synthesis procedure conforming with the general structure of a network of threshold-logic elements is given. A simplified synthesis procedure, which needs less cut and trial than the original procedure and is, therefore, more practical, is also presented. Some special situations are discussed. Examples are worked out to show the decomposition of functions, and to illustrate the synthesis method.

A new method for detection and identification of totally symmetric Boolean functions is developed. Instead of using maps, charts, or truth tables, with permutations and complementations of variables, this method is based on the number of vertices contained in symmetric functions and the concept of distance between two vertices. Since the distance between any two vertices remains invariant under any number of permutations and complementations of the variables, this method lends itself equally conveniently to the detection of total symmetry of Boolean functions with respect to purely uncomplemented, purely complemented, or mixed variables. A procedure is presented, and an example is worked out to illustrate the application of the method.

A standard method for testing and realizing a threshold function is to solve a set of linear inequalities in which the unknowns are the n weights to be assigned to the n variables. In this paper a simple method of solving this set of inequalities is presented. Instead of using the weights themselves as the unknowns, a set of n new unknowns, the incremental weights Δa1, Δa2, . . ., Δan-1, together with the lowest weight an, is used. This change of unknowns results in a simpler set of inequalities which, in turn, furnishes direct information on 1-realizability1of the function and on the assignment of weights for realization, often without the necessity for trial and adjustment.