Some computer-generated improved designs for the digital summation threshold logic gate, first described by Hurst, are presented. These designs require considerably fewer logic elements than the original designs and have improved gate-delay parameters. The improved designs are shown to have delay times comparable to those of the ‘bifurcated’ gates of Reddy and Swamy, whilst enjoying a more elegant structure. Some implementations for these circuits are given.

A new mathematically based method of synthesis for combinational digital networks is presented, based upon the recognition of symmetry patterns in the functions being synthesized. The design procedure is structured such that at each stage of the synthesis a relevant gate assembly is indicated, which leaves a simplified "remainder" function to be synthesized at the next stage of the realization. This decomposition procedure optimally uses gates with a fan in of two, but post-synthesis reconfiguration for amalgamation into larger fan-in gates is available.

I wish to thank C. K. Yuen for calling attention to a previous paper on the application of the Rademacher-Walsh transform to threshold logic synthesis [1].

Five operations are defined in the Rademacher-Walsh transform domain. It is shown that these operations allow Boolean functions to be classified in a very concise way. The result of applying this classification to Boolean functions of up to fourth order indicates that threshold functions play an important part in the composition of Boolean functions. A synthesis method is developed for the synthesis of any Boolean function using, as a basic element, an "optimized universal threshold logic gate." This gate overcomes the analog threshold tolerancing problems encountered in other threshold gate designs and is readily fabricated in an integrated circuit form. The use of this gate in logic design is expected to provide a considerable cost-saving over conventional methods.

A Boolean matrix algebra is described which enables many logical functions to be manipulated simultaneously. The algebra is similar to conventional matrix algebra and its structure includes a topological description of logic circuits. The matrices are readily manipulated by the digital computer.