SLAC-PUB-806 (Revised) January 1971 PREVIOUS INVESTIGA.TION OF REALIZATION OF AN ARBITRARY SWITCHING FUNCTION WITH A NETWORK OF THRESHOLD A.ND PARITY ELEMENTS

Attention is called to previous research on realization of an arbitrary switching function by a network of threshold gates (or threshold elements) and modulo 2 gates (or parity elements), establishment of greatest lower bounds on the number of gates needed, artifices that lead to further network reduction in special cases, and systematic minimization of the number of module 2 gates required. Although not widely published, a report describing the research in detail is readily available. (Submitted to Correspondence Section. IEEE Trans. on Computers) Index terms : arbitrary switching function, linearly separable function, matrix partitioning, minimal network, modulo 2 gate, optimal realization, parity element, threshold element, threshold gate, Walsh matrix, Walsh series. (Proposed revision of IEEETC L2421) Ostapko and Yau have recently published a short note on the realization of an arbitrary switching function by a two-level network of threshold and parity elements [l]. N early a decade ago this writer and others also investigated the realization of an arbitrary switching function by a more general network of these same elements, but followed a very different approach and sought a different form of optimization [2]: Unfortunately it was not feasible then to publish our results more widely, even though they were not classified. However, they are readily available at a nominal price. It is now well known that any m-ary function (not merely binary functions) of n binary variables and 2n arguments can be expanded into a finite Walsh series. The search for linearly separable threshold functions is simply a search for switching functions that are, or can be, represented by the constant and linear terms of their Walsh expansion. Section III of 1’ is simply a redevelopment of the widely known Walsh matrix and its binary equivalent 3 , r4a. In view of the great decrease in the fraction of switching functions that are linearly separable as n is increased, the writer, like Ostapko and Yau, became aware that the search for switching functions realizable by threshold devices alone promised very limited rewards, and that a more fruitful search might be one for optimal means of realizing any switching function by a combination of threshold gates (corresponding to the threshold elements of [I]> for the constant and linear terms in the Walsh expansion, and module-2 gates (corresponding to the parity elements of [l]) for the nonlinear terms. Instead of endeavoring to optimize the realization by programming methods, the writer applied the well known method of partitioning matrices to the recursive Walsh matrix derived by Lechner [3], [4], and showed that any switching function could be realized by a three-level network, the first consisting of at most