Minimization and Convexity in Threshold Logic

The problem of deciding whether or not an arbitrary N-variable switching function is realizable with a single-threshold-element device is known to be convertible to the global minimization of a functional derivable from the given switching function. This functional is shown to consist of a structure of intersecting hyper-planes each of which is related to some threshold function. It is further shown that the functional is a convex function of its arguments so that test synthesis by minimization is a valid procedure. Two such computer-implemented minimization techniques are discussed. Finally, it is shown that this approach cannot be directly extended to threshold-element network synthesis, since there exists no convex functional with a global extremum at a network realization.