PI k mass production and an optimal circuit for the Ne?iporuk slice

Abstract

Let D: {0,1}n $ {0, 1}m be an m-output Boolean function in n variables. The k-slice of D is the monotone function Dsl-k defined component-wise by D (i)sl-k = (D (i) [ Tn k ) Z Tn k +1 , where Tnk is the kth threshold function. Wegener showed that certain ''PIk - set circuits'' are at the heart of any optimum Boolean circuit for a k-slice D. We prove that, in PIk - set circuits, savings are possible for the mass production of any F l X, i.e., any collection F of ms output-sets given any collection X of ns input-sets, if their PIk - set complexity satisfies SCm (F lX) 3 3ns + 2ms. This PIk mass production, which can be used also in monotone circuits for slice functions, is exploited then in different ways to obtain a monotone circuit of complexity 3n + o(n) for the canonical slice of the Nechiporuk sums, thus disproving a conjecture by Wegener that this slice has monotone complexity O(n3/2). Finally, we prove that this new circuit for the Nechiporuk slice is asymptotically optimal, not only with respect to monotone, but also with respect to combinational, complexity.