PIK MASS-PRODUCTION AND AN OPTIMAL CIRCUIT FOR THE NECIPORUK SLICE

Let f : {0, 1}(n) --> {0, 1}(m) be an m-output Boolean function in n variables. f is called a k-slice if f(x) equals the all-zero vector for all x with Hamming weight less than k and f(x) equals the all-one vector for all x with Hamming weight more than k. Wegener showed that ''PIk-set circuits'' (set circuits over prime implicants of length k) are at the heart of any optimum Boolean circuit for a k-slice f. We prove that, in PIk-set circuits, savings are possible for the mass production of any F\X, i.e., any collection F of m output-sets given any collection X of n input-sets, if their PIk-set complexity satisfies SCm(F\X) greater than or equal to 3n + 2m. This PIk mass production, which can be used in monotone circuits for slice functions, is then exploited in different ways to obtain a monotone circuit of complexity 3n + o(n) for the Neciporuk slice, thus disproving a conjecture by Wegener that this slice has monotone complexity Theta(n(3/2)). Finally, the new circuit for the Neciporuk slice is proven to be asymptotically optimal, not only with respect to monotone complexity, but also with respect to combinational complexity