A Note on Nečiporuk’s Method for Nondeterministic Branching Programs

Nečiporuk [Nec̆66] gave a method based on counting subfunctions to lower bound the formula size over an arbitrary binary basis and contact scheme (or Boolean branching program) size required to compute explicit n-input Boolean functions. Pudlak [Pud87] observed that Nečiporuk’s method also yields an Ω(n/ log n) lower bound on switching-and-rectifier network size as well as nondeterministic branching program size [Raz91], and Karchmer and Wigderson [KW93] extended this lower bound to apply to span program size. In this note, we show that any lower bound achievable by Nečiporuk’s method is at most O(n/ log n) for any of these three models. As part of our argument we show that every n-input Boolean function can be computed by a nondeterministic branching program of size O(2) and this is asymptotically optimal.