An log lower bound on synchronous combinational complexity

Synchronous combinational machines are combinational ma- chines such that the length of all paths from inputs to a logic element are the same. In this paper is is shown that any Boolean function of n variables satisfying certain subfunction conditions (which are satisfied by "almost all" such functions) must have synchronous combinational complexity at least nlogn. 0. Introduction. In this paper is is shown that if a Boolean function of n variables satisfies certain subfunction conditions (which are satisfied by "almost all" such functions), then its synchronous combinational complexity must be at least n log n.2 This bound appears to be unique at this time in being nonlinear and applying to a large class of single-valued functions. The restriction on the definition of combinational complexity, i.e., that the machines be synchronous, is very natural from an engineering point of view, and seems to change the general theory of combinational complexity very little. For background on combinational complexity the reader is referred to a survey article by the present author and J. E. Savage entitled Complexity made simple. 1. Definitions. A synchronous combinational machine (s.c.m.), 911, is