The Boolean Functions Computed by Random Boolean Formulas OR How to Grow the Right Function

How to Grow the Right Function probabilistic amplification, random Boolean functions were used for constructing reliable networks from unreliable components, and deriving complexity bounds of various classes of functions. Hence, determining the initial conditions for such processes is an important and challenging problem. In this paper we characterize growth processes by their initial conditions and derive conditions under which results such as Valiant's (Valiant, 1984) hold. First, we completely characterize growth processes that use linear connectives. Second, by extending Savický's (Savický, 1990) analysis, via "Restriction Lemmas", we characterize growth processes that use monotone connectives, and show that our technique is applicable to growth processes that use other connectives as well.