  Among their many uses, growth processes (probabilistic amplification), 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 (1984)
hold. First, we completely characterize growth processes that use linear
connectives. Second, by extending Savick\'y's (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. Additionally, we obtain explicit bounds on the
convergence rates of several growth processes, including the growth process
studied by Savick\'y (1990).

Discrete Mathematics;Computational Complexity