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

TL;DR

This paper characterizes the initial conditions and convergence behaviors of growth processes in Boolean formulas, extending previous analyses to linear and monotone connectives, and providing explicit bounds on their convergence rates.

Contribution

It offers a complete characterization of growth processes with linear and monotone connectives and extends analysis techniques to other connectives, including explicit convergence bounds.

Findings

Characterization of growth processes with linear connectives.

Extension of analysis to monotone connectives using Restriction Lemmas.

Explicit bounds on convergence rates for various growth processes.

Abstract

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).