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

Alex Brodsky·Nicholas Pippenger
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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 Savický'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 Savický (1990).

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Alex Brodsky, Nicholas Pippenger. 2003-02-19. The Boolean Functions Computed by Random Boolean Formulas OR How to Grow the Right Function. https://arxiv.org/abs/cs/0302028

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