# Proving lower bounds on the monotone complexity of Boolean functions

• Ingo Wegener
Section VI: Complexity Of Boolean Functions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 171)

## Abstract

The hardware of computers may be well modelled by Boolean networks computing Boolean functions f:{0, 1}n → {0, 1}m. Though by counting arguments it is easy to show that nearly all Boolean functions may be computed only by networks of exponential size until now one is not able to prove nonlinear lower bounds for explicitly defined functions. Therefore one has restricted oneself to monotone networks where only -and ∨-gates are available. We repeat shortly the known methods for the proof of nonlinear lower bounds for the monotone network complexity of functions of n inputs and outputs. Afterwards we describe the method of using value functions, which was introduced by the author [13] to prove a bound of size n2/logn, until now the largest bound of this kind. For the Boolean convolution we present a new Ω(n3/2)-lower bound proved recently by one of my students (Weiß [14]). There one combines the elimination method and some information flow arguments. Finally we discuss how one may combine the above mentioned methods in order to get perhaps better bounds for the Boolean convolution.

## Keywords

Boolean Function Boolean Network Elimination Method World Record Boolean Matrix

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