m](n([summation])) such that each proper binary

boolean function ([L.

If there are n variables, then the

Boolean function is a logical mapping from a set of {[v.

c) The procedures of translations of a

Boolean function f([k.

Definition 6 Balanced

Boolean Function: If the Hamming weight of a

Boolean function of n variables is 2n-1, it is called a balanced

Boolean function.

Graph-based algorithms for

Boolean function manipulation.

Control inputs are used to specify the functionality of the circuit and remaining four inputs are used to implement

Boolean function of four variables.

A]) for f a

Boolean function, let Vars(f) denote the set of variables appearing in f.

A

Boolean function involves any number of variables whose values and arguments are binary (for example, taking values 0 or 1).

However, we prefer not to complicate the notation, instead relying on an understanding that, at any time, there is a set of variables of interest, say, VI = (x, y, z) so that a formula [Psi] denotes the

Boolean function [Lambda]xyz : [Psi].

A simple counting argument shows that, for a randomly chosen

boolean function of kn variables, the k-party communication complexity is [OMEGA](n) with high probability, regardless of the dependence of k on n.

k] (f) denote the density of and/or trees defining a

boolean function f within the set of and/or trees with fixed number of variables k.

2]} or [beta](v) is a 0-ary

boolean function from B.

The idea of BDDs is similar to decision trees: A

Boolean function is represented as a rooted acyclic-directed graph.

But for an arbitrary

Boolean function the best known upper bound on block sensitivity in terms of sensitivity is exponential.

is given by a non trivial

Boolean function f : [{0,1}.