Dynamics of BN is introduced by assigning a Boolean state variable and

a Boolean function to each node.

In 1988, Xiao and Massey introduced (by using properties of Walsh spectra) the notion of correlation immunity as an important cryptographic measure of

a Boolean function with respect to its resistance against the correlation attack (which can be seen as solving a system of multivariate linear equations) [7].

The future state of a node at each time step is determined by the current states of all its input nodes (parents) through

a Boolean function. For each Boolean function F, it can be represented as a mapping B : [{0, 1}.sup.k] [right arrow] {0,1}.

Furthermore, since the behavior of GRNs is stochastic by the effects of noise, it is appropriate that

a Boolean function is randomly decided at each time among the candidates of Boolean functions.

We observe that, by using these annotations, any data request ([C.sub.i] [member of] C) from BIGD can be written as

a Boolean function of keywords like:

Definition 4 Affine Function:

A Boolean function which can be expressed as 'xor' ([direct sum]) of some or all of its input variables and a Boolean constant is an affine function.

We note that a slight modification of Rubinstein's construction (Example 2.15) gives

a Boolean function, invariant under the cyclic shift of the variables, which still shows the quadratic gap between sensitivity and block sensitivity.

Thus, a boolean circuit C will compute

a boolean function f:{0,1}2??{0,1}.

The idea of BDDs is similar to decision trees:

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

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

The key idea applied in this article is to view the sets of variables in an element of the Sharing domain as the models of

a Boolean function. Because the elements of the Sharing domain always contain the empty set, we are concerned with the class of Boolean functions which are satisfied by assigning the value false to all of the variables.

In particular, the (global) evolution operator of a PDS can be given by means of

a Boolean function of n variables: