Some comments on infinite Boolean functions

Some Comments on Infinite Boolean Functions

Some Comments on Infinite Sets

Suppose that A is an infinite set. Constructing a denumerable subset {a1, a2, a3, . . .} of A, which is the same as showing that there is an injection f : Z −→ A, requires more than just mathematical induction. In these notes we explore some consequences of basic definitions con cerning infinite sets with the aid of mathematical induction only. We show that the set of rational numbers, and more...

On Some Iteration Properties of Boolean Functions

This paper introduces and describes an iterative model for the study of some properties of Boolean functions. Several kinds of iterative functions are studied: they are obtained by substituting some of the independent variables x, off (x) by the function f itself. Algebraic formulas relating classical representation modes of Boolean functions to the iterative forms are introduced. Several possi...

Comments and Corrections Comments on “Canalizing Boolean Functions Maximize Mutual Information”

In their recent paper “Canalizing Boolean Functions Maximize Mutual Information,” Klotz et al. argued that canalizing Boolean functions maximize certain mutual informations by an argument involving Fourier analysis on the hypercube. This note supplies short new proofs of their results based on a coupling argument and also clarifies a point on the necessity of considering randomized functions.

On Analysis and Generation of some Biologically Important Boolean Functions

Boolean networks are used to model biological networks such as gene regulatory networks. Often Boolean networks show very chaotic behaviour which is sensitive to any small perturbations. In order to reduce the chaotic behaviour and to attain stability in the gene regulatory network, nested Canalizing Functions (NCFs) are best suited. NCFs and its variants have a wide range of applications in sy...