A function f : {−1, 1} → R is called pseudo-Boolean. It is wellknown that each pseudo-Boolean function f can be written as f(x) =

The correlation of a Boolean functions with its spectral coefficients is closely related to the some cryptosystems and applications in engineering and computer sciences. By analysis of Boolean functions can be investigated whether a given function belongs to some standard class (linear, non-linear self-dual, threshold etc.) functions. For n variable Boolean function the number of nonzero spectr...

Interaction graphs provide an important qualitative modeling approach for System Biology. This paper presents a novel approach for construction of interaction graph with the help of Boolean function decomposition. Each decomposition part (Consisting of 2-bits) of the Boolean functions has some important significance. In the dynamics of a biological system, each variable or node is nothing but g...

We consider an n-ary random Boolean function f such that Pr[f(α̃) = 1] = p for α̃ ∈ {0, 1} and study its geometric model, the so called interval graph. The interval graph of a Boolean function was introduced by Sapozhenko and has been used in construction of schemes realizing Boolean functions. Using this model, we estimate the number of maximal intervals intersecting a given maximal interval of ...

For a Boolean Matrix A a binary vector v is called tfrequent if Av has at least t entries of value supp (v). Given two parameters t1 < t2 the t1-frequent but t2-infrequent vectors of a matrix represent a Boolean function that has two domains of (opposite) monotonicity. These functions were studied for the purpose of data analysis and abstract concept discovery in (Eisenschmidt et al. 2010). In ...

A Boolean circuit C on n inputs x1, . . . , xn is a directed acyclic graph (DAG) with n nodes of in-degree 0 (the inputs x1, . . . , xn), one node of out-degree 0 (the output), and every node of the graph except the input nodes is labeled by AND, OR, or NOT; it has in-degree 2 (for AND and OR), or 1 (for NOT). The Boolean circuit C computes a Boolean function f(x1, . . . , xn) in the obvious wa...

We say that a reversible boolean function on n bits has alternation depth d if it can be written as the sequential composition of d reversible boolean functions, each of which acts only on the top n − 1 bits or on the bottom n− 1 bits. We show that every reversible boolean function of n > 4 bits has alternation depth 9.

This paper argues that the ideas underlying the renormalization group technique applicable to phase transitions in physical systems are useful in distinguishing computational complexity classes. The paper presents a renormalization group transformation that maps an arbitrary Boolean function of N Boolean variables to one of N − 1 variables and shows that when this transformation is applied repe...

We prove a lower bound of 5n − o(n) for the circuit complexity of an explicit (constructible in deterministic polynomial time) Boolean function , over the basis U2. That is, we obtain a lower bound of 5n−o(n) for the number of {and, or} gates needed to compute a certain Boolean function, over the basis {and, or, not} (where the not gates are not counted). Our proof is based on a new combinatori...

We propose numerical schemes to determine whether a given function is convex, polyconvex, quasiconvex, and rank one convex. These notions are of fundamental importance in the vectorial problems of the calculus of variations.