We identify a new restriction on Boolean circuits called bijectivity and prove that bijective Boolean circuits require exponential size to compute the Boolean permanent function. As consequences of this lower bound, we show exponential size lower bounds for: (a) computing the Boolean permanent using monotone multilinear circuits ; (b) computing the 0-1 permanent function using monotone arithmet...

We study the length of polynomials over nite simple non-Abelian groups needed to realize Boolean functions. We apply the results for bounding the length of 5-permutation branching programs recognizing a Boolean set. Moreover, for Boolean and general functions on these groups, we present upper bounds on the length of shortest polynomials computing an arbitrary nary Boolean or general function, o...

The conceptions of χ-value and K-rotation symmetric Boolean functions are introduced by Cusick. K-rotation symmetric Boolean functions are a special rotation symmetric functions, which are invariant under the k − th power of ρ. In this paper, we discuss cubic 2-value 2-rotation symmetric Boolean function with 2n variables, which denoted by F2n(x2n). We give the recursive formula of weight of F2...

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.