It is well known that free Boolean algebra on n generators isomorphic to the of functions variables. The distributive lattice monotone In this paper we introduce concept De Morgan function and prove Keywords: Antichain, function, algebra.

1. Closed ideals. (1.1) Definition. Let 5 be an arbitrary set, and 2 a Boolean ring of subsets of 5. The algebra B(S, 2) consists of all uniform limits of finite linear combinations of characteristic functions of sets in 2. The norm in B(S, 2) is given by the formula ||/|| =SupseS \f(s) | • This definition is slightly more general than that given in [l ] where 2 is required to be a field of set...

It is well known that Exclusive Sum-Of-Products (ESOP) expressions for Boolean functions require on average the smallest number of cubes. Thus, a simple complexity measure for a Boolean function is the number of cubes in its simplest ESOP. It will be shown that this structure-oriented measure of the complexity can be improved by a unique complexity measure which is based on the function. Thus, ...

It has been pointed out that an n-variable Boolean function f has optimal resistance against fast algebraic attacks if and only if there does not exist a nonzero n-variable Boolean function g of degree lower than n 2 such that fg = h and deg(g) + deg(h) < n. In this corresponding, we show that there does not exist an n-variable Boolean function with optimal resistance against fast algebraic att...

| The Boolean vector function synthesis problem can be stated as follows: Given a truth table with n input variables and m output variables, synthesize a Boolean vector function that describes the table. In this paper we describe a new formulation of the Boolean vector function synthesis problem as a particular type of Satissability Problem. The Satissability Problem is translated into an integ...

A boolean function f(x1, ..., xn) is weakly symmetric if it is invariant under a transitive permutation group on its variables. A boolean function f(x1, ..., xn) is elusive if we have to check all x1,..., xn to determine the output of f(x1, ..., xn) in the worst-case. It is conjectured that every nontrivial monotone weakly symmetric boolean function is elusive, which has been open for a long ti...

In this work we consider a quantum single-error-correcting scheme derived from a oneway entanglement purification protocol in purifying one Bell state from a finite block of five Bell states. The main issue to be concerned with in the theory of the present errorcorrection is to create specific linear Boolean functions which can transform the sixteen error syndromes occurring in the error-correc...

A nearest neighbor representation of a Boolean function is set positive and negative prototypes in Rn such that the has value 1 on an input iff closest prototype positive. For k-nearest majority classification k considered. The complexity minimal number needed to represent function. We give several bounds for this measure. Separations are given between cases when can be real or required Boolean...

This paper argues that the ideas underlying the renormalization group technique used to characterize phase transitions in condensed matter systems could be useful for 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. When this transformation is applie...

Suppose that Y n is obtained by observing a uniform Bernoulli random vector X through a binary symmetric channel with crossover probability α. The “most informative Boolean function” conjecture postulates that the maximal mutual information between Y n and any Boolean function b(X) is attained by a dictator function. In this paper, we consider the “complementary” case in which the Boolean funct...