The aim of this paper is to completely classify all aggregation functions based on the notion of arity gap. We first establish explicit descriptions of the arity gap of the Lovász extensions of pseudo-Boolean functions and, in particular, of the Choquet integrals. Then we consider the wider class of order-preserving functions between arbitrary, possibly different, posets, and show that similar ...

in this paper the concept of an $omega$- almost boolean ring is introduced and illistrated how a sheaf of algebras can be constructed from an $omega$- almost boolean ring over a locally boolean space.

The minimum number of NOT gates in a Boolean circuit computing a Boolean function f is called the inversion complexity of f . In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that ⌈log 2 (n+ 1)⌉ NOT gates are sufficient to compute any Boolean function on n variables. In this paper, we consider circuits computing non-deterministically and dete...

The dynamics of discrete mathematical systems can be related to the values of certain ordering parameters. Experiments with the total set of possible structures of a boolean network with six elements and extended sets of two-valued boolean functions show that three simple parameters describing the topology, the bias of boolean functions, and their canalyzing potential yield necessary conditions...

Higher order differential cryptanalysis is based on the property of higher order derivatives of Boolean functions that the degree of a Boolean function can be reduced by at least 1 by taking a derivative on the function at any point. We define fast point as the point at which the degree can be reduced by at least 2. In this paper, we show that the fast points of a n-variable Boolean function fo...

In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone’s representation theorem. Some special classes of Boolean algebras are also concerned with and, in particular, the relations of Boolean algebras to propositional logic and to set theory are studied in more details. We mention here that all what we do for propositional logic can also be extende...

A classical construction of stream ciphers is to combine several LFSRs and a highly non-linear Boolean function f . Their security is usually studied in terms of correlation attacks, that can be seen as solving a system of multivariate linear equations, true with some probability. At ICISC’02 this approach is extended to systems of higher-degree multivariate equations, and gives an attack in 2 ...

The paper addresses the problem of synthesizing programs using a restricted set of input-output data. The program to be synthesized consists of assigning to individual Boolean variables x′, the result of evaluating functions involving other Boolean variables x. The program can be initially viewed as a black box having n inputs corresponding to the variables x and n outputs corresponding to the ...

Properties valid on the classical theory (Boolean laws) have been extended to fuzzy set theory and so-called Boolean-like laws. The fact that they are not always valid in any standard fuzzy set theory induced a wide investigation. In this paper we show the sufficient and necessary conditions that the Boolean-like law y ≤ I(x, y) holds in fuzzy logic. We focus the investigation on the following ...