In this paper, we introduce two pairs of rough operations on Boolean algebras. First we define a pair of rough approximations based on a partition of the unity of a Boolean algebra. We then propose a pair of generalized rough approximations on Boolean algebras after defining a basic assignment function between two different Boolean algebras. Finally, some discussions on the relationship between...

Traditionally Sum-Of-Products (SOPs) have been studied as expressions of Boolean functions. It has been shown by Sasao in [7] that Exclusive-or-Sum-Of-Products (ESOPs) are more compact representations of Boolean functions than SOPs. The problem to identify an exact minimal ESOP of a Boolean function is despite of some partial results still unsolved. Our new results rely on recently detected pro...

We study the class of polynomial threshold functions using harmonic analysis and apply the results to derive lower bounds related to ACo functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). Our main result is that the class of polynomial threshold functions can be characteriz...

Many problems in computer vision, such as stereo, segmentation and denoising can be formulated as pseudo-boolean optimization problems. Over the last decade, graphs cuts have become a standard tool for solving such problems. The last couple of years have seen a great advancement in the methods used to minimize pseudoboolean functions of higher order than quadratic. In this paper, we give an ove...

A Boolean function is called normal if it is constant on flats of certain dimensions. This property is relevant for the construction and analysis of cryptosystems. This paper presents an asymmetric Monte Carlo algorithm to determine whether a given Boolean function is normal. Our algorithm is far faster than the best known (deterministic) algorithm of Daum et al. In a first phase, it checks for...

Nonlinearity of rotation symmetric Boolean functions is an important topic on cryptography algorithm. Let e ≥ 1 be any given integer. In this paper, we investigate the following question: Is the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial x0xex2ex3e equal to its weight? We introduce some new simple sub-functions and develop new technique to get seve...

The average sensitivity of a Boolean function is the expectation, given a uniformly random input, of the number of input bits which when flipped change the output of the function. Answering a question by O’Donnell, we show that every Boolean function represented by a k-CNF (or a k-DNF) has average sensitivity at most k. This bound is tight since the parity function on k variables has average se...