In this paper we give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following: • Almost every Boolean function has PTF degree at most n2 + O( √ n log n). Together with results of Anthony and Alon, this establishes a conjecture of Wang and Williams [32] and Aspnes, Beigel, Furst, and Rudich [4] up to lower order terms. • ...

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.

Every Boolean function is uniquely defined by a polynomial modulo 2. The degree of a Boolean function is the degree of its defining polynomial. In cryptography, the Boolean functions of fixed degree played important role, for example, 1 or 2 degrees. Therefore, in finding algorithms that recognize properties of Boolean functions polynomials by their values vectors, it makes sense consider only ...

The expressiveness of Boolean Algebras is significantly extended by the Boolean Differential Calculus (BDC). The additionally defined differentials of Boolean variables, differentials and further differential operators of Boolean functions as well as several derivative operations of Boolean functions allow to model changes of function values together with changes of the values of variables and ...

Stochastic Boolean satisfiability (SSAT) is a formalism allowing decision-making for optimization under quantitative constraints. Although SSAT solvers are active development, existing do not provide Skolem-function witnesses, which crucial practical applications. In this work, we develop new witness-generating solver, SharpSSAT, integrates techniques, including component caching, clause learni...