Calculus of pseudo-Boolean functions

The purpose of this paper is to propose a framework for the calculus of pseudo-Boolean function (PBF). Using semitensor product (STP) of matrices, this paper generalizes the calculus of Boolean functions (BF) to PBF. First, the derivative of PBF is defined. Using the algebraic form of PBFs, the formula for the derivative of a PBF is obtained. Then, the integral of a PBF is also defined as the inverse of the derivative. By generalizing the MacLaurin expansion of BF to PBF, a necessary and sufficient condition for the existence of the indefinite integral of a PBF is proved.