Complexity of Boolean functions' representations in classes of extended pair-generated operator forms

In this paper, we study the problem of receiving the complexity's value of Boolean functions' representations in some classes of polynomial normal forms or exclusive-or sum-of-products expressions (ESOPs). These classes are extensions of known classes of polarized Zhegalkin polynomials or Reed-Muller forms and the Kronecker forms' class. An operator approach for the ESOPs classes' description is used in the work. A Boolean function is represented as a sum of operator images with respect to some basis function. If we consider the product's function as the basic function, then the classes of operator forms are becoming the ESOPs. In this paper, we received estimates of the complexity's value in various classes of extended pair-generated operator forms. The lower bound of the complexity's value to the class of all extended pair-generated operator forms (A. Baliuk and S. Vinokourov, 2001) was improved.