We use combinatorial methods and permutation groups to classify homogeneous boolean functions. The property of symmetry of a boolean function limits the size of the function’s class. We exhaustively searched for all boolean functions on V6. We found two interesting classes of degree 3 homogeneous boolean functions: the first class is degree 3 homogeneous bent boolean functions; and the second i...

In the present paper we study properties of perfectly balanced Boolean functions. Based on the concept of Boolean function barrier, we propose a novel approach to construct large classes of perfectly balanced Boolean functions.

We establish a lower bound for deciding the satisfiability of the conjunction of any two Boolean formulas from a set called a full representation of Boolean functions of n variables a set containing a Boolean formula to represent each Boolean function of n variables. The contradiction proof first assumes that there exists a Turing machine with k symbols in its tape alphabet that correctly decid...

We establish a lower bound of 2 conditional branches for deciding the satisfiability of the conjunction of any two Boolean formulas from a set called a full representation of Boolean functions of n variables a set containing a Boolean formula to represent each Boolean function of n variables. The contradiction proof first assumes that there exists a Post machine (Post’s Formulation 1) that corr...

We study the relations between boolean functions and symmetric groups. We consider elements of a symmetric group as variable transformation operators for boolean functions. Boolean function may be xed or permuted by these operators. We give some properties relating the symmetric group Sn and boolean functions on Vn.

The Specialized Normal Form (SNF) is a unique (canonical) representation of Exclusive Sum-Of-Products (ESOP) expressions of a Boolean function. Hence this expansion of a Boolean function abstracts the form of the given ESOP expression and directs the focus to the properties of the function itself. This paper shows, how information about the given function can be detected from the shape of the S...

This paper argues that the renormalization group technique used to characterize phase transitions in condensed matter systems can be used to classify Boolean functions. A renormalization group transformation is presented that maps an arbitrary Boolean function ofN Boolean variables to one ofN−1 variables. Applying this transformation to a generic Boolean function (one whose output for each inpu...