Integer invariants of joint action of Boolean variables and their properties

The sufficient causes theory is a generally accepted way to describe causality in the biomedical sciences. This theory can be adequately formalized in the Boolean algebra framework. An important question of biomedical research is determination of a joint (combined) action type of the acting factors. In Boolean interpretation of sufficient causes theory, this question is treated as computation of the orbits of the action of a certain group of automorphisms on the algebra of Boolean functions. Boolean formalization allows us to introduce additional concepts that help to study in more detail various questions of combined action of binary factors. In particular, the article provides a definition of combined action of a set of Boolean variables in a Boolean function and its geometric interpretation. The invariance of this concept is proved under the action of the automorphism group of the Boolean cube. These automorphisms are formal representation of the experimental symmetries. An integer number is introduced, called the degree of joint action. This number can be considered as a characteristic of the strength of the joint action of a number of variables in a Boolean function. This number is proved to be an invariant under the action of the Boolean cube automorphism group. The notion of a spectrum of combined action of a set of variables in a Boolean function is proposed. This concept allows us put all types of joint action in an order. An upper bound is found for the number of all possible spectra of joint action of variables in Boolean functions depending on a given number of variables.