Joint action of binary factors in the sufficient causes theory and its classification

Abstract

We consider a problem of formal definition of joint action in the binary sufficient causes framework based on the theory of Boolean algebras. This theory is one of the general causality concepts in epidemiology, environmental sciences, medicine and biology. Its correct mathematical form allows us to regard the binary version of this theory as a specific application of Boolean functions theory. Within the formalism of Boolean functions, a strict definition of the joint action is given and various criteria for the presence of joint action of factors in a Boolean function are obtained. The methods previously developed for analyzing joint action in binary sufficient causes framework allows us to split all the Boolean functions into disjoint equivalence classes. The relationships among these classes however remain uncertain. In the present paper, an integer invariant is introduced which allows one to order joint action types in a certain way. We consider examples of two-and three-factor theories of sufficient causes with the ordinary epidemiological symmetry group. Estimation of the time complexity of determining the type of joint action are considered as well.