On Decomposability and Interaction Functions

A formal notion of a Boolean-function decomposition was introduced recently and used to provide lower bounds on various representations of Boolean functions, which are subsets of decomposable negation normal form (DNNF). This notion has introduced a fundamental optimization problem for DNNF representations, which calls for computing decompositions of minimal size for a given partition of the function variables. We consider the problem of computing optimal decompositions in this paper for general Boolean functions and those represented using CNFs. We introduce the notion of an interaction function, which characterizes the relationship between two sets of variables and can form the basis of obtaining such decompositions. We contrast the use of these functions to the current practice of computing decompositions, which is based on heuristic methods that can be viewed as using approximations of interaction functions. We show that current methods can lead to decompositions that are exponentially larger than optimal decompositions, pinpoint the specific reasons for this lack of optimality, and finally present empirical results that illustrate some characteristics of interaction functions in contrast to their approximations.