The complexity of modular decomposition of Boolean functions

Modular decomposition is a thoroughly investigated topic in many areas such as switching theory, reliability theory, game theory and graph theory. We propose an O(mn)-algorithm for the recognition of a modular set of a monotone Boolean function f with m prime implicants and n variables. Using this result we show that the computation of the modular closure of a set can be done in time O(mn2). On the other hand, we prove that the recognition problem for general Boolean functions is coNP-complete. Moreover, we introduce the so-called generalized Shannon decomposition of a Boolean function as an efficient tool for proving theorems on Boolean function decompositions.