Criteria for universality of quantum gates

We consider the problem of deciding if a set of quantum one-qudit gates S = {U-1,...,U-n} is universal. We provide the compact-form criteria leading to a simple algorithm that allows deciding the universality of any given set of gates in a finite number of steps. Moreover, for a nonuniversal S our criteria indicate what types of gates can be added to S to turn it into a universal set.