It is shown that the problem whether an arbitrary regular language has a non-empty decomposable sublanguage with respect to a fixed distribution is decidable if and only if the independence relation induced by the distribution is transitive. A sufficient condition on the distributed control architecture is then derived, under which there exist some fixed non-blocking local generators such that the distributed supervisor synthesis problem is undecidable. We also show that a natural formulation of the parameterized supervisor synthesis problem is undecidable for a fixed non-blocking generator template, so long as the template alphabet has at least two private events and one global event that are controllable. In particular, all the undecidability results are still valid even if star free specification languages are considered.